# Existence of global and explosive mild solutions of fractional reaction-
diffusion system of semilinear SPDEs
with fractional noise

S. Sankar1, Manil T. Mohan2, and S. Karthikeyan1
1Department of Mathematics, Periyar University, Salem 636 011, India
2Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee
247 667, India Corresponding author email<EMAIL_ADDRESS>
###### Abstract

In this paper, we investigate the existence and finite-time blow-up for the
solution of a reaction-diffusion system of semilinear stochastic partial
differential equations (SPDEs) subjected to a two-dimensional fractional
Brownian motion given by

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
du_{1}(t,x)&=\mathopen{}\mathclose{{}\left[\Delta_{\alpha}u_{1}(t,x)+\gamma_{1}u_{1}(t,x)+u^{1+\beta_{1}}_{2}(t,x)}\right]dt\\\
&\qquad\ \ +k_{11}u_{1}(t,x)dB^{H}_{1}(t)+k_{12}u_{1}(t,x)dB^{H}_{2}(t),\\\
du_{2}(t,x)&=\mathopen{}\mathclose{{}\left[\Delta_{\alpha}u_{2}(t,x)+\gamma_{2}u_{2}(t,x)+u^{1+\beta_{2}}_{1}(t,x)}\right]dt\\\
&\qquad\ \ +k_{21}u_{2}(t,x)dB^{H}_{1}(t)+k_{22}u_{2}(t,x)dB^{H}_{2}(t),\\\
\end{aligned}}\right.$

for $x\in\mathbb{R}^{d},\ t\geq 0$, along with

$\begin{array}[]{ll}u_{i}(0,x)=f_{i}(x),&x\in\mathbb{R}^{d},\\\ \end{array}$

where $\Delta_{\alpha}$ is the fractional power
$-(-\Delta)^{\frac{\alpha}{2}}$ of the Laplacian, $0<\alpha\leq 2$ and
$\beta_{i}>0,\ \gamma_{i}>0$ and $k_{ij}\neq 0,i,j=1,2$ are constants. We
provide sufficient conditions for the existence of a global weak solution.
Under the assumption that $\beta_{1}\geq\beta_{2}>0$ with Hurst index $1/2\leq
H<1,$ we obtain the blow-up times for an associated system of random partial
differential equations in terms of an integral representation of exponential
functions of Brownian motions. Moreover, we provide lower and upper bounds for
the finite-time blow-up of the above system of SPDEs and obtain the lower and
upper bounds for the probability of non-explosive solutions to our considered
system.

Keywords: Semilinear SPDEs; fractional Brownian motion; blow-up times;
stopping times; lower and upper bounds.

MSC: 35R60; 60H15; 74H35

##  1 Introduction

A considerable effort has been devoted to exploring the problem of finite-time
blow-up solutions to a numerous families of deterministic partial differential
equations (PDEs). In particular, motivated by the primary and seminal papers
by Kaplan [13] and Fujita [10, 11], new techniques were developed for studying
the blow-up behavior to a more general class of nonlinear PDEs. As far as the
stochastic counterpart is concerned, there are only a limited results
available on the non-existence of solutions of stochastic partial differential
equations (SPDEs). For SPDEs driven by Brownian motions, Mueller [18] proved
that the heat equation

$u_{t}=u_{xx}+u^{\gamma}\dot{W}$

with continuous and nonnegative initial data blow-up in a finite time with
positive probability when $\gamma>3/2$ where $\dot{W}=\dot{W}(\cdot,\cdot)$ is
a two-dimensional white noise process. Chow [1] proposed the notion of blow-up
solution to nonlinear stochastic wave equations whose $L^{2}$-norms explode at
a finite time in the mean-square sense under some appropriate conditions on
the initial data and the noise term. The existence of global positive
solutions of semilinear stochastic partial differential equations (SPDEs) has
been extensively investigated in [2, 3, 9], etc. Dozzi and Lòpez-Mimbelab [6]
investigated the blow-up problem for the following SPDE:

$\displaystyle du(t,x)=[\Delta u(t,x)+G(u(t,x))]dt+\kappa u(t,x)dW(t),$ (1.1)

with the Dirichlet data, on a smooth bounded domain, perturbed by a standard
one-dimensional Brownian motion $\\{W(t):t\geq 0\\}$ defined on some
probability space
$\mathopen{}\mathclose{{}\left(\Omega,\mathscr{F},\mathbb{P}}\right)$. Here
$G:\mathbb{R}\to\mathbb{R}^{+}$ is a locally Lipschitz function satisfying

$\displaystyle G(z)\leq Cz^{1+\beta},\ \mbox{for all}\ z>0,$ (1.2)

where $C\ \mbox{and}\ \beta>0$ are given numbers. The bounds for the explosion
time are estimated in terms of exponential functionals of $W(\cdot)$ by
transforming the original equation into a random PDE. For the case,
$G(u)=u^{1+\beta}$ with $\beta>0$, the upper and lower bounds for the
explosion time are estimated in [15] by using a formula developed by Yor (cf.
[12, 23, 29], etc.).

The impact of Gaussian noises on the time of blow-up of solutions to the
following nonlinear SPDE:

$du(t,x)=[\Delta
u(t,x)+G(u(t,x))]dt+k_{1}u(t,x)dW_{1}(t)+k_{2}u(t,x)dW_{2}(t),$

with the Dirichlet boundary condition, where $G$ satisfying (1.2) and
$\mathopen{}\mathclose{{}\left(W_{1}(t),W_{2}(t)}\right)_{t\geq 0}$ is a two-
dimensional Brownian motion was investigated by Niu and Xie [19]. Dozzi [4]
extended the results of (1.1) to a system of semilinear SPDEs with the
Dirichlet boundary data on a bounded smooth domain defined on some probability
space $\mathopen{}\mathclose{{}\left(\Omega,\mathscr{F},\mathbb{P}}\right)$.

Concerning semilinear SPDEs driven by fractional Brownian motions, Dozzi [7]
derived lower and upper bounds for the explosion time to a semilinear SPDE
driven by a one-dimensional fractional Brownian motion $\\{B^{H}(t)\\}_{t\geq
0}$ with Hurst index $H>\frac{1}{2}$. Further, sufficient conditions for blow-
up in finite time and the existence of a global solution are deduced. Dung [8]
proposed a new approach to obtain the upper and lower bounds for the
probability of the finite-time blow-up of solution in terms of the Hurst
parameter $H\in(0,1)$. Note that for the case $H=\frac{1}{2}$,
$B^{H}(t)=W(t)$, a standard Brownian motion. In [24], lower and upper bounds
for the explosion times of a system of semilinear SPDEs with standard Brownian
motion are obtained. Sankar et. al., [25] established the lower and upper
bounds for the non-global solution for a system of semilinear SPDEs driven by
two-dimensional fractional Brownian motion.

Only a few authors have addressed the explosive behavior of SPDEs with
fractional Laplacian operators of the nonlocal type. Wang [27] proved the
positivity of solutions to such a stochastic nonlocal heat equation under some
assumptions and proved that the solution blow-up in finite time. In a recent
paper, Dozzi [5] provided sufficient conditions for finite-time blow-up of
positive weak solutions of a fractional reaction-diffusion equation perturbed
by a fractional Brownian motion (fBm). Moreover, lower and upper bounds for
the probability that the solution does not explode in a finite time are also
provided. However, for many classes of SPDEs, the problem of existence and
non-existence of solutions is not yet completely explored.

Motivated by the above facts, in the present paper, our aim is to obtain the
global existence, and lower and upper bounds for the blow-up time to the
following system of semilinear SPDEs:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
du_{1}(t,x)&=\mathopen{}\mathclose{{}\left[\Delta_{\alpha}u_{1}(t,x)+\gamma_{1}u_{1}(t,x)+u^{1+\beta_{1}}_{2}(t,x)}\right]dt\\\
&\qquad\ \ +k_{11}u_{1}(t,x)dB^{H}_{1}(t)+k_{12}u_{1}(t,x)dB^{H}_{2}(t),\\\
du_{2}(t,x)&=\mathopen{}\mathclose{{}\left[\Delta_{\alpha}u_{2}(t,x)+\gamma_{2}u_{2}(t,x)+u^{1+\beta_{2}}_{1}(t,x)}\right]dt\\\
&\qquad\ \ +k_{21}u_{2}(t,x)dB^{H}_{1}(t)+k_{22}u_{2}(t,x)dB^{H}_{2}(t),\\\
\end{aligned}}\right.$ (1.3)

for $x\in\mathbb{R}^{d},\ t\geq 0$, along with the initial condition

$\begin{array}[]{ll}u_{i}(0,x)=f_{i}(x),&x\in\mathbb{R}^{d},\
i=1,2,\end{array}$

where $\Delta_{\alpha}$ is the fractional power
$-(-\Delta)^{\frac{\alpha}{2}}$ of the Laplacian. Here, $0<\alpha\leq 2$, and
$\beta_{1}\geq\beta_{2}>0,\ \gamma_{i}>0$ and $k_{ij}\neq 0,i,j=1,2$ are real
constants. Moreover, $f_{1},\ f_{2}$ are $C^{2}$-functions which are assumed
to be non-negative and not identically zero. Further,
$\mathopen{}\mathclose{{}\left(B_{1}^{H}(t),B_{2}^{H}(t)}\right)_{t\geq 0}$ is
a standard independent two-dimensional fractional Brownian motion (fBm) having
Hurst index $\frac{1}{2}\leq H<1$ with respect to a filtered probability space
$\mathopen{}\mathclose{{}\left(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\geq
0},\mathbb{P}}\right)$. The operator $\Delta_{\alpha}$ is the usual Laplacian
when $\alpha=2$ and is the infinitesimal generator of $d$-dimensional Brownian
motion with variance parameter 2. For $\alpha\in(0,2),$

$\displaystyle\Delta_{\alpha}u(x)=C(d,\alpha)PV\mathopen{}\mathclose{{}\left(\int_{\mathbb{R}^{d}}\frac{u(x+y)-u(x)}{\arrowvert
y\arrowvert^{d+\alpha}}dy}\right),\ \ u\in C^{2}_{c}(\mathbb{R}^{d}),$

where $C(d,\alpha)$ is a constant and $PV$ denotes the principal value of the
integral, and in this case $\Delta_{\alpha}$ is the generator of the symmetric
$\alpha$-stable Lévy process $\\{{X}_{t}\\}_{t\geq 0}$ on $\mathbb{R}^{d}$
satisfying

$\mathbb{E}\mathopen{}\mathclose{{}\left[\exp(i\boldsymbol{u}\cdot{X}_{t})|{X}_{0}=0}\right]=\exp(-t|\boldsymbol{u}|^{\alpha}),\
\boldsymbol{u}\in\mathbb{R}^{d},\ t\geq 0.$

The structure of the remaining paper is as follows: In section 2, we obtain an
associated system of random PDEs and derive a weak solution by using a random
transformation which is useful to obtain lower and upper bounds for the blow-
up of solutions for the system (1.3). Further, we prove that weak and mild
solutions of the system (1.3) are equivalent. The lower bounds for the finite
time blow-up of the system (1.3) with $1/2<H<1$ and suitable parameters are
obtained in section 3 (cf. Theorem 3.1). For appropriate initial data, upper
bounds for the blow-up time of the solution $u=(u_{1},u_{2})^{\top}$ of the
system (1.3) for $1/2<H<1$, with the parameters
$\rho_{1}=(1+\beta_{1})k_{21}-k_{11}=(1+\beta_{2})k_{11}-k_{21}$ and
$\rho_{2}=(1+\beta_{1})k_{22}-k_{12}=(1+\beta_{2})k_{12}-k_{22}$ are obtained
in section 4 (cf. Theorem 4.1). Furthermore, we establish upper bounds for the
probability of non-explosive positive solution $u=(u_{1},u_{2})^{\top}$ of the
system (1.3) (cf. Theorem 4.2). In section 5, we obtain upper bounds for the
probability of the existence of a global mild solution of the system (1.3) for
the case $H=\frac{1}{2}$ (Theorem 5.1). In section 6, we provide conditions
for the global existence in time of the mild solution $u=(u_{1},u_{2})^{\top}$
to the system (1.3) for the case $1/2\leq H<1$ and establish the lower and
upper bounds for the probability of the non-explosion solution
$u=(u_{1},u_{2})^{\top}$ to the system (1.3). In the final section, we find
lower and upper bounds for the blow-up time of the system (1.3) for a more
general case.

##  2 A System of Random PDEs

In this section, we obtain a system of random PDEs by making use of the random
transformations

$v_{i}(t,x)=\exp\\{-k_{i1}B_{1}^{H}(t)-k_{i2}B_{2}^{H}(t)\\}u_{i}(t,x),\
i=1,2,$

for $t\geq 0,\ x\in\mathbb{R}^{d}$. The system $\eqref{b1}$ is transformed
into the following system of random PDEs:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
v_{i}(t,x)}{\partial
t}&=\mathopen{}\mathclose{{}\left(\Delta_{\alpha}+\gamma_{i}}\right)v_{i}(t,x)+e^{-k_{i1}B_{1}^{H}(t)-k_{i2}B_{2}^{H}(t)}\mathopen{}\mathclose{{}\left(e^{k_{j1}B_{1}^{H}(t)+k_{j2}B_{2}^{H}(t)}v_{j}(t,x)}\right)^{1+\beta_{i}},\\\
v_{i}(0,x)&=f_{i}(x),\ \ x\in\mathbb{R}^{d},\end{aligned}}\right.$ (2.1)

for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
This system is understood in the pathwise sense and thus the classical results
for parabolic PDEs can be applied to show existence, uniqueness and positivity
of solution $v=(v_{1},v_{2})^{\top}\in
C([{0,\tau}];L^{2}(\mathbb{R}^{d};\mathbb{R}^{2}))$ up to an eventual blow-up
(Theorem 2.1, [30]).

Next theorem helps us to prove that the system (1.3) is transformed into the
system of random PDEs (2.1) when $\frac{1}{2}<H<1$.

######  Theorem 2.1.

Let $u=(u_{1},u_{2})^{\top}$ be the weak solution of $\eqref{b1}$. Then the
function $v=(v_{1},v_{2})^{\top}$ defined by

$\displaystyle
v_{i}(t,x)=\exp\\{-k_{i1}B_{1}^{H}(t)-k_{i2}B_{2}^{H}(t)\\}u_{i}(t,x),\ t\geq
0,\ x\in\mathbb{R}^{d},\ i=1,2,$

solves the system $\eqref{s1}$.

###### Proof.

By using Ito’s formula (Lemma 2.7.1 in [17]), we have

$\displaystyle
e^{-k_{i1}B^{H}_{1}(t)-k_{i2}B^{H}_{2}(t)}=1-\int_{0}^{t}e^{-k_{i1}B^{H}_{1}(s)-k_{i2}B^{H}_{2}(s)}(k_{i1}dB^{H}_{1}(s)+k_{i2}dB^{H}_{2}(s)),\
i=1,2.$

For any smooth functions $\varphi_{1}$ and $\varphi_{2}$ with compact support,
we denote

$u_{i}(t,\varphi_{i})=\int_{\mathbb{R}^{d}}u_{i}(t,x)\varphi_{i}(x)dx,\ \
i=1,2.$

Let us recall the notion of weak solution of $\eqref{b1}$. Let
$0\leq\tau\leq+\infty$ be a stopping time. A continuous random vector field
$u=(u_{1},u_{2})^{\top}=\mathopen{}\mathclose{{}\left\\{(u_{1}(t,x),u_{2}(t,x))^{\top}:\
t\geq 0,\ x\in\mathbb{R}^{d}}\right\\}$ is a weak solution of $\eqref{b1}$ on
the interval $(0,\tau)$ provided that for every $\varphi_{i}\in
C^{\infty}_{c}(\mathbb{R}^{d}),$ there holds

$\displaystyle u_{i}\mathopen{}\mathclose{{}\left(t,\varphi_{i}}\right)$
$\displaystyle=u_{i}\mathopen{}\mathclose{{}\left(0,\varphi_{i}}\right)+{\int_{0}^{t}u_{i}\mathopen{}\mathclose{{}\left(s,\Delta_{\alpha}\varphi_{i}}\right)ds}+\gamma_{i}\int_{0}^{t}u_{i}\mathopen{}\mathclose{{}\left(s,\varphi_{i}}\right)ds+\int_{0}^{t}u^{1+\beta_{i}}_{j}\mathopen{}\mathclose{{}\left(s,\varphi_{i}}\right)ds$
$\displaystyle\quad+k_{i1}\int_{0}^{t}u_{i}\mathopen{}\mathclose{{}\left(s,\varphi_{i}}\right)dB^{H}_{1}(s)+k_{i2}\int_{0}^{t}u_{i}\mathopen{}\mathclose{{}\left(s,\varphi_{i}}\right)dB^{H}_{2}(s),$
(2.2)

where $i=1,2,\
\mathopen{}\mathclose{{}\left\\{j}\right\\}=\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\\{i\\}.$
By applying the integration by parts formula (Chapter 8 in [14]), we obtain

$\displaystyle v_{i}(t,\varphi_{i}):$
$\displaystyle=\int_{\mathbb{R}^{d}}v_{i}(t,x)\varphi_{i}(x)dx$
$\displaystyle=v_{i}(0,\varphi_{i})+\int_{0}^{t}e^{-k_{i1}W^{H}_{1}(t)-k_{i2}B^{H}_{2}(t)}du_{i}(s,\varphi_{i})$
$\displaystyle\quad+\int_{0}^{t}u_{i}(s,\varphi_{i})\mathopen{}\mathclose{{}\left(e^{-k_{i1}B^{H}_{1}(s)-k_{i2}B^{H}_{2}(s)}(k_{i1}dB^{H}_{1}(s)+k_{i2}dB^{H}_{2}(s))ds}\right).$

Therefore,

$\displaystyle v_{i}(t,\varphi_{i})$
$\displaystyle=v_{i}(0,\varphi_{i})+\int_{0}^{t}v_{i}\mathopen{}\mathclose{{}\left(s,\Delta_{\alpha}\varphi_{i}}\right)ds+\gamma_{i}\int_{0}^{t}v_{i}\mathopen{}\mathclose{{}\left(s,\varphi_{i}}\right)ds$
$\displaystyle\qquad+\int_{0}^{t}e^{-k_{i1}B^{H}_{1}(t)-k_{i2}B^{H}_{2}(t)}\mathopen{}\mathclose{{}\left(e^{k_{j1}B^{H}_{1}(t)+k_{j2}B^{H}_{2}(t)}v_{j}}\right)^{1+\beta_{i}}(s,\varphi_{i})ds$
$\displaystyle=v_{i}(0,\varphi_{i})+\int_{0}^{t}[v_{i}\mathopen{}\mathclose{{}\left(s,\Delta_{\alpha}\varphi_{i}}\right)+\gamma_{i}v_{i}\mathopen{}\mathclose{{}\left(s,\varphi_{i}}\right)]ds$
$\displaystyle\qquad+\int_{0}^{t}e^{-k_{i1}B^{H}_{1}(t)-k_{i2}B^{H}_{2}(t)}\mathopen{}\mathclose{{}\left(e^{k_{j1}B^{H}_{1}(t)+k_{j2}B^{H}_{2}(t)}v_{j}}\right)^{1+\beta_{i}}(s,\varphi_{i})ds.$
(2.3)

The preceding equalities mean that $v=(v_{1},v_{2})^{\top}$ is a weak solution
of $\eqref{s1},$ which is differentiable in time $t.$ From the uniqueness of
weak solutions, $v=(v_{1},v_{2})^{\top}$ solves $\eqref{s1}$. ∎

Let us denote by $\mathopen{}\mathclose{{}\left\\{S_{t}}\right\\}_{t\geq 0}$,
the $\alpha$-stable semigroup of bounded linear operators with generator
$\Delta_{\alpha}$ and
$\mathopen{}\mathclose{{}\left\\{p(t,x):t>0,x\in\mathbb{R}^{d}}\right\\}$ to
indicate the family of spherically symmetric $\alpha$-stable transition
densities. Hence, for any $t\geq 0$ and bounded measurable function
$f:\mathbb{R}^{d}\rightarrow\mathbb{R}$, the bounded linear operator is
defined by

$\displaystyle
S_{t}f(x)=\mathbb{E}\mathopen{}\mathclose{{}\left[f(X_{t})|X_{0}=x}\right]=\int_{\mathbb{R}^{d}}p(t,y-x)f(y)dy,\quad
x\in\mathbb{R}^{d},$

where $\mathopen{}\mathclose{{}\left\\{X_{t}}\right\\}_{t\geq 0}$ is the
symmetric $\alpha$-stable Lévy process in $\mathbb{R}^{d}.$ For every $t\geq
0,$ the linear operator $S_{t}$ is a contraction that preserves positivity and
therefore the bounded linear operators

$\displaystyle T_{i}(t)=\exp\\{\gamma_{i}t\\}S_{t},\ \ i=1,2,$ (2.4)

are also positivity preserving. Moreover,
$\mathopen{}\mathclose{{}\left\\{T_{1}(t)}\right\\}_{t\geq 0}$ and
$\mathopen{}\mathclose{{}\left\\{T_{2}(t)}\right\\}_{t\geq 0}$are linear
semigroups with infinitesimal generators $\Delta_{\alpha}+\gamma_{1}$ and
$\Delta_{\alpha}+\gamma_{2}$ respectively.

From the classical results available in semigroup theory [22], we note that
for any bounded measurable initial data $f_{i}\geq 0,$ there exists a unique
local mild solution $v=(v_{1},v_{2})^{\top}$ of the random PDE (2.1).
Precisely, there exists a number $0<\tau=\tau(\omega)\leq\infty$ such that
$v=(v_{1},v_{2})^{\top}$ satisfies the integral equation

$\displaystyle
v_{i}(t)=T_{i}(t)f_{i}(x)+\int_{0}^{t}T_{i}(t-r)\mathopen{}\mathclose{{}\left[e^{-k_{i1}B^{H}_{1}(r)-k_{i2}B^{H}_{2}(r)}\mathopen{}\mathclose{{}\left(e^{k_{j1}B^{H}_{1}(r)+k_{j2}B^{H}_{2}(r)}v_{j}(r,\cdot)}\right)^{1+\beta_{i}}}\right](x)dr,$
(2.5)

or, equivalently,

$\displaystyle v_{i}(t,x)$ $\displaystyle=S_{t}f_{i}(x)$
$\displaystyle\quad+\int_{0}^{t}S_{t-r}\mathopen{}\mathclose{{}\left[\gamma_{i}v_{i}(s,\cdot)+e^{-k_{i1}B^{H}_{1}(r)-k_{i2}B^{H}_{2}(r)}\mathopen{}\mathclose{{}\left(e^{k_{j1}B^{H}_{1}(r)+k_{j2}B^{H}_{2}(r)}v_{j}(r,\cdot)}\right)^{1+\beta_{i}}}\right](x)dr,$
(2.6)

for $i=1,2,$ on $[0,t]$ for each $0\leq t<\tau.$

######  Lemma 2.1.

$v=(v_{1},v_{2})^{\top}$ is a weak solution of (2.1) on
$\mathopen{}\mathclose{{}\left[0,\tau}\right]$ if and only if
$v=(v_{1},v_{2})^{\top}$ is a mild solution of (2.1) on
$\mathopen{}\mathclose{{}\left[0,\tau}\right].$

###### Proof.

Assume that $v=(v_{1},v_{2})^{\top}$ is a weak solution of (2.1). Let
$h_{i}\in C^{1}(\mathbb{R}),\varphi_{i}\in C_{c}^{2}(\mathbb{R}^{d}),$ for
$i=1,2$. For convenience, let us take

$g_{i,j}(t,x)=\gamma_{i}v_{i}(t,x)+e^{-k_{i1}B_{1}^{H}(t)-k_{i2}B_{2}^{H}(t)}\mathopen{}\mathclose{{}\left(e^{k_{j1}B_{1}^{H}(t)+k_{j2}B_{2}^{H}(t)}v_{j}(t,x)}\right)^{1+\beta_{i}},$

for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
Using integration by parts formula for any $t<\tau,$ and by denoting
$\langle\cdot,\cdot\rangle$, the scalar product in $L^{2}(\mathbb{R}^{d}),$ we
have

$\displaystyle\langle h_{i}(t)\varphi_{i}(\cdot),v_{i}(t,\cdot)\rangle$
$\displaystyle=\langle
h_{i}(0)\varphi_{i}(\cdot),v_{i}(0,\cdot)\rangle+\int_{0}^{t}\langle
h_{i}(s)\Delta_{\alpha}\varphi_{i}(\cdot),v_{i}(s,\cdot)\rangle ds$
$\displaystyle\qquad+\int_{0}^{t}\mathopen{}\mathclose{{}\left\langle\frac{d}{ds}h_{i}(s)\varphi_{i}(\cdot),v_{i}(s,\cdot)}\right\rangle
ds+\int_{0}^{t}\langle h_{i}(s)\varphi_{i}(\cdot),g_{i,j}(s,\cdot)\rangle ds.$

Using a density argument, the following is valid for $\Psi_{i}\in
C^{1,2}_{c}\mathopen{}\mathclose{{}\left(\mathbb{R}\times\mathbb{R}^{d}}\right),\
i=1,2,$

$\displaystyle\langle\Psi_{i}(t,\cdot),v_{i}(t,\cdot)\rangle$
$\displaystyle=\langle\Psi_{i}(0,\cdot),v_{i}(0,\cdot)\rangle+\int_{0}^{t}\mathopen{}\mathclose{{}\left\langle\frac{d}{ds}\Psi_{i}(s,\cdot)+\Delta_{\alpha}\Psi_{i}(s,\cdot),v_{i}(s,\cdot)}\right\rangle
ds$
$\displaystyle\qquad+\int_{0}^{t}\langle\Psi_{i}(s,\cdot),g_{i,j}(s,\cdot)\rangle
ds.$ (2.7)

Moreover, by a density argument, (2) is valid for all $\Psi_{i}\in
L^{1}\mathopen{}\mathclose{{}\left([0,\tau]\times\mathbb{R}^{d}}\right)$. For
$\phi_{i}\in C_{c}^{2}(\mathbb{R}^{d})$, let $\Psi_{i},\ i=1,2$ be defined by

$\displaystyle\Psi_{i}(s,x)=S_{t-s}\phi_{i}(x)=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{ccc}\langle
p(t-s,\cdot-x),\phi_{i}(\cdot)\rangle,&\ \text{ if }\ \ s<t,\\\ \\\
\phi_{i}(x),&\ \text{ if }\ \ s=t.\end{array}}\right.$

Since, $p(s,x)$ satisfies the Kolmogorov equation
$\displaystyle\frac{d}{ds}p(s,x)=\Delta_{\alpha}p(s,x),$ we have

$\displaystyle\Delta_{\alpha}\Psi_{i}(s,x)+\frac{d}{ds}\Psi_{i}(s,x)$
$\displaystyle=\Delta_{\alpha}\mathopen{}\mathclose{{}\left(\int_{\mathbb{R}^{d}}p(t-s,y-x)\phi_{i}(y)dy}\right)+\frac{d}{ds}\mathopen{}\mathclose{{}\left(\int_{\mathbb{R}^{d}}p(t-s,y-x)\phi_{i}(y)dy}\right)$
$\displaystyle=\int_{\mathbb{R}^{d}}\mathopen{}\mathclose{{}\left[\Delta_{\alpha}p(t-s,y-x)-\frac{d}{ds}p(t-s,y-x)}\right]\phi_{i}(y)dy=0.$

Moreover,

$\displaystyle\langle\Psi_{i}(0,\cdot),v_{i}(0,\cdot)\rangle$
$\displaystyle=\int_{\mathbb{R}^{d}}\Psi_{i}(0,x)v_{i}(0,x)dx=\int_{\mathbb{R}^{d}}f_{i}(x)\mathopen{}\mathclose{{}\left(\int_{\mathbb{R}^{d}}p(t,y-x)\phi_{i}(y)dy}\right)dx$
$\displaystyle=\int_{\mathbb{R}^{d}}\phi_{i}(y)\mathopen{}\mathclose{{}\left(\int_{\mathbb{R}^{d}}p(t,y-x)f_{i}(x)dx}\right)dy$
$\displaystyle=\int_{\mathbb{R}^{d}}\phi_{i}(y)\mathopen{}\mathclose{{}\left(S_{t}f_{i}}\right)(y)dy.$

From (2), we have

$\displaystyle\langle\phi_{i}(\cdot),v_{i}(t,\cdot)\rangle=\langle\phi_{i}(\cdot),S_{t}f_{i}(\cdot)\rangle+\int_{0}^{t}\langle\phi_{i}(\cdot),S_{t-s}g_{i,j}(s,\cdot)\rangle
ds,\ i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$

Since $v_{i}$ and $S_{t}v_{i}$ are locally integrable, and the above equality
holds for any $\phi_{i}\in C_{c}^{2}(\mathbb{R}^{d})$, it follows that $v_{i}$
solves (2). Hence $v=(v_{1},v_{2})^{\top}$ is a mild solution of (2.1).

Conversely, assume that $v=(v_{1},v_{2})^{\top}$ is a mild solution of (2.1).
Let $\varphi_{1},\varphi_{2}\in C_{c}^{2}(\mathbb{R}^{d})$. Then for any
$t<\tau$, we have

$\displaystyle\int_{0}^{t}\langle\varphi_{i}(\cdot),\Delta_{\alpha}v_{i}(s,\cdot)\rangle
ds$
$\displaystyle=\int_{0}^{t}\langle\Delta_{\alpha}\varphi_{i}(\cdot),v_{i}(s,\cdot)\rangle
ds$
$\displaystyle=\int_{0}^{t}\mathopen{}\mathclose{{}\left\langle\Delta_{\alpha}\varphi_{i}(\cdot),S_{s}f_{i}(\cdot)+\int_{0}^{s}S_{s-r}g_{i,j}(r,\cdot)dr}\right\rangle
ds$
$\displaystyle=\int_{0}^{t}\langle\Delta_{\alpha}\varphi_{i}(\cdot),S_{s}f_{i}(\cdot)\rangle
ds+\int_{0}^{t}\mathopen{}\mathclose{{}\left\langle\Delta_{\alpha}\varphi_{i}(\cdot),\int_{0}^{s}S_{s-r}g_{i,j}(r,\cdot)dr}\right\rangle
ds$ $\displaystyle=\int_{0}^{t}\langle
S_{s}\Delta_{\alpha}\varphi_{i}(\cdot),f_{i}(\cdot)\rangle
ds+\int_{0}^{t}\mathopen{}\mathclose{{}\left\langle\int_{r}^{t}S_{s-r}\Delta_{\alpha}\varphi_{i}(\cdot),g_{i,j}(r,\cdot)dr}\right\rangle
ds$
$\displaystyle=\int_{0}^{t}\mathopen{}\mathclose{{}\left\langle\frac{d}{ds}S_{s}\varphi_{i}(\cdot),f_{i}(\cdot)}\right\rangle
ds+\int_{0}^{t}\mathopen{}\mathclose{{}\left\langle\int_{r}^{t}\frac{d}{ds}S_{s-r}\varphi_{i}(\cdot)ds,g_{i,j}(r,\cdot)}\right\rangle
dr$ $\displaystyle=\langle
S_{t}\varphi_{i}(\cdot),f_{i}(\cdot)\rangle-\langle\varphi_{i}(\cdot),f_{i}(\cdot)\rangle+\int_{0}^{t}\langle
S_{t-r}\varphi_{i}(\cdot),g_{i,j}(r,\cdot)\rangle
dr-\int_{0}^{t}\langle\varphi_{i}(\cdot),g_{i,j}(r,\cdot)\rangle dr,$ (2.8)

for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
Since $v=(v_{1},v_{2})^{\top}$ is a mild solution of (2.1), we have

$\displaystyle\langle\varphi_{i}(\cdot),v_{i}(t,\cdot)\rangle=\langle
S_{t}\varphi_{i}(\cdot),f_{i}(\cdot)\rangle+\int_{0}^{t}\langle
S_{t-s}\varphi_{i}(\cdot),g_{i,j}(s,\cdot)\rangle ds.$

From (2), we obtain

$\displaystyle\int_{0}^{t}\langle\varphi_{i}(\cdot),\Delta_{\alpha}v_{i}(s,\cdot)\rangle
ds=\langle\varphi_{i}(\cdot),v_{i}(t,\cdot)\rangle-\langle\varphi_{i}(\cdot),f_{i}(\cdot)\rangle-\int_{0}^{t}\langle\varphi_{i}(\cdot),g_{i,j}(r,\cdot)\rangle
dr.$

Therefore,

$\displaystyle\langle\varphi_{i}(\cdot),v_{i}(t,\cdot)\rangle=\langle\varphi_{i}(\cdot),f_{i}(\cdot)\rangle+\int_{0}^{t}\langle\Delta_{\alpha}\varphi_{i}(\cdot),v_{i}(s,\cdot)\rangle
ds+\int_{0}^{t}\langle\varphi_{i}(\cdot),g_{i,j}(r,\cdot)\rangle dr,$

for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
Hence, $v=(v_{1},v_{2})^{\top}$ is a weak solution of (2.1). ∎

Let $\tau$ be the blow-up time of the system (2.1) with the initial values of
the above form. Due to Theorem 2.1 and a.s continuity of
$B_{i}^{H}(\cdot),i=1,2$, $\tau$ is also the blow-up time for the system (1.3)
(Corollary 1 in [7]).

Next, we recall the following properties of $p(\cdot,\cdot)$ which will be
useful for our further discussions (cf. Section 2 in [26]).

######  Lemma 2.2.

For any $t>0,$ the function $p(t,\cdot)$ is continuous and strictly positive.
Moreover, for any $t>0$ and $x\in\mathbb{R}^{d}$,

  * 1.

$p(ts,x)=t^{-d/\alpha}p(s,t^{-1/\alpha}x),$ for any $s>0,$

  * 2.

$p(t,0)\geq p(t,x)$. More generally, $p(t,y)\geq p(t,x)$ if $\lVert
x\rVert\geq\lVert y\rVert,$

  * 3.

$p(t,x)\geq\mathopen{}\mathclose{{}\left(\frac{s}{t}}\right)^{d/\alpha}p(s,x),$
for all $t\geq s,$

  * 4.

$p\mathopen{}\mathclose{{}\left(t,\frac{1}{r}(x-y)}\right)\geq p(t,x)p(t,y),\
x,y\in\mathbb{R}^{d},$ provided that $p(t,0)\leq 1$ and $r\geq 2.$

In the sequel, $r_{0}>0$ denotes a fixed constant such that
$p(r_{0},0)=r_{0}^{-d/\alpha}p(1,0)<1.$

##  3 A Lower Bound for $\tau$

This section aims to find the lower bounds $\tau_{\ast}$ to the blow-up times
such that $\tau_{\ast}\leq\tau$ when $\gamma_{1}=\gamma_{2}=\lambda.$ Hence
$T_{1}(t)=T_{2}(t)=T(t)\ {\rm(say)}$ defined in (2.4) and $\frac{1}{2}<H<1$.
First, we consider the equation (1.3) with parameters

$\displaystyle(1+\beta_{1})k_{21}-k_{11}$
$\displaystyle=(1+\beta_{2})k_{11}-k_{21}=:\rho_{1},$ (3.1)
$\displaystyle(1+\beta_{1})k_{22}-k_{12}$
$\displaystyle=(1+\beta_{2})k_{12}-k_{22}=:\rho_{2}.$

The following result provides the lower bounds for the finite time blow-up
solution of the system (1.3).

######  Theorem 3.1.

Assume that the conditions (3.1) holds and $\frac{1}{2}<H<1.$ If the initial
values are of the form $f_{1}=C_{1}\psi,f_{2}=C_{2}\psi,$ where $C_{1}$ and
$C_{2}$ are any positive constants with $C_{1}\leq C_{2}$ and $\psi\in
C_{c}^{\infty}({\mathbb{R}}^{d}).$ Then $\tau_{\ast}\leq\tau$, where
$\tau_{\ast}$ is given by

$\displaystyle\tau_{\ast}=\inf\Bigg{\\{}t\geq 0:$
$\displaystyle\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\lambda\beta_{1}r\\}r^{-d\beta_{1}/\alpha}dr\geq\frac{1}{\beta_{1}p(1,0)^{\beta_{1}}C_{1}^{\beta_{1}}\mathopen{}\mathclose{{}\left\|\psi}\right\|_{\infty}^{\beta_{1}}},$
$\displaystyle(or)$
$\displaystyle\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\lambda\beta_{2}r\\}r^{-d\beta_{2}/\alpha}dr\geq\frac{1}{\beta_{2}p(1,0)^{\beta_{2}}C_{2}^{\beta_{2}}\mathopen{}\mathclose{{}\left\|\psi}\right\|_{\infty}^{\beta_{2}}}\Bigg{\\}}.$
(3.2)

###### Proof.

Let $v=(v_{1},v_{2})^{\top}$ solve (2.5). Then, we have

$\displaystyle v_{1}(t,x)$
$\displaystyle=T(t)f_{1}(x)+\int_{0}^{t}T(t-r)\mathopen{}\mathclose{{}\left(\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}v^{1+\beta_{1}}_{2}(r,x)}\right)dr,$
$\displaystyle v_{2}(t,x)$
$\displaystyle=T(t)f_{2}(x)+\int_{0}^{t}T(t-r)\mathopen{}\mathclose{{}\left(\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}v^{1+\beta_{2}}_{1}(r,x)}\right)dr.$

For all $x\in\mathbb{R}^{d},\ t\geq 0,$ let us define the operators
$\mathcal{J}_{1}$ and $\mathcal{J}_{2}$ as follows:

$\displaystyle\mathcal{J}_{1}v(t,x)$
$\displaystyle=T(t)f_{1}(x)+\int_{0}^{t}\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}\mathopen{}\mathclose{{}\left(T(t-r)v}\right)^{1+\beta_{1}}dr,$
$\displaystyle\mathcal{J}_{2}v(t,x)$
$\displaystyle=T(t)f_{2}(x)+\int_{0}^{t}\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}\mathopen{}\mathclose{{}\left(T(t-r)v}\right)^{1+\beta_{2}}dr,$

where $v$ is any non-negative, bounded and measurable function.

First we shall prove that

$\displaystyle v_{1}(t,x)=\mathcal{J}_{1}v_{2}(t,x),\
v_{2}(t,x)=\mathcal{J}_{2}v_{1}(t,x),\ \ x\in D,\ \ 0\leq t<\tau_{*},$

for some non-negative, bounded and measurable functions $v_{1}$ and $v_{2}$.
Moreover, on the set $t<\tau_{*},$ we set

$\displaystyle\mathscr{G}_{1}(t)$
$\displaystyle=\mathopen{}\mathclose{{}\left[1-\beta_{1}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(r)f_{1}}\right\|_{\infty}^{\beta_{1}}dr}\right]^{\frac{-1}{\beta_{1}}},$
$\displaystyle\mathscr{G}_{2}(t)$
$\displaystyle=\mathopen{}\mathclose{{}\left[1-\beta_{2}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(r)f_{2}}\right\|_{\infty}^{\beta_{2}}dr}\right]^{\frac{-1}{\beta_{2}}}.$

By (3.1), it is immediate that $\mathscr{G}_{1}(t)$ and $\mathscr{G}_{2}(t)$
are defined. Then, it can be easily seen that

$\displaystyle\frac{d\mathscr{G}_{1}(t)}{dt}=\exp\\{\rho_{1}B^{H}_{1}(t)+\rho_{2}B^{H}_{2}(t)\\}\mathopen{}\mathclose{{}\left\|T(t)f_{1}}\right\|_{\infty}^{\beta_{1}}\mathscr{G}^{1+\beta_{1}}_{1}(t),\
\ \mathscr{G}_{1}(0)=1,$

so that

$\displaystyle\mathscr{G}_{1}(t)=1+\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(r)f_{1}}\right\|_{\infty}^{\beta_{1}}\mathscr{G}^{1+\beta_{1}}_{1}(r)dr.$

Similarly, we have

$\displaystyle\mathscr{G}_{2}(t)=1+\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(r)f_{2}}\right\|_{\infty}^{\beta_{2}}\mathscr{G}^{1+\beta_{2}}_{2}(r)dr.$

Let us choose $v\geq 0$ such that

$v(t,x)\leq T(t)f_{1}(x)\mathscr{G}_{1}(t),$

for $x\in\mathbb{R}^{d}$ and $t<\tau_{*}.$ Then
$T(t)f_{1}(x)\leq\mathcal{J}_{1}v(t,x)$ and

$\displaystyle\mathcal{J}_{1}v(t,x)$
$\displaystyle=T(t)f_{1}(x)+\int_{0}^{t}\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}\mathopen{}\mathclose{{}\left(T(t-r)v}\right)^{1+\beta_{1}}dr$
$\displaystyle\leq
T(t)f_{1}(x)+\int_{0}^{t}\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}$
$\displaystyle\qquad\times\mathopen{}\mathclose{{}\left[\mathscr{G}^{1+\beta_{1}}_{1}(r)\mathopen{}\mathclose{{}\left\|T(r)f_{1}}\right\|_{\infty}^{\beta_{1}}T(t-r)(T(r)f_{1}(x))}\right]dr$
$\displaystyle=T(t)f_{1}(x)+\int_{0}^{t}\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}$
$\displaystyle\qquad\times\mathopen{}\mathclose{{}\left[\mathscr{G}^{1+\beta_{1}}_{1}(r)\mathopen{}\mathclose{{}\left\|T(r)f_{1}}\right\|_{\infty}^{\beta_{1}}T(t)f_{1}(x)}\right]dr$
$\displaystyle=T(t)f_{1}(x)\Bigg{\\{}1+\int_{0}^{t}\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}\mathopen{}\mathclose{{}\left\|T(r)f_{1}}\right\|_{\infty}^{\beta_{1}}\mathscr{G}^{1+\beta_{1}}_{1}(r)dr\Bigg{\\}}$
$\displaystyle=T(t)f_{1}(x)\mathscr{G}_{1}(t).$

Thus, we have

$\displaystyle T(t)f_{1}(x)\leq\mathcal{J}_{1}v(t,x)\leq
T(t)f_{1}(x)\mathscr{G}_{1}(t).$ (3.3)

Similarly, we get

$\displaystyle T(t)f_{2}(x)\leq\mathcal{J}_{2}w(t,x)\leq
T(t)f_{2}(x)\mathscr{G}_{2}(t),$ (3.4)

for all $w$ such that $0\leq w(t,x)\leq T(t)f_{2}(x)\mathscr{G}_{2}(t).$

Let us take,

$\displaystyle u^{(0)}_{1}(t,x)=T(t)f_{1}(x),\ \
u^{(0)}_{2}(t,x)=\frac{C_{1}}{C_{2}}T(t)f_{2}(x),$ (3.5)

and

$\displaystyle u^{(n)}_{1}(t,x)=\mathcal{J}_{1}u^{(n-1)}_{2}(t,x),\ \
u^{(n)}_{2}(t,x)=\mathcal{J}_{2}u^{(n-1)}_{1}(t,x),\ \ n\geq 1,$ (3.6)

for $x\in\mathbb{R}^{d},\ 0\leq t\leq\tau_{*}.$

Our aim is to show that the sequences of functions $\\{u^{(n)}_{1}\\}\
\mbox{and}\ \\{u^{(n)}_{2}\\}$ are monotonically increasing. Let us consider

$\displaystyle u^{(0)}_{1}(t,x)$ $\displaystyle\leq
T(t)f_{1}(x)+\int_{0}^{t}\exp\\{{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)}\\}\mathopen{}\mathclose{{}\left(T(t-r)u^{(0)}_{2}(r,x)}\right)^{1+\beta_{1}}dr$
$\displaystyle=\mathcal{J}_{1}u^{(0)}_{2}(t,x)=u^{(1)}_{1}(t,x).$

Now assume that $u^{(n)}_{i}\geq u^{(n-1)}_{i},\ i=1,2,$ for some $n\geq 1.$
Then the monotonicity of $\mathcal{J}_{1}$ leads to the inequality

$\displaystyle
u^{(n+1)}_{1}=\mathcal{J}_{1}u^{(n)}_{2}(t,x)\geq\mathcal{J}_{1}u^{(n-1)}_{2}(t,x)=u^{(n)}_{1},$

and the monotonicity of $\mathcal{J}_{2}$ results in

$\displaystyle
u^{(n+1)}_{2}=\mathcal{J}_{2}u^{(n)}_{1}(t,x)\geq\mathcal{J}_{2}u^{(n-1)}_{1}(t,x)=u^{(n)}_{2}.$

Therefore the limits

$\displaystyle v_{1}(t,x)=\lim_{n\rightarrow\infty}u^{(n)}_{1}(t,x),\ \
v_{2}(t,x)=\lim_{n\rightarrow\infty}u^{(n)}_{2}(t,x),$

exist for $x\in\mathbb{R}^{d}$ and $0\leq t<\tau_{*}.$ Then by the monotone
convergence theorem, we obtain

$\displaystyle v_{1}(t,x)=\mathcal{J}_{1}v_{2}(t,x),\
v_{2}(t,x)=\mathcal{J}_{2}v_{1}(t,x),\ \ x\in\mathbb{R}^{d},\ \ 0\leq
t<\tau_{*}.$

Since $f_{1}=C_{1}\psi$ and $f_{2}=C_{2}\psi,$ for some positive constants
$C_{1}$ and $C_{2}$ with $C_{1}\leq C_{2}$, from (3.5), it follows that

$\displaystyle\mathopen{}\mathclose{{}\left\\{\begin{aligned} 0&\leq
u_{1}^{(0)}(t,x)=T(t)\mathopen{}\mathclose{{}\left(\frac{C_{1}}{C_{2}}f_{2}(x)}\right)=\frac{C_{1}}{C_{2}}T(t)f_{2}(x)\leq
T(t)f_{2}(x)\mathscr{G}_{2}(t),\\\
0&\leq\frac{C_{2}}{C_{1}}u_{2}^{(0)}(t,x)=T(t)\mathopen{}\mathclose{{}\left(\frac{C_{2}}{C_{1}}f_{1}(x)}\right)=\frac{C_{2}}{C_{1}}T(t)f_{1}(x),\\\
&u_{2}^{(0)}(t,x)=T(t)f_{1}(x)\leq
T(t)f_{1}(x)\mathscr{G}_{1}(t).\end{aligned}}\right.$ (3.7)

From (3.6), we infer

$\displaystyle u^{(1)}_{1}(t,x)$
$\displaystyle=\mathcal{J}_{1}u_{2}^{(0)}(t,x),$ (3.8)

so that (3.7) gives $u_{2}^{(0)}(t,x)\leq T(t)f_{1}(x)\mathscr{G}_{1}(t).$
Therefore from (3.8), we obtain

$\displaystyle T(t)f_{1}(x)\leq u^{(1)}_{1}(t,x)\leq
T(t)f_{1}(x)\mathscr{G}_{1}(t).$ (3.9)

Similarly, from (3.6), we deduce

$\displaystyle u^{(1)}_{2}(t,x)=\mathcal{J}_{2}u^{(0)}_{1}(t,x),$ (3.10)

so that from (3.7), we get $u_{1}^{(0)}(t,x)\leq
T(t)f_{2}(x)\mathscr{G}_{2}(t)$. Therefore from (3.10), we have

$\displaystyle T(t)S_{t}f_{2}(x)\leq u^{(1)}_{2}(t,x)\leq
T(t)f_{2}(x)\mathscr{G}_{2}(t).$

Note that (3.6) leads to

$\displaystyle u^{(2)}_{1}(t,x)$
$\displaystyle=\mathcal{J}_{1}u^{(1)}_{2}(t,x),$ (3.11)

where $u^{(1)}_{2}(t,x)=\mathcal{J}_{2}u^{(0)}_{1}(t,x)$. We know that
$u_{1}^{(0)}(t,x)\leq T(t)f_{1}(x)\mathscr{G}_{1}(t)$, so that

$\displaystyle u^{(1)}_{2}(t,x)\leq T(t)f_{1}(x)\mathscr{G}_{1}(t).$ (3.12)

By using (3.12) in (3.11), we have

$\displaystyle T(t)f_{1}(x)\leq u^{(2)}_{1}(t,x)\leq
T(t)f_{1}(x)\mathscr{G}_{1}(t).$ (3.13)

Also (3.6) results in

$\displaystyle u^{(2)}_{2}(t,x)=\mathcal{J}_{2}u^{(1)}_{1}(t,x),$ (3.14)

where $u^{(1)}_{1}(t,x)=\mathcal{J}_{1}u^{(0)}_{1}(t,x)$. From (3.7) we know
that $u^{(0)}_{1}(t,x)\leq T(t)f_{2}(x)\mathscr{G}_{2}(t)$. Therefore $0\leq
u^{(1)}_{1}(t,x)\leq T(t)f_{2}(x)\mathscr{G}_{2}(t).$ From (3.14), we obtain

$\displaystyle T(t)f_{1}(x)\leq u^{(2)}_{2}(t,x)\leq
T(t)f_{2}(x)\mathscr{G}_{2}(t).$ (3.15)

Continuing like this, for each $n\geq 1$, one gets

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} T(t)f_{1}(x)&\leq
u^{(n)}_{1}(t,x)\leq T(t)f_{1}(x)\mathscr{G}_{1}(t),\\\ T(t)f_{2}(x)&\leq
u^{(n)}_{2}(t,x)\leq T(t)f_{2}(x)\mathscr{G}_{2}(t).\end{aligned}}\right.$

Moreover, we deduce

$\displaystyle\mathopen{}\mathclose{{}\left\\{\begin{aligned}
v_{1}(t,x)&\leq\frac{T(t)f_{1}(x)}{\mathopen{}\mathclose{{}\left[1-\beta_{1}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(r)f_{1}}\right\|_{\infty}^{\beta_{1}}dr}\right]^{\frac{1}{\beta_{1}}}},\\\
v_{2}(t,x)&\leq\frac{T(t)f_{2}(x)}{\mathopen{}\mathclose{{}\left[1-\beta_{2}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(r)f_{2}}\right\|_{\infty}^{\beta_{2}}dr}\right]^{\frac{1}{\beta_{2}}}}.\end{aligned}}\right.$
(3.16)

By Lemma 2.2 (1), we have

$\displaystyle\mathopen{}\mathclose{{}\left\|T(t)f_{i}}\right\|_{\infty}^{\beta_{i}}=\mathopen{}\mathclose{{}\left\|e^{\lambda
t}S_{t}f_{i}}\right\|_{\infty}^{\beta_{i}}$
$\displaystyle=e^{\lambda\beta_{i}t}\mathopen{}\mathclose{{}\left(\sup_{x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}p(r,y-x)f_{i}(y)dy}\right)^{\beta_{i}}$
$\displaystyle=e^{\lambda\beta_{i}t}r^{-d\beta_{i}/\alpha}\mathopen{}\mathclose{{}\left(\sup_{x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}p(1,r^{-1/\alpha}(y-x)f_{i}(y)dy}\right)^{\beta_{i}}$
$\displaystyle=e^{\lambda\beta_{i}t}r^{-d\beta_{i}/\alpha}p(1,0)^{\beta_{i}}\mathopen{}\mathclose{{}\left\|f_{i}}\right\|_{\infty}^{\beta_{i}},$

for $i=1,2$. Therefore,

$\displaystyle\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(r)f_{i}}\right\|_{\infty}^{\beta_{i}}dr$
$\displaystyle\qquad\qquad=p(1,0)^{\beta_{i}}\mathopen{}\mathclose{{}\left\|f_{i}}\right\|_{\infty}^{\beta_{i}}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\lambda\beta_{i}r\\}r^{-d\beta_{i}/\alpha}dr.$

From (3.16), we have

$\displaystyle v_{1}(t,x)$
$\displaystyle\leq\frac{T(t)f_{1}(x)}{\mathopen{}\mathclose{{}\left[1-\beta_{1}p(1,0)^{\beta_{1}}\mathopen{}\mathclose{{}\left\|f_{1}}\right\|_{\infty}^{\beta_{1}}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\lambda\beta_{1}r\\}r^{-d\beta_{1}/\alpha}dr}\right]^{\frac{1}{\beta_{1}}}},$
$\displaystyle v_{2}(t,x)$
$\displaystyle\leq\frac{T(t)f_{2}(x)}{\mathopen{}\mathclose{{}\left[1-\beta_{2}p(1,0)^{\beta_{2}}\mathopen{}\mathclose{{}\left\|f_{2}}\right\|_{\infty}^{\beta_{2}}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\lambda\beta_{2}r\\}r^{-d\beta_{2}/\alpha}dr}\right]^{\frac{1}{\beta_{2}}}}.$

By the choice of initial values, one can complete the proof. ∎

######  Remark 3.1.

Since the initial data $f_{1}$ and $f_{2}$ are non-negative, the weak
solutions $u=(u_{1},u_{2})^{\top}$ of (1.3) and $v=(v_{1},v_{2})^{\top}$ of
(2.1) are non-negative as well.

##  4 Finite time explosion of positive solutions

In this section, we obtain upper bounds for the blow-up time of the system
(1.3) with parameters given in (3.1). Also we find upper bounds for the
probability of non-explosive positive solution $u=(u_{1},u_{2})^{\top}$ of the
system (1.3) with $\frac{1}{2}<H<1$.

######  Lemma 4.1.

Suppose $v=(v_{1},v_{2})^{\top}$ is the mild solution of (2.1),

$m_{i}(t)=\mathbb{E}\mathopen{}\mathclose{{}\left[v_{i}(t,X_{t})}\right]=\int_{\mathbb{R}^{d}}p(t,y)v_{i}(t,y)dy=S_{t}v_{i}(t,0),\
\ i=1,2,$

where $\mathopen{}\mathclose{{}\left\\{X_{t},t\geq 0}\right\\}$ is a
spherically symmetric $\alpha$-stable process in $\mathbb{R}^{d}$ starting
from $0,$ and (3.1) holds. For $\omega\in\Omega,$ if $m_{i}$ blows-up in a
finite time $\tau_{m_{i}}=\tau_{m_{i}}(\omega),\ i=1,2,$ then
$v=(v_{1},v_{2})^{\top}$ blows-up in a finite time $\tau_{v}$. Moreover,

$\displaystyle\tau_{v}=\min\\{\tau_{v_{1}},\tau_{v_{2}}\\}\leq\min\mathopen{}\mathclose{{}\left\\{(r_{0}+\tau_{m_{1}})(1+2^{\alpha}),(r_{0}+\tau_{m_{2}})(1+2^{\alpha})}\right\\}.$
(4.1)

###### Proof.

For $i=1,2,\
\mathopen{}\mathclose{{}\left\\{j}\right\\}=\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\\{i\\}$,
let $\tau_{m_{i}}<\infty$ be the blow-up time of $m_{i},$ and
$t\geq\tau_{m_{j}}+2^{\alpha}(r_{0}+\tau_{m_{j}}).$ Then, for
$s\in[0,\tau_{m_{j}}],$

$\displaystyle t-s\geq t-\tau_{m_{j}}\geq 2^{\alpha}(r_{0}+\tau_{m_{j}})\geq
2^{\alpha}(r_{0}+s).$

In the above inequality, we define

$\displaystyle\mathopen{}\mathclose{{}\left(\frac{t-s}{r_{0}+s}}\right)^{1/\alpha}=:\rho\geq
2.$ (4.2)

Since $p(r_{0},0)<1,$ we get by Lemma 2.2 (1)

$\displaystyle
p(r_{0}+s,0)=p\mathopen{}\mathclose{{}\left(\frac{r_{0}+s}{r_{0}}r_{0},0}\right)=\mathopen{}\mathclose{{}\left(\frac{r_{0}+s}{r_{0}}}\right)^{-d/\alpha}p(r_{0},0)<1,\
\mbox{ for all }\ s\in[0,\tau_{m_{j}}].$ (4.3)

Therefore, for any $x,y\in\mathbb{R}^{d},$ by Lemma 2.2 (1), we obtain

$\displaystyle p(t-s,y-x)$
$\displaystyle=p\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{t-s}{r_{0}+s}}\right)r_{0}+s,y-x}\right)$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{t-s}{r_{0}+s}}\right)^{-d/\alpha}p\mathopen{}\mathclose{{}\left(r_{0}+s,\mathopen{}\mathclose{{}\left(\frac{t-s}{r_{0}+s}}\right)^{-1/\alpha}(y-x)}\right).$

From (4.2),(4.3) and Lemma 2.2 (4), we deduce the inequality

$\displaystyle p(t-s,y-x)$
$\displaystyle=\rho^{-d}p\mathopen{}\mathclose{{}\left(r_{0}+s,\frac{y-x}{\rho}}\right)\geq\rho^{-d}p\mathopen{}\mathclose{{}\left(r_{0}+s,x}\right)p\mathopen{}\mathclose{{}\left(r_{0}+s,y}\right).$
(4.4)

Therefore, from (2.5) and (4.4), it follows that for each
$t\geq\tau_{m_{j}}+2^{\alpha}(r_{0}+\tau_{m_{j}}),\ i=1,2,\
\mathopen{}\mathclose{{}\left\\{j}\right\\}=\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\\{i\\}$,
all $x\in\mathbb{R}^{d},$

$\displaystyle v_{i}(r_{0}+t,x)$
$\displaystyle=\exp\\{\gamma_{i}t\\}\int_{\mathbb{R}^{d}}p(t,x-y)v_{i}(r_{0},y)dy$
$\displaystyle\quad+\int_{0}^{t}\int_{\mathbb{R}^{d}}e^{\gamma_{i}(t-s)}p(t-s,x-y)v_{j}^{1+\beta_{i}}(r_{0}+s,y)e^{\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}dyds$
$\displaystyle\geq\rho^{-d}\int_{0}^{\tau_{m_{j}}}\int_{\mathbb{R}^{d}}e^{\gamma_{i}(t-s)+\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}p\mathopen{}\mathclose{{}\left(r_{0}+s,x}\right)p\mathopen{}\mathclose{{}\left(r_{0}+s,y}\right)v_{j}^{1+\beta_{i}}(r_{0}+s,y)dyds$
$\displaystyle=\rho^{-d}\int_{0}^{\tau_{m_{j}}}e^{\gamma_{i}(t-s)+\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}p\mathopen{}\mathclose{{}\left(r_{0}+s,x}\right)\mathopen{}\mathclose{{}\left[\int_{\mathbb{R}^{d}}p\mathopen{}\mathclose{{}\left(r_{0}+s,y}\right)v_{j}^{1+\beta_{i}}(r_{0}+s,y)dy}\right]ds.$
(4.5)

Here $p(r_{0}+s,y)$ is the probability density and by using Jensen’s
inequality, we have

$\displaystyle\mathopen{}\mathclose{{}\left[\int_{\mathbb{R}^{d}}p\mathopen{}\mathclose{{}\left(r_{0}+s,y}\right)v^{1+\beta_{i}}_{j}(r_{0}+s,y)dy}\right]$
$\displaystyle\geq\mathopen{}\mathclose{{}\left[\int_{\mathbb{R}^{d}}p\mathopen{}\mathclose{{}\left(r_{0}+s,y}\right)v_{j}(r_{0}+s,y)dy}\right]^{1+\beta_{i}}=m_{j}^{1+\beta_{i}}(r_{0}+s).$

Therefore (4) becomes

$\displaystyle v_{i}(r_{0}+t,x)$
$\displaystyle\geq\rho^{-d}\int_{0}^{\tau_{m_{j}}}e^{\gamma_{i}(t-s)+\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}p\mathopen{}\mathclose{{}\left(r_{0}+s,x}\right)m_{j}^{1+\beta_{i}}(r_{0}+s)ds$
$\displaystyle=\rho^{-d}\int_{r_{0}}^{\tau_{m_{j}}}e^{\gamma_{i}(t+r_{0}-\ell)+\rho_{1}B_{1}^{H}(\ell)+\rho_{2}B_{2}^{H}(\ell)}p\mathopen{}\mathclose{{}\left(\ell,x}\right)m_{j}^{1+\beta_{i}}(\ell)d\ell$
$\displaystyle\quad+\rho^{-d}\int_{\tau_{m_{j}}}^{r_{0}+\tau_{m_{j}}}e^{\gamma_{i}(t+r_{0}-\ell)+\rho_{1}B_{1}^{H}(\ell)+\rho_{2}B_{2}^{H}(\ell)}p\mathopen{}\mathclose{{}\left(\ell,x}\right)m_{j}^{1+\beta_{i}}(\ell)d\ell.$

Since $r_{0}>0,$ the second integral in the RHS of the above inequality is
infinite. Hence $\tau_{v_{i}}\leq
r_{0}+\tau_{m_{j}}+2^{\alpha}(r_{0}+\tau_{m_{j}})=(r_{0}+\tau_{m_{j}})(1+2^{\alpha}),$
for $i=1,2,\
\mathopen{}\mathclose{{}\left\\{j}\right\\}=\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\\{i\\}$,
which completes the proof. ∎

The following theorem gives an upper bound for the finite-time blow-up of
solution $u=(u_{1},u_{2})^{\top}$ of the system (1.3).

######  Theorem 4.1.

Let $\frac{1}{2}<H<1$ and let $u=(u_{1},u_{2})^{\top}$ be a weak solution of
(1.3) with nonnegative initial data $f_{1}$ and $f_{2}$ which are bounded
measurable functions and (3.1) holds.

If $\beta_{1}=\beta_{2}=\beta>0.$ Then $u=(u_{1},u_{2})^{\top}$ blows-up in
finite time which is given by

$\displaystyle\theta_{1}=\inf\mathopen{}\mathclose{{}\left\\{t>r_{0}:\int_{r_{0}}^{t}\min\\{g_{1,2}(s),g_{2,1}(s)\\}ds\geq\frac{2^{1+\beta}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}}\right\\}.$

Moreover, if $\tau$ is the blow-up time of (1.3), then

$\displaystyle\tau$
$\displaystyle\leq\min\mathopen{}\mathclose{{}\left\\{(r_{0}+\tau_{m_{1}})(1+2^{\alpha}),(r_{0}+\tau_{m_{2}})(1+2^{\alpha})}\right\\}\leq(r_{0}+\theta_{1})(1+2^{\alpha}),$

where
$r_{i}=2^{-2d}p(1,0)\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right],\
i=1,2$ and

$g_{i,j}(s)=2^{\frac{-d(1+\beta_{i})}{\alpha}}e^{k_{i,j}s+\rho_{1}B_{1}^{H}(s)+\rho_{2}B_{2}^{H}(s)}s^{\frac{-d\beta_{i}}{\alpha}},$

where $\rho_{1},\rho_{2}$ are defined in (3.1) and
$k_{i,j}=-\gamma_{i}+(1+\beta_{i})\gamma_{j}$ for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$

If $\beta_{1}>\beta_{2}>0$ and let
$D_{1}=\mathopen{}\mathclose{{}\left(\frac{\beta_{1}-\beta_{2}}{1+\beta_{1}}}\right)\mathopen{}\mathclose{{}\left(\frac{1+\beta_{1}}{1+\beta_{2}}}\right)^{\frac{1+\beta_{2}}{\beta_{1}-\beta_{2}}},$

$\displaystyle\epsilon_{0}$
$\displaystyle=\min\Bigg{\\{}1,\mathopen{}\mathclose{{}\left(h_{1}(r_{0})/D_{1}^{1/1+\beta_{2}}}\right)^{\beta_{1}-\beta_{2}},\mathopen{}\mathclose{{}\left(2^{-(1+\beta_{2})}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}/D_{1}}\right)^{\frac{\beta_{1}-\beta_{2}}{1+\beta_{1}}}\Bigg{\\}}.$
(4.6)

Assume that

$\displaystyle
2^{-(1+\beta_{2})}\epsilon_{0}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}\geq\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}.$
(4.7)

Then $u=(u_{1},u_{2})^{\top}$ blow-up in finite time which is given by

$\displaystyle\theta_{2}=$
$\displaystyle\inf\Bigg{\\{}t>r_{0}:\int_{r_{0}}^{t}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds\geq$
$\displaystyle\qquad\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1}\Bigg{\\}},$

Moreover, if $\tau$ is the blow-up time of (1.3), then

$\displaystyle\tau$
$\displaystyle\leq\min\mathopen{}\mathclose{{}\left\\{(r_{0}+\tau_{m_{1}})(1+2^{\alpha}),(r_{0}+\tau_{m_{2}})(1+2^{\alpha})}\right\\}\leq(r_{0}+\theta_{2})(1+2^{\alpha}),$

where
$r_{i}=p(1,0)2^{-2d}\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right],\
i=1,2$ and for any $r>0,$

$\displaystyle
g_{i,j}(s)=2^{\frac{-d(1+\beta_{i})}{\alpha}}e^{k_{i,j}s+\rho_{1}B_{1}^{H}(s)+\rho_{2}B_{2}^{H}(s)}s^{\frac{-d\beta_{i}}{\alpha}},$
(4.8)

where $\rho_{1},\rho_{2}$ are defined in (3.1) and

$\displaystyle k_{i,j}=-\gamma_{i}+(1+\beta_{i})\gamma_{j}\ \text{ for }\
i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
(4.9)

###### Proof.

By using the property of semigroup for $v_{i},\ i=1,2$, we have

$\displaystyle
v_{i}(t+r_{0})=T_{i}(t)v_{i}(r_{0})+\int_{0}^{t}T_{i}(t-s)v_{j}^{1+\beta_{i}}(r_{0}+s)e^{\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}ds,\
t\geq 0,$

for
$j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
Therefore

$\displaystyle m_{i}(t+r_{0})$ $\displaystyle=S_{t+r_{0}}v_{i}(t+r_{0})(0)$
$\displaystyle=S_{t+r_{0}}\mathopen{}\mathclose{{}\left(e^{\gamma_{i}t}S_{t}v_{i}(r_{0})+\int_{0}^{t}e^{\gamma_{i}(t-s)}S_{t-s}v_{j}^{1+\beta_{i}}(r_{0}+s)e^{\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}ds}\right)(0)$
$\displaystyle=S_{2t+r_{0}}e^{\gamma_{i}t}v_{i}(r_{0},0)+\int_{0}^{t}e^{\gamma_{i}(t-s)}S_{2t+r_{0}-s}v_{j}^{1+\beta_{i}}(r_{0}+s,0)e^{\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}ds.$
(4.10)

By using Lemma 2.2 (1) and Jensen’s inequality, we obtain

$\displaystyle S_{2t+r_{0}-s}v_{j}^{1+\beta}(r_{0}+s,0)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(\frac{r_{0}+s}{2t+r_{0}-s}}\right)^{d/\alpha}S_{r_{0}+s}v_{j}^{1+\beta_{i}}(r_{0}+s,0)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(\frac{r_{0}+s}{2t+2r_{0}}}\right)^{d/\alpha}\mathopen{}\mathclose{{}\left(S_{r_{0}+s}v_{j}(r_{0}+s,0)}\right)^{1+\beta_{i}}$
$\displaystyle=\mathopen{}\mathclose{{}\left(\frac{r_{0}+s}{2t+2r_{0}}}\right)^{d/\alpha}m_{j}^{1+\beta_{i}}(r_{0}+s),\
0\leq s\leq t,$ (4.11)

for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
Also, we have

$\displaystyle
v_{i}(r_{0},y)\geq\exp\\{\gamma_{i}r_{0}\\}\int_{\mathbb{R}^{d}}p(r_{0},x-y)f_{i}(x)dx,$

for all $y\in\mathbb{R}^{d}.$ Since $p(r_{0},0)<1$ and by Lemma 2.2 (1) and
(4), we get

$\displaystyle\exp\\{\gamma_{i}r_{0}\\}$
$\displaystyle\int_{\mathbb{R}^{d}}p\mathopen{}\mathclose{{}\left(r_{0},\frac{2x-2y}{2}}\right)f_{i}(x)dx$
$\displaystyle\geq\exp\\{\gamma_{i}r_{0}\\}p(r_{0},2y)\int_{\mathbb{R}^{d}}p(r_{0},2x)f_{i}(x)dx$
$\displaystyle=2^{-2d}\exp\\{\gamma_{i}r_{0}\\}p(2^{-\alpha}r_{0},y)\int_{\mathbb{R}^{d}}p(2^{-\alpha}r_{0},x)f_{i}(x)dx$
$\displaystyle=2^{-2d}\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right]p(2^{-\alpha}r_{0},y),$

for $i=1,2.$ Therefore we have

$v_{i}(r_{0},y)\geq
2^{-2d}\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right]p(2^{-\alpha}r_{0},y).$

By Lemma 2.2 (4) and semigroup property, we deduce

$\displaystyle S_{2t+r_{0}}v_{i}(r_{0},0)$
$\displaystyle=\int_{\mathbb{R}^{d}}p(2t+r_{0},y)v_{i}(r_{0},y)dy$
$\displaystyle\geq
2^{-2d}\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right]p\mathopen{}\mathclose{{}\left(2t+(1+2^{-\alpha})r_{0},0}\right)$
$\displaystyle=2^{-2d}\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right]\mathopen{}\mathclose{{}\left(2t+(1+2^{-\alpha})r_{0}}\right)^{-d/\alpha}p(1,0)$
$\displaystyle\geq
2^{-2d}\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right](2t+2r_{0})^{-d/\alpha}p(1,0)$
$\displaystyle=r_{i}(2t+2r_{0})^{-d/\alpha},$ (4.12)

where

$\displaystyle
r_{i}=2^{-2d}p(1,0)\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right],\
i=1,2.$ (4.13)

Plugging (4) and (4) into (4), and multiplying both sides of (4) by
$\exp\\{-\gamma_{i}(r_{0}+t)\\}(2t+2r_{0})^{d/\alpha}$ for $i=1,2,$ we have

$\displaystyle m_{i}(t+r_{0})$ $\displaystyle\geq
r_{i}(2t+2r_{0})^{-d/\alpha}e^{\gamma_{i}t}$
$\displaystyle\quad+\int_{0}^{t}e^{\gamma_{i}(t-s)}\mathopen{}\mathclose{{}\left(\frac{r_{0}+s}{2t+2r_{0}}}\right)^{d/\alpha}m_{j}^{1+\beta_{i}}(r_{0}+s)e^{\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}ds.$

That is,

$\displaystyle\exp\\{-\gamma_{i}(r_{0}+t)\\}(2t+2r_{0})^{d/\alpha}m_{i}(t+r_{0})$
$\displaystyle\geq
e^{-\gamma_{i}r_{0}}r_{i}+\int_{0}^{t}e^{-\gamma_{i}(r_{0}+s)}(r_{0}+s)^{d/\alpha}m_{j}^{1+\beta_{i}}(r_{0}+s)e^{\rho_{1}B_{1}^{H}(r_{0}+s)+\rho_{2}B_{2}^{H}(r_{0}+s)}ds.$

Let

$\displaystyle w_{i}(r)$
$\displaystyle=e^{-\gamma_{i}r}(2r)^{d/\alpha}m_{i}(r),\ \ \mbox{and}$
$\displaystyle g_{i,j}(r)$
$\displaystyle=2^{\frac{-d(1+\beta_{i})}{\alpha}}e^{k_{i,j}r+\rho_{1}B_{1}^{H}(r)+\rho_{2}B_{2}^{H}(r)}r^{\frac{-d\beta_{i}}{\alpha}},\
\ r>0,$ (4.14)

where $k_{i,j}=-\gamma_{i}+(1+\beta_{i})\gamma_{j}$ for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
Then, we have

$\displaystyle w_{i}(r_{0}+t)$
$\displaystyle=e^{-\gamma_{i}(r_{0}+t)}(2(r_{0}+t))^{d/\alpha}m_{i}(r_{0}+t)$
$\displaystyle\geq
r_{i}e^{-\gamma_{i,j}r_{0}}+\int_{0}^{t}g_{i,j}(r_{0}+s)w_{j}^{1+\beta_{i}}(r_{0}+s)ds$
$\displaystyle=r_{i}e^{-\gamma_{i}r_{0}}+\int_{r_{0}}^{r_{0}+t}g_{i,j}(r)w_{j}^{1+\beta_{i}}(r)dr,$

which can be equivalently written as

$\displaystyle w_{i}(t)$ $\displaystyle\geq
r_{i}e^{-\gamma_{i}r_{0}}+\int_{r_{0}}^{t}g_{i,j}(r)w_{j}^{1+\beta_{i}}(r)dr,\
\ t\geq r_{0}.$

Notice that $m_{i}$ and $w_{i}$ explode at the same finite time due to (4),
and so do $v_{i}$ and $u_{i}.$ Moreover, let $w_{i}(t)\geq h_{i}(t),\ i=1,2$
for all $t\geq r_{0},$ where $h_{1}$ and $h_{2}$ are given by

$\displaystyle h_{1}(t)$
$\displaystyle=r_{1}e^{-\gamma_{1}r_{0}}+\int_{r_{0}}^{t}g_{1,2}(r)h_{2}^{1+\beta_{1}}(r)dr,$
$\displaystyle h_{2}(t)$
$\displaystyle=r_{2}e^{-\gamma_{2}r_{0}}+\int_{r_{0}}^{t}g_{2,1}(r)h_{1}^{1+\beta_{2}}(r)dr.$

Define $E(t)=h_{1}(t)+h_{2}(t),$ so that

$\displaystyle
E(t)=r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}+\int_{r_{0}}^{t}g_{1,2}(r)h_{2}^{1+\beta_{1}}(r)dr+\int_{r_{0}}^{t}g_{2,1}(r)h_{1}^{1+\beta_{2}}(r)dr.$

Therefore,

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{dE(t)}{dt}&=g_{1,2}(t)h_{2}^{1+\beta_{1}}(t)+g_{2,1}(t)h_{1}^{1+\beta_{2}}(t),\\\
E(r_{0})&=r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}.\end{aligned}}\right.$
(4.15)

Case 1: Suppose that $\beta_{1}=\beta_{2}=\beta>0$ in (4.15). Then, we have

$\displaystyle\frac{dE(t)}{dt}$
$\displaystyle=g_{1,2}(t)h_{2}^{1+\beta}(t)+g_{2,1}(t)h_{1}^{1+\beta}(t)$
$\displaystyle\geq\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(t),g_{2,1}(t)}\right\\}\mathopen{}\mathclose{{}\left[h_{2}^{1+\beta}(t)+h_{1}^{1+\beta}(t)}\right].$
(4.16)

By substituting $a=1,\ b=\frac{h_{1}}{h_{2}},\ n=1+\beta$ into the inequality

$\displaystyle a^{n}+b^{n}\geq 2^{-n}{(a+b)}^{n},$ (4.17)

which is valid for $a,\ b\geq 0,$ we obtain

$\displaystyle h_{1}^{1+\beta}(t)+h_{2}^{1+\beta}(t)\geq
2^{-(1+\beta)}\mathopen{}\mathclose{{}\left(h_{1}(t)+h_{2}(t)}\right)^{1+\beta}\geq
2^{-(1+\beta)}E^{1+\beta}(t).$

Form (4), we have

$\displaystyle\frac{dE(t)}{dt}\geq\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(t),g_{2,1}(t)}\right\\}2^{-(1+\beta)}E^{1+\beta}(t).$

Thus, $E(t)$ blows-up not later than the solution $I(t)$ of the differential
equation

$\displaystyle\frac{dI(t)}{dt}=\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(t),g_{2,1}(t)}\right\\}2^{-(1+\beta)}I^{1+\beta}(t),\
\ I(r_{0})=E(r_{0}),$

which gives that

$\displaystyle
I(t)=\mathopen{}\mathclose{{}\left\\{E^{-\beta}(r_{0})-2^{-(1+\beta)}\beta\int_{r_{0}}^{t}\min\\{g_{1,2}(s),g_{2,1}(s)\\}ds}\right\\}^{-\frac{1}{\beta}}.$

For the above equation, the explosion time is given by

$\displaystyle\tau_{m_{i}}=\tau_{w_{i}}\leq\theta_{1}:=\inf\mathopen{}\mathclose{{}\left\\{t>r_{0}:\int_{r_{0}}^{t}\min\\{g_{1,2}(s),g_{2,1}(s)\\}ds\geq\frac{2^{1+\beta}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}}\right\\},$

for $i=1,2,$ where $r_{1}$ and $r_{2}$ are given in (4.13). If $\tau$ is the
blow-up time of $u=(u_{1},u_{2})^{\top},$ then by Lemma 4.1, we have

$\displaystyle\tau$
$\displaystyle\leq\min\mathopen{}\mathclose{{}\left\\{(r_{0}+\tau_{m_{1}})(1+2^{\alpha}),(r_{0}+\tau_{m_{2}})(1+2^{\alpha})}\right\\}\leq(r_{0}+\theta_{1})(1+2^{\alpha}).$

Case 2: Suppose that $\beta_{1}>\beta_{2}>0$ in (4.15). Then, we have

$\displaystyle\frac{dE(t)}{dt}\geq\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(t),g_{2,1}(t)}\right\\}\mathopen{}\mathclose{{}\left[h_{2}^{1+\beta_{1}}(t)+h_{1}^{1+\beta_{2}}(t)}\right].$
(4.18)

The Young inequality states that if $1<b<\infty$, $\delta>0$, and
$a=\frac{b}{b-1}$, then

$\displaystyle xy\leq\frac{\delta^{a}x^{a}}{a}+\frac{\delta^{-b}y^{b}}{b},\
x,y\geq 0.$ (4.19)

By setting $b=\frac{1+\beta_{1}}{1+\beta_{2}},\ y=h_{2}^{1+\beta_{2}}(t),\
x=\epsilon,\
\delta=\mathopen{}\mathclose{{}\left(\frac{1+\beta_{1}}{1+\beta_{2}}}\right)^{\frac{1+\beta_{2}}{1+\beta_{1}}},$
and using the fact that $\beta_{2}<\beta_{1}$ in (4.19), it follows that for
any $\epsilon>0,$

$\displaystyle h_{2}^{1+\beta_{1}}(t)\geq\epsilon
h_{2}^{1+\beta_{2}}(t)-D_{1}\epsilon^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}},$

where
$D_{1}=\mathopen{}\mathclose{{}\left(\frac{\beta_{1}-\beta_{2}}{1+\beta_{1}}}\right)\mathopen{}\mathclose{{}\left(\frac{1+\beta_{1}}{1+\beta_{2}}}\right)^{\frac{1+\beta_{2}}{\beta_{1}-\beta_{2}}}.$
Since $\epsilon_{0}$ is the minimum of the quantities given in (4.6), in
particular, we have
$\epsilon_{0}\leq(h_{1}(r_{0})/D_{1}^{1/1+\beta_{2}})^{\beta_{1}-\beta_{2}},$
so that

$\displaystyle\epsilon_{0}h_{2}^{1+\beta_{2}}(r_{0})-D_{1}\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}\geq
0.$

From (4.18), we have

$\displaystyle\frac{dE(t)}{dt}\geq\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(t),g_{2,1}(t)}\right\\}\mathopen{}\mathclose{{}\left[h_{1}^{1+\beta_{2}}(t)+\epsilon_{0}h_{2}^{1+\beta_{2}}(t)-D_{1}\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}}\right].$
(4.20)

Utilizing the inequality (4.17) again with $n=1+\beta_{2},\ a=1$ and
$b=\epsilon_{0}^{\frac{1}{1+\beta_{2}}}\frac{h_{1}(t)}{h_{2}(t)}$, and by
using (4.6), we see that

$\displaystyle h_{2}^{1+\beta_{2}}(t)+\epsilon_{0}h_{1}^{1+\beta_{2}}(t)\geq
2^{-(1+\beta_{2})}\mathopen{}\mathclose{{}\left(h_{2}(t)+\epsilon_{0}^{\frac{1}{1+\beta_{2}}}h_{1}(t)}\right)^{1+\beta_{2}}\geq
2^{-(1+\beta_{2})}\epsilon_{0}E^{1+\beta_{2}}(t).$

From (4.20), we deduce

$\displaystyle\frac{dE(t)}{dt}$
$\displaystyle\geq\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(t),g_{2,1}(t)}\right\\}\mathopen{}\mathclose{{}\left[2^{-(1+\beta_{2})}\epsilon_{0}E^{1+\beta_{2}}(t)-D_{1}\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}}\right].$

Note that the condition (4.7) gives $E(t)\geq E(r_{0})>0,$ and so

$\displaystyle\frac{dE(t)}{E^{1+\beta_{2}}(t)}$
$\displaystyle\geq\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(t),g_{2,1}(t)}\right\\}\mathopen{}\mathclose{{}\left[\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{D_{1}\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}}{E^{1+\beta_{2}}(r_{0})}}\right]dt.$

Thus, $E(t)$ blows-up not later than the solution $I(t)$ of the differential
equation

$\displaystyle\frac{dI(t)}{I^{1+\beta_{2}}(t)}=\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(t),g_{2,1}(t)}\right\\}\mathopen{}\mathclose{{}\left[\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{D_{1}\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}}{I^{1+\beta_{2}}(r_{0})}}\right]dt.$

Therefore,

$\displaystyle
I(t)=\mathopen{}\mathclose{{}\left\\{I^{-\beta_{2}}(r_{0})-\beta_{2}\mathopen{}\mathclose{{}\left[\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{I^{1+\beta_{2}}(r_{0})}}\right]\displaystyle\int_{r_{0}}^{t}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds}\right\\}^{\frac{-1}{\beta_{2}}}.$

For the above inequality, the explosion time is given by

$\displaystyle\tau_{m_{i}}$
$\displaystyle=\tau_{w_{i}}\leq\theta_{2}:=\inf\Bigg{\\{}t>r_{0}:\int_{r_{0}}^{t}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds\geq$
$\displaystyle\qquad\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1}\Bigg{\\}},$

for $i=1,2,$ where $r_{1}$ and $r_{2}$ are given in (4.13). If $\tau$ is the
blow-up time of $u=(u_{1},u_{2})^{\top},$ then by Lemma 4.1, we have

$\displaystyle\tau$
$\displaystyle\leq\min\mathopen{}\mathclose{{}\left\\{(r_{0}+\tau_{m_{1}})(1+2^{\alpha}),(r_{0}+\tau_{m_{2}})(1+2^{\alpha})}\right\\}\leq(r_{0}+\theta_{2})(1+2^{\alpha}),$

which completes the proof. ∎

Next, we extend the results obtained in Lemma 1, [5] on one dimensional
fractional Brownian motion to two dimensions.

######  Lemma 4.2.

Let $0<H<1$ and let
$\mathopen{}\mathclose{{}\left(B_{1}^{H}(t),B_{2}^{H}(t)}\right)_{t\geq 0}$ be
a two-dimensional fractional Brownian motion with Hurst parameter $H$ defined
on a probability space
$\mathopen{}\mathclose{{}\left(\Omega,\mathscr{F},\mathbb{P}}\right).$ If
$\nu>0,$ then

$\mathbb{P}\mathopen{}\mathclose{{}\left(\int_{0}^{\infty}e^{B^{H}_{1}(s)+B^{H}_{2}(s)-\nu
s}ds<\infty}\right)=1.$

If $\nu<0,$ then

$\mathbb{P}\mathopen{}\mathclose{{}\left(\int_{0}^{\infty}e^{B^{H}_{1}(s)+B^{H}_{2}(s)-\nu
s}ds=\infty}\right)=1.$

###### Proof.

As a consequence of the law of iterated logarithm for fractional Brownian
motion ([21]), we have

$\displaystyle\limsup_{t\rightarrow\infty}\frac{B_{i}^{H}(t)}{t^{H}\sqrt{2\log\log
t}}=1,\ \mathbb{P}\text{-a.s.,}\ $

for $i=1,2.$ This means that for some $t_{0}>e$ and for all $t\geq t_{0},$ we
have

$B^{H}_{i}(t)\leq 2t^{H}\sqrt{2\log\log t},\ i=1,2.$

Assume that $\nu>0.$ Since $H<1,$ there exists $t_{1}>t_{0}$ such that for
each $t\geq t_{1}$

$2t^{H}\sqrt{2\log\log t}\leq\frac{\nu t}{4}.$

Therefore, for all $t\geq t_{1},$ we have

$\displaystyle\int_{t}^{\infty}e^{B^{H}_{1}(s)+B^{H}_{2}(s)-\nu
s}ds<\int_{t}^{\infty}e^{\frac{\nu s}{4}+\frac{\nu s}{4}-\nu
s}ds=\int_{t}^{\infty}e^{-\frac{\nu s}{2}}ds\leq\frac{2}{\nu}<\infty,\
\mathbb{P}\text{-a.s.}$

Since $B_{1}^{H}$ and $B_{2}^{H}$ have continuous paths, it follows that

$\int_{0}^{\infty}e^{B^{H}_{1}(s)+B^{H}_{2}(s)-\nu s}ds<\infty,\
\mathbb{P}\text{-a.s.}$

If $\nu<0,$ using the fact that $B_{i}^{H}(t)=-B_{i}^{H}(t)$ in distribution,
for all $t>t_{0},$ we have

$B^{H}_{i}(t)\geq-2t^{H}\sqrt{2\log\log t},\ \mathbb{P}\text{-a.s.}.$

Taking $t_{1}>t_{0}$ such that $-\frac{\nu t}{4}>2t^{H}\sqrt{2\log\log t},$
for all $t>t_{1},$ we have

$\displaystyle\int_{t}^{\infty}e^{B^{H}_{1}(s)+B^{H}_{2}(s)-\nu s}ds$
$\displaystyle>\int_{t_{1}}^{\infty}e^{B^{H}_{1}(s)+B^{H}_{2}(s)-\nu
s}ds>\int_{t_{1}}^{\infty}e^{-2t^{H}\sqrt{2\log\log t}-\nu t}ds$
$\displaystyle>\int_{t_{1}}^{\infty}e^{-\frac{\nu s}{2}}ds=\infty,\
\mathbb{P}\text{-a.s.}$

which completes the proof. ∎

In the following theorem, we find upper bounds for the probability of non-
explosive positive solution $u=(u_{1},u_{2})^{\top}$ of the system (1.3).

######  Theorem 4.2.

Let $1/2<H<1$ and $\tau$ be the blow-up time of the system (1.3) with non-
negative initial values $f_{i},\ \rho_{i}>0$ in (3.1) for $i=1,2,$ and
$\gamma_{1}\geq\gamma_{2}>0.$ If $k_{1,2}>0$ (cf. (4.9)), then any positive
nontrivial solution of the system (1.3) is local $\mathbb{P}$-a.s., that is,
$\mathbb{P}\mathopen{}\mathclose{{}\left[\tau<\infty}\right]=1$ for all
bounded measurable functions $f_{i},\ i=1,2,$ which does not vanish. Further,

  1. 1.

If $k_{1,2}<0$ and $\beta_{1}=\beta_{2}=\beta,$ there exits $\eta>r_{0}$ such
that for all $t>5(\eta+r_{0}),$ the probability of non-explosion of the system
(1.3) is upper bounded by

$\mathbb{P}[\tau\geq
t]\leq\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{\eta}^{\frac{t}{1+2^{\alpha}}-r_{0}}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds<\frac{2^{(1+\beta)}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}}\right],$

  2. 2.

If $k_{1,2}<0$ and $\beta_{1}>\beta_{2}>0,$ there exits $\eta>r_{0}$ such that
for all $t>5(\eta+r_{0}),$ the probability of non-explosion of the system
(1.3) is upper bounded by

$\displaystyle\mathbb{P}[\tau\geq t]$ (4.21)
$\displaystyle\leq\mathbb{P}\Bigg{\\{}\int_{\eta}^{\frac{t}{1+2^{\alpha}}-r_{0}}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds$
$\displaystyle\qquad<\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1}\Bigg{\\}},$

where the functions $g_{1,2},\ g_{2,1}$ and $r_{1},\ r_{2}$ are defined in
(4.8) and (4.13) respectively.

###### Proof.

For $\beta_{1}\geq\beta_{2}>0$ and $k_{1,2}>0,$ we have

$\exp\\{sk_{1,2}\\}\leq\exp\\{sk_{2,1}\\},\ \mbox{for all}\ s\geq 0.$

We can choose $t_{1}>r_{0}$ for all $s>t_{1}$ such that

$s^{-d\beta_{1}/\alpha}\exp\mathopen{}\mathclose{{}\left\\{sk_{1,2}}\right\\}>\exp\mathopen{}\mathclose{{}\left\\{\frac{sk_{1,2}}{2}}\right\\}.$

Therefore

$\displaystyle\int_{r_{0}}^{t}2^{-d(1+\beta_{1})/\alpha}e^{sk_{1,2}+\rho_{1}B_{1}^{H}(s)+\rho_{2}B_{2}^{H}(s)}s^{-d\beta_{1}/\alpha}\wedge
2^{-d(1+\beta_{2})/\alpha}e^{sk_{2,1}+\rho_{1}B_{1}^{H}(s)+\rho_{2}B_{2}^{H}(s)}s^{-d\beta_{2}/\alpha}ds$
$\displaystyle=2^{-d(1+\beta_{1})/\alpha}\int_{r_{0}}^{t}e^{sk_{1,2}+\rho_{1}B_{1}^{H}(s)+\rho_{2}B_{2}^{H}(s)}s^{-d\beta_{1}/\alpha}ds$
$\displaystyle>2^{-d(1+\beta_{1})/\alpha}\int_{t_{1}}^{t}e^{sk_{1,2}/2+\rho_{1}B_{1}^{H}(s)+\rho_{2}B_{2}^{H}(s)}ds.$

By Lemma 4.2, we have

$\displaystyle\int_{t_{1}}^{t}e^{sk_{1,2}/2+\rho_{1}B_{1}^{H}(s)+\rho_{2}B_{2}^{H}(s)}ds\rightarrow\infty\
\ \mbox{ as }\ t\rightarrow\infty,\ \mathbb{P}\text{-a.s. }$

Hence by Theorem 4.1,
$\mathbb{P}\mathopen{}\mathclose{{}\left[\tau<\infty}\right]=1$ and therefore
for any nontrivial, positive initial conditions, the solution of system (2.1)
exhibits finite time blow-up $\mathbb{P}$-a.s.

Case 1: If $k_{1,2}<0,$ then we can choose $\eta>r_{0}$ such that for all
$s>\eta,$

$\ s^{-d\beta/\alpha}\exp\\{sk_{1,2}\\}>\exp\\{2sk_{1,2}\\}.$

Therefore, by Theorem 4.1 we have for all $t>5(\eta+r_{0}),$

$\displaystyle\mathbb{P}[\tau\geq t]$
$\displaystyle\leq\mathbb{P}\mathopen{}\mathclose{{}\left[\theta_{1}>\frac{t}{1+2^{\alpha}}-r_{0}}\right]$
$\displaystyle=\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{r_{0}}^{\frac{t}{1+2^{\alpha}}-r_{0}}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds<\frac{2^{(1+\beta)}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}}\right]$
$\displaystyle\leq\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{\eta}^{\frac{t}{1+2^{\alpha}}-r_{0}}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds<\frac{2^{(1+\beta)}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}}\right],$

where the functions $g_{1,2},\ g_{2,1}$ and $r_{1},\ r_{2}$ are defined in
(4.8) and (4.13) respectively.

Case 2: If $k_{1,2}<0,$ then we can choose $\eta>r_{0}$ such that for all
$s>\eta,$

$s^{-d\beta_{1}/\alpha}\exp\\{sk_{1,2}\\}>\exp\\{2sk_{1,2}\\}.$

Therefore by Theorem 4.1 we have for all $t>5(\eta+r_{0}),$

$\displaystyle\mathbb{P}[\tau\geq t]$
$\displaystyle\leq\mathbb{P}\mathopen{}\mathclose{{}\left[\theta_{2}>\frac{t}{1+2^{\alpha}}-r_{0}}\right]$
$\displaystyle=\mathbb{P}\Bigg{[}\int_{r_{0}}^{\frac{t}{1+2^{\alpha}}-r_{0}}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds$
$\displaystyle\qquad<\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1}\Bigg{]}$
$\displaystyle\leq\mathbb{P}\Bigg{[}\int_{\eta}^{\frac{t}{1+2^{\alpha}}-r_{0}}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds$
$\displaystyle\qquad<\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1}\Bigg{]},$

where $g_{1,2},\ g_{2,1}$ and $r_{1},\ r_{2}$ are defined in (4.8) and (4.13)
respectively. ∎

##  5 The case $H=\frac{1}{2}$

In this section, we consider the case $H=\frac{1}{2}$ and
$(B_{1}^{\frac{1}{2}}(\cdot),B_{1}^{\frac{1}{2}}(\cdot))=(W_{1}(\cdot),W_{2}(\cdot))$,
where $W_{1}(\cdot)$ and $W_{2}(\cdot)$ are two independent standard Brownian
motions. We are devoted to establish the upper bounds for the probability of
non-explosive positive solution $u=(u_{1},u_{2})^{\top}$ of the system (5.1).
By using random transformations

$v_{i}(t,x)=\exp\\{-k_{i1}W_{1}(t)-k_{i2}W_{2}(t)\\}u_{i}(t,x),\ i=1,2,$

for $t\geq 0,\ x\in\mathbb{R}^{d}$, the system $\eqref{b1}$ is transformed
into the following system of random PDEs:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
v_{i}(t,x)}{\partial
t}&=\mathopen{}\mathclose{{}\left(\Delta_{\alpha}+\gamma_{i}-\frac{k_{i1}^{2}+k_{i2}^{2}}{2}}\right)v_{i}(t,x)+e^{-k_{i1}W_{1}(t)-k_{i2}W_{2}(t)}\mathopen{}\mathclose{{}\left(e^{-k_{j1}W_{1}(t)-k_{j2}W_{2}(t)}v_{j}(t,x)}\right)^{1+\beta_{i}},\\\
v_{i}(0,x)&=f_{i}(x),\ \ x\in\mathbb{R}^{d},\end{aligned}}\right.$ (5.1)

for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
Let $\mathit{{\Gamma}}_{i}=\gamma_{i}-\frac{k_{i1}^{2}+k_{i2}^{2}}{2},$ for
$i=1,2.$ The results obtained in previous sections are valid by replacing the
constants $\gamma_{1}$ and $\gamma_{2}$ by $\mathit{\Gamma}_{1}$ and
$\mathit{\Gamma}_{2}$ respectively. For $\mathit{\Gamma}_{i}\neq 0,i=1,2,$ the
blow-up results are quite similar to the case of $H>\frac{1}{2}.$

######  Remark 5.1.

For $H=1/2$, by replacing the constants $\gamma_{1}$ and $\gamma_{2}$ by
$\mathit{\Gamma}_{1}$ and $\mathit{\Gamma}_{2}$ respectively and proceeding in
the same way as in the proof of Theorem 4.1, we obtain the upper bounds for
the blow-up of the system (5.1),

Next, we obtain upper bounds for the probability of non-explosive positive
solution $u=(u_{1},u_{2})^{\top}$ of the system (5.1).

######  Theorem 5.1.

Let $\mathopen{}\mathclose{{}\left(W_{1}(t),W_{2}(t)}\right)_{t\geq 0}$ be a
two-dimensional Brownian motion and $u=(u_{1},u_{2})^{\top}$ be a mild
solution of the system $\eqref{b1}$ with bounded measurable functions
$f_{i}\geq 0$ for $i=1,2,$ as initial data. If
$\mathit{\Gamma}_{1}\geq\mathit{\Gamma}_{2}>0$ and $k_{1,2}>0,$ then any
positive nontrivial solution of the system (1.3) is $\mathbb{P}$-a.s. local,
that is, $\mathbb{P}\mathopen{}\mathclose{{}\left[\tau<\infty}\right]=1$ for
all $f_{i}\geq 0,i=1,2,$ which does not vanish.

  * 1.

If $\beta_{1}=\beta_{2}=\beta,\ \mathit{\Gamma}_{1}\geq\mathit{\Gamma}_{2}>0$
and $k_{1,2}<0,$ there exits $\eta>r_{0}$ such that for all $t>5(\eta+r_{0}),$
the probability of the non-explosion time of the system (1.3) is upper bounded
by

$\mathbb{P}[\tau\geq
t]\leq\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{\eta}^{\frac{t}{1+2^{\alpha}}-r_{0}}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds<\frac{2^{(1+\beta)}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}}\right].$

In particular,

$\mathbb{P}[\tau=\infty]\leq\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{\eta}^{\infty}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds<\frac{2^{(1+\beta)}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}}\right].$

  * 2.

If $\beta_{1}>\beta_{2}>0,\ \mathit{\Gamma}_{1}\geq\mathit{\Gamma}_{2}>0$ and
$k_{1,2}<0,$ there exits $\eta>r_{0}$ such that for all $t>5(\eta+r_{0}),$ the
probability of non-explosion time of the system (1.3) is upper bounded by

$\displaystyle\mathbb{P}[\tau\geq t]$
$\displaystyle\leq\mathbb{P}\Bigg{\\{}\int_{\eta}^{\frac{t}{1+2^{\alpha}}-r_{0}}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds<\mathit{N}\Bigg{\\}}.$

In particular,

$\displaystyle\mathbb{P}[\tau=\infty]\leq\mathbb{P}\Bigg{\\{}\int_{\eta}^{\infty}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds<\mathit{N}\Bigg{\\}},$

where

$\displaystyle\mathit{N}=\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1},$

$r_{1},\ r_{2}$ are defined in Theorem 4.1 and

$g_{i,j}(s)=2^{\frac{-d(1+\beta_{i})}{\alpha}}e^{k_{i,j}s+\rho_{1}W_{1}(s)+\rho_{2}W_{2}(s)}s^{\frac{-d\beta_{i}}{\alpha}},$

where $\rho_{1},\rho_{2}$ are defined in (3.1) and

$\displaystyle k_{i,j}=-\mathit{\Gamma}_{i}+(1+\beta_{i})\mathit{\Gamma}_{j}\
\ \mbox{for}\ i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$
(5.2)

  * 3.

    * (a)

If $\beta_{1}=\beta_{2}=\beta$ and
$\mathit{\Gamma}_{1}=\mathit{\Gamma}_{2}=0,$ then any non trivial solution
$u=(u_{1},u_{2})^{\top}$ of (1.3) exhibits finite time blow-up in dimension
$d\leq\frac{\alpha}{\beta}.$

    * (b)

Suppose $\beta_{1}>\beta_{2}>0$ and
$\mathit{\Gamma}_{1}=\mathit{\Gamma}_{2}=0,$ then any non trivial solution
$u=(u_{1},u_{2})^{\top}$ of the system (1.3) exhibits finite time blow-up in
dimension $d\leq\frac{\alpha}{\beta_{1}}.$

###### Proof.

The proof of parts 1 and 2 in Theorem 5.1 can be established by proceeding in
the same way as in the proof of Theorem 4.2. An upper bound for
$\mathbb{P}[\tau=\infty]$ in the case of $k_{1,2}<0$ (cf. (5.2)), can be
obtained in a similar way. Also, for the case $k_{1,2}>0,$ (cf. (5.2)), the
solution $u=(u_{1},u_{2})^{\top}$ of (1.3) with $H=\frac{1}{2}$ exhibits blow-
up in finite time $\mathbb{P}$-a.s.

Now we prove part 3 of Theorem 5.1. Let us assume that $\mathit{\Gamma}_{i}=0$
for $i=1,2,$ so that the system of random PDE in (5.1) becomes:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
v_{i}(t,x)}{\partial
t}&=\Delta_{\alpha}v_{i}(t,x)+e^{-k_{i1}W_{1}(t)-k_{i2}W_{2}(t)}\mathopen{}\mathclose{{}\left(e^{-k_{j1}W_{1}(t)-k_{j2}W_{2}(t)}v_{j}(t,x)}\right)^{1+\beta_{i}},\\\
v_{i}(0,x)&=f_{i}(x),\ \ x\in\mathbb{R}^{d},\end{aligned}}\right.$ (5.3)

for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$

The following two Lemmas are useful for the proof of part 3 of Theorem 5.1.

######  Lemma 5.1 (Lemma 16, [5]).

Let $W=\mathopen{}\mathclose{{}\left(W_{1}(t),W_{2}(t)}\right)_{t\geq 0}$ be a
standard two-dimensional Brownian motion defined on a filtered probability
space
$\mathopen{}\mathclose{{}\left(\Omega,\mathscr{F},(\mathscr{F}_{t})_{t\geq
0},\mathbb{P}}\right).$ For $T>0$, the process

$\displaystyle X\equiv\mathopen{}\mathclose{{}\left\\{X(t)=X_{1}(t)+X_{2}(t),\
X_{i}(t)=\exp\\{-t/2\\}W_{i}(e^{t}-1),i=1,2,\ 0\leq t\leq T}\right\\}$ (5.4)

is a Brownian motion which is equivalent to
$W=\mathopen{}\mathclose{{}\left(W_{1}(t)+W_{2}(t)}\right)_{t\geq 0},$ that
is, there exists a probability measure $\mathbb{Q}$ on $(\Omega,\mathscr{F})$
having the same null set as $\mathbb{P}$ such that
$\mathopen{}\mathclose{{}\left(X(t),\mathcal{M}(t),\mathbb{Q}}\right)$ is a
Wiener process, where
$\mathcal{M}(t)=\sigma\mathopen{}\mathclose{{}\left\\{\sigma\mathopen{}\mathclose{{}\left\\{X(s),\
s\leq t}\right\\}\cup\mathcal{N}}\right\\},\ 0\leq t\leq T,$ with
$\mathcal{N}=\mathopen{}\mathclose{{}\left\\{B\in\mathscr{F}:\
\mathbb{P}(B)=0}\right\\}.$

###### Proof.

For each $s\geq u,$ let $h_{i}(s,u)=\exp\\{(u-s)/2\\},$ for $i=1,2.$ Then

$\displaystyle
Y\equiv\mathopen{}\mathclose{{}\left\\{Y(t)=\int_{0}^{t}2\mathopen{}\mathclose{{}\left[h_{1}(s,u)dW_{1}(u)+h_{2}(s,u)dW_{2}(u)}\right],\
0\leq t\leq T}\right\\}$ (5.5)

is a centered Gaussian process with covariance function

$\displaystyle\mathbb{E}[Y(t)Y(s)]$
$\displaystyle=\mathbb{E}\Bigg{[}\mathopen{}\mathclose{{}\left(\int_{0}^{t}2\mathopen{}\mathclose{{}\left[h_{1}(s,u)dW_{1}(u)+h_{2}(s,u)dW_{2}(u)}\right]}\right)$
$\displaystyle\quad\times\mathopen{}\mathclose{{}\left(\int_{0}^{s}2\mathopen{}\mathclose{{}\left[h_{1}(s,u)dW_{1}(u)+h_{2}(s,u)dW_{2}(u)}\right]}\right)\Bigg{]}$
$\displaystyle=e^{\frac{-t-s}{2}}\mathbb{E}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\int_{0}^{t}e^{u/2}\mathopen{}\mathclose{{}\left(dW_{1}(u)+dW_{2}(u)}\right)}\right)\mathopen{}\mathclose{{}\left(\int_{0}^{s}e^{u/2}\mathopen{}\mathclose{{}\left(dW_{1}(u)+dW_{2}(u)}\right)}\right)}\right]$
$\displaystyle=e^{\frac{-t-s}{2}}\mathbb{E}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\int_{0}^{s}e^{u/2}\mathopen{}\mathclose{{}\left(dW_{1}(u)+dW_{2}(u)}\right)}\right)^{2}}\right]$
$\displaystyle\quad+\mathbb{E}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(\int_{s}^{t}e^{u/2}\mathopen{}\mathclose{{}\left(dW_{1}(u)+dW_{2}(u)}\right)}\right)\mathopen{}\mathclose{{}\left(\int_{0}^{s}e^{u/2}\mathopen{}\mathclose{{}\left(dW_{1}(u)+dW_{2}(u)}\right)}\right)}\right]$
$\displaystyle=2e^{\frac{-t-s}{2}}\mathopen{}\mathclose{{}\left(e^{s}-1}\right)$
$\displaystyle=2\mathopen{}\mathclose{{}\left[\exp\mathopen{}\mathclose{{}\left\\{-\frac{t}{2}+\frac{s}{2}}\right\\}-\exp\mathopen{}\mathclose{{}\left\\{-\frac{t}{2}-\frac{s}{2}}\right\\}}\right],\
s\leq t,$

which is the same as the covariance function of $X$, that is, the covariance
function of $X$ is

$\displaystyle\mathbb{E}[X(t)X(s)]$
$\displaystyle=\mathbb{E}\mathopen{}\mathclose{{}\left[\mathopen{}\mathclose{{}\left(X_{1}(t)+X_{2}(t)}\right)\mathopen{}\mathclose{{}\left(X_{1}(s)+X_{2}(s)}\right)}\right]$
$\displaystyle=\mathbb{E}\mathopen{}\mathclose{{}\left[e^{-t/2}W_{1}(e^{t}-1)e^{-s/2}W_{1}(e^{s}-1)}\right]+\mathbb{E}\mathopen{}\mathclose{{}\left[e^{-t/2}W_{2}(e^{t}-1)e^{-s/2}W_{2}(e^{s}-1)}\right]$
$\displaystyle=e^{\frac{-t-s}{2}}[e^{s}-1]+e^{\frac{-t-s}{2}}[e^{s}-1]$
$\displaystyle=2\mathopen{}\mathclose{{}\left[\exp\mathopen{}\mathclose{{}\left\\{-\frac{t}{2}+\frac{s}{2}}\right\\}-\exp\mathopen{}\mathclose{{}\left\\{-\frac{t}{2}-\frac{s}{2}}\right\\}}\right],\
s\leq t.$

Therefore $X$ and $Y$ are equivalent. We have to prove that $Y$ and $W$ are
equivalent. From (5.5), we have that for $0\leq t\leq T,$

$\displaystyle
Y(t)=W_{1}(t)+W_{2}(t)-\int_{0}^{t}\mathopen{}\mathclose{{}\left(\int_{0}^{s}h_{1}(s,u)dW_{1}(u)+h_{2}(s,u)dW_{2}(u)}\right)ds.$

For $0\leq t\leq T,$ let

$\displaystyle M(t)=$
$\displaystyle\exp\Bigg{\\{}\int_{0}^{t}\int_{0}^{s}\mathopen{}\mathclose{{}\left[h_{1}(s,u)dW_{1}(u)+h_{2}(s,u)dW_{2}(u)}\right]\mathopen{}\mathclose{{}\left(dW_{1}(s)+dW_{2}(s)}\right)$
$\displaystyle\quad-\frac{1}{2}\int_{0}^{t}2\mathopen{}\mathclose{{}\left(\int_{0}^{s}h_{1}(s,u)dW_{1}(u)+h_{2}(s,u)dW_{2}(u)}\right)^{2}ds\Bigg{\\}}.$

Therefore the process $\mathopen{}\mathclose{{}\left\\{M(t),\ t\geq
0}\right\\}$ is a local martingale. Then by using the same argument as in
Lemma 16, [5], we have $\mathopen{}\mathclose{{}\left\\{M(t),\ t\geq
0}\right\\}$ is a martingale and $Y$ and $W$ are equivalent. Hence
$\mathbb{P}$ and $\mathbb{Q}$ are equivalent probability measures. ∎

######  Lemma 5.2 (Lemma 17, [5]).

Let $\rho_{i}>0$ be given in (3.1) for $i=1,2$ and $Y$ be given by

$Y(t)=\rho_{1}\exp\\{-t/2\\}W_{1}(e^{t}-1)+\rho_{2}\exp\\{-t/2\\}W_{2}(e^{t}-1),\
0\leq t\leq T.$

Then

$\displaystyle\int^{\ln(T+1)}_{\ln(r_{0}+1)}\exp\mathopen{}\mathclose{{}\left\\{Y(t)}\right\\}\mathbf{1}_{\\{Y(t)\geq
0\\}}dt\rightarrow\infty\ \mbox{ as }\ T\rightarrow\infty.$

_Proof of Theorem 5.1 (3)._ If $\beta_{1}=\beta_{2}=\beta,$ we obtain

$\displaystyle\int_{r_{0}}^{T}2^{-d(1+\beta)/\alpha}e^{\rho_{1}W_{1}(s)+\rho_{2}W_{2}(s)}s^{-d\beta/\alpha}\wedge
2^{-d(1+\beta)/\alpha}e^{\rho_{1}W_{1}(s)+\rho_{2}W_{2}(s)}s^{-d\beta/\alpha}ds$
$\displaystyle=2^{-d(1+\beta)/\alpha}\int_{r_{0}}^{T}e^{\rho_{1}W_{1}(s)+\rho_{2}W_{2}(s)}s^{-d\beta/\alpha}ds.$
(5.6)

By using change of variables $s=e^{t}-1$ in (5), we have

$\displaystyle\int_{\ln(r_{0}+1)}^{\ln(T+1)}\frac{e^{\rho_{1}e^{t/2}X_{1}(t)+\rho_{2}e^{t/2}X_{2}(t)}}{(e^{t}-1)^{d\beta/\alpha}}e^{t}dt$
$\displaystyle=\int_{\ln(r_{0}+1)}^{\ln(T+1)}\frac{e^{\rho_{1}e^{t/2}X_{1}(t)+\rho_{2}e^{t/2}X_{2}(t)+t}}{(e^{t}-1)^{d\beta/\alpha}}dt$
(5.7)
$\displaystyle\geq\int_{\ln(r_{0}+1)}^{\ln(T+1)}\frac{e^{\rho_{1}X_{1}(t)+\rho_{2}X_{2}(t)+t}}{e^{td\beta/\alpha}}\mathbf{1}_{\\{Y(t)=\rho_{1}X_{1}(t)+\rho_{2}X_{2}(t)\geq
0\\}}dt$
$\displaystyle=\int_{\ln(r_{0}+1)}^{\ln(T+1)}e^{Y(t)-t(d\beta/\alpha-1)}\mathbf{1}_{\\{Y(t)\geq
0\\}}dt$
$\displaystyle\geq\int_{\ln(r_{0}+1)}^{\ln(T+1)}e^{Y(t)}\mathbf{1}_{\\{Y(t)\geq
0\\}}dt,$

provided $d\beta/\alpha\leq 1.$ By Lemma 5.2, we have

$\displaystyle\int_{\ln(r_{0}+1)}^{\ln(T+1)}e^{Y(t)}\mathbf{1}_{\\{Y(t)\geq
0\\}}dt\rightarrow\infty\ \mbox{ as }\ T\rightarrow\infty,\
\mathbb{P}\mbox{-a.s.}$

Therefore, for each $N_{0}>0$ there exists $r_{0}>0$ such that ,

$\displaystyle\mathit{A}_{T}^{N_{0}}=\mathopen{}\mathclose{{}\left[\int_{\ln(r_{0}+1)}^{\ln(T+1)}e^{Y(t)}\mathbf{1}_{\\{Y(t)\geq
0\\}}dt>N_{0}}\right].$

for all $T>r_{0}$ and

$\displaystyle\mathbb{Q}\mathopen{}\mathclose{{}\left[\liminf_{T\rightarrow\infty}\mathopen{}\mathclose{{}\left(\int_{\ln(r_{0}+1)}^{\ln(T+1)}\frac{e^{\rho_{1}e^{t/2}X_{1}(t)+\rho_{2}e^{t/2}X_{2}(t)+t}}{(e^{t}-1)^{d\beta/\alpha}}dt>N_{0}}\right)}\right]\geq\mathbb{Q}\mathopen{}\mathclose{{}\left[\liminf_{T\rightarrow\infty}\mathopen{}\mathclose{{}\left(\mathit{A}_{T}^{N_{0}}}\right)}\right]=1.$

By Lemma 5.1, the probability measures $\mathbb{P}$ and $\mathbb{Q}$ are
equivalent. Therefore

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left[\liminf_{T\rightarrow\infty}\mathopen{}\mathclose{{}\left(\int_{r_{0}}^{T}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds>N_{0}}\right)}\right]=1,$

and we conclude that for any nontrivial positive initial values, the system of
random PDEs (5.3) exhibits a finite time blow-up for dimension
$d\leq\alpha/\beta.$

If $\beta_{1}>\beta_{2},$ then we obtain

$\displaystyle\int_{r_{0}}^{T}2^{-d(1+\beta_{1})/\alpha}e^{\rho_{1}W_{1}(s)+\rho_{2}W_{2}(s)}s^{-d\beta_{1}/\alpha}\wedge
2^{-d(1+\beta_{2})/\alpha}e^{\rho_{1}W_{1}(s)+\rho_{2}W_{2}(s)}s^{-d\beta_{2}/\alpha}ds$
$\displaystyle=2^{-d(1+\beta_{1})/\alpha}\int_{r_{0}}^{T}e^{\rho_{1}W_{1}(s)+\rho_{2}W_{2}(s)}s^{-d\beta_{1}/\alpha}ds.$
(5.8)

Following similar arguments as above, for $N_{1}>0$, we obtain

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left[\liminf_{T\rightarrow\infty}\mathopen{}\mathclose{{}\left(\int_{r_{0}}^{T}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds>N_{1}}\right)}\right]=1,$

so that for any nontrivial positive initial values, the system of random PDEs
(5.3) exhibits a finite time blow-up for dimension $d\leq\alpha/\beta_{1}.$ ∎

##  6 Existence of global solutions

In this section, we provide a sufficient condition for the existence of global
mild solution of the system (1.3) for $1/2\leq H<1$ such that (3.1) holds. For
$i=1,2,$ let

$\displaystyle\mathit{N}_{i}=\mathopen{}\mathclose{{}\left\\{\begin{array}[]{rcl}\gamma_{i},&H>1/2,\\\
\\\ \gamma_{i}-\frac{k_{i1}^{2}+k_{i2}^{2}}{2},&H=1/2.\end{array}}\right.$
(6.4)

######  Theorem 6.1.

Let $1/2\leq H<1.$ If the initial values of the form
$f_{1}=C_{1}\psi,f_{2}=C_{2}\psi,$ where $C_{1}$ and $C_{2}$ are any positive
constants with $C_{1}\leq C_{2}$ and $\psi\in
C_{c}^{\infty}({\mathbb{R}}^{d})$ and the inequalities

$\beta_{i}\int_{0}^{\infty}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|\exp\\{\mathit{N}_{i}t\\}S_{t}f_{i}}\right\|_{\infty}^{\beta_{i}}dr<1,\
i=1,2,$

hold then the mild solution $u=(u_{1},u_{2})^{\top}$ of (1.3) is global for
all $(t,x)\in[0,\infty)\times\mathbb{R}^{d}.$ Moreover, for all
$(t,x)\in[0,\infty)\times\mathbb{R}^{d}$

$\displaystyle
u_{i}(t,x)\leq\frac{\exp\\{k_{i1}B^{H}_{1}(t)+k_{i2}B^{H}_{2}(t)+\mathit{N}_{i}t\\}S_{t}f_{i}(x)}{\mathopen{}\mathclose{{}\left(1-\beta_{i}\int_{0}^{\infty}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|\exp\\{\mathit{N}_{i}r\\}S_{r}f_{i}}\right\|_{\infty}^{\beta_{i}}dr}\right)^{\frac{1}{\beta_{i}}}},\
i=1,2.$

where $\rho_{1}$ and $\rho_{2}$ are defined in (3.1) and
$\mathit{N}_{1},\mathit{N}_{2}$ are given in (6.4).

The proof of Theorem 6.1 follows from Theorem 3.1 and Theorem 3.a in [28].

######  Remark 6.1.

We have to provide the upper bounds for the probability that the solution
$u=(u_{1},u_{2})^{\top}$ does not blow-up for $1/2\leq H<1$ with the parameter
(6.4). First, we find the upper bounds by using the same argument as in
Theorem 4.1.

By using Theorem 4.2, if $\beta_{1}=\beta_{2}=\beta,$ then an upper bound of
the probability that the solution $u=(u_{1},u_{2})^{\top}$ of the system (1.3)
does not blow-up is given by

$\displaystyle\mathbb{P}[\tau=+\infty]$ $\displaystyle=\mathbb{P}[\tau\geq t\
\mbox{ for all }\ t]$
$\displaystyle\leq\mathbb{P}\mathopen{}\mathclose{{}\left[\theta_{1}>\frac{t}{1+2^{\alpha}}-r_{0}\
\ \mbox{ for all }\ t}\right]$
$\displaystyle=\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{r_{0}}^{\infty}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}s^{-d\beta/\alpha}ds<\frac{2^{(1+\beta)}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}}\right],$

where $g_{i,j}$ and $r_{i}$ are defined in (4.8) and (4.13), respectively, and
one has to replace $\gamma_{i}$ by $\mathit{N}_{i}$ in $g_{i,j}$ for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$

If $\beta_{1}>\beta_{2},$ then the probability that the solution
$u=(u_{1},u_{2})^{\top}$ of (1.3) does not blow-up is upper bounded by

$\displaystyle\mathbb{P}[\tau=+\infty]$ $\displaystyle=\mathbb{P}[\tau\geq t\
\mbox{ for all }\ t]$
$\displaystyle\leq\mathbb{P}\mathopen{}\mathclose{{}\left[\theta_{2}>\frac{t}{1+2^{\alpha}}-r_{0}\
\ \mbox{ for all }\ t}\right]$
$\displaystyle=\mathbb{P}\Bigg{[}\int_{r_{0}}^{+\infty}\min\mathopen{}\mathclose{{}\left\\{g_{1,2}(s),g_{2,1}(s)}\right\\}ds$
$\displaystyle\qquad<\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1}\Bigg{]},$

where $g_{i,j}$ and $r_{i}$ are defined in (4.8) and (4.13), respectively, and
$\gamma_{i}$ in $g_{i,j}$ has to be replaced by $\mathit{N}_{i}$ for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$

Next, we find the lower bound for the probability that the solution
$u=(u_{1},u_{2})^{\top}$ does not blow-up. By Lemma 2.2 (1), we have

$\displaystyle\mathopen{}\mathclose{{}\left\|S_{t}f_{i}}\right\|_{\infty}^{\beta_{i}}$
$\displaystyle=\mathopen{}\mathclose{{}\left(\sup_{x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}p(r,y-x)f_{i}(y)dy}\right)^{\beta_{i}}$
$\displaystyle=r^{-d\beta_{i}/\alpha}\mathopen{}\mathclose{{}\left(\sup_{x\in\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}p(1,r^{-1/\alpha}(y-x)f_{i}(y)dy}\right)^{\beta_{i}}$
$\displaystyle\leq
r^{-d\beta_{i}/\alpha}p(1,0)^{\beta_{i}}\mathopen{}\mathclose{{}\left\|f_{i}}\right\|_{1}^{\beta_{i}},$

for $i=1,2$. Therefore,

$\displaystyle\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\mathit{N}_{i}\beta_{1}r\\}\mathopen{}\mathclose{{}\left\|S_{t}f_{i}}\right\|_{\infty}^{\beta_{i}}dr$
$\displaystyle\qquad\qquad\leq
p(1,0)^{\beta_{i}}\mathopen{}\mathclose{{}\left\|f_{i}}\right\|_{1}^{\beta_{i}}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\mathit{N}_{i}\beta_{1}r\\}r^{-d\beta_{i}/\alpha}dr.$

By Theorem 3.1, we have

$\displaystyle\mathbb{P}[\tau_{\ast}=+\infty]$
$\displaystyle=\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\mathit{N}_{i}\beta_{i}r\\}dr<\min\Bigg{\\{}\frac{1}{\beta_{1}\mathopen{}\mathclose{{}\left\|S_{r}f_{1}}\right\|_{\infty}^{\beta_{1}}},\frac{1}{\beta_{2}\mathopen{}\mathclose{{}\left\|S_{r}f_{2}}\right\|_{\infty}^{\beta_{2}}}\Bigg{\\}}}\right]$
$\displaystyle\geq\mathbb{P}\Bigg{[}\int_{0}^{t}\exp\\{\rho_{1}B^{H}_{1}(r)+\rho_{2}B^{H}_{2}(r)+\mathit{N}_{i}r\\}r^{\frac{-d\beta_{i}}{\alpha}}dr$
$\displaystyle\qquad\qquad<\min\Bigg{\\{}\frac{1}{p(1,0)^{\beta_{1}}\mathopen{}\mathclose{{}\left\|f_{1}}\right\|_{1}^{\beta_{1}}\beta_{1}},\frac{1}{p(1,0)^{\beta_{2}}\mathopen{}\mathclose{{}\left\|f_{2}}\right\|_{1}^{\beta_{2}}\beta_{2}}\Bigg{\\}}\Bigg{]}.$

##  7 A More General Case

In this section, we consider the system (1.3) with parameters $1/2\leq H<1$
and (6.4). We find the lower and upper bounds to the system (1.3) without the
conditions given in (3.1). From (2.5), we have

$\displaystyle v_{i}(t,x)$ $\displaystyle=S_{t}f_{i}(x)$
$\displaystyle\quad+\int_{0}^{t}S_{t-r}\mathopen{}\mathclose{{}\left[\mathit{N}_{i}v_{i}(s,\cdot)+e^{-k_{i1}B^{H}_{1}(r)-k_{i2}B^{H}_{2}(r)}\mathopen{}\mathclose{{}\left(e^{k_{j1}B^{H}_{1}(r)+k_{j2}B^{H}_{2}(r)}v_{j}(r,\cdot)}\right)^{1+\beta_{i}}}\right](x)dr.$

for $i=1,2,\
\mathopen{}\mathclose{{}\left\\{j}\right\\}=\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\\{i\\}$
and $\mathit{N}_{i}$ are defined in (6.4).

Next result gives the lower bounds for finite-time blow-up of the solution
$u=(u_{1},u_{2})^{\top}$ of the system (1.3) when
$\gamma_{1}=\gamma_{2}=\lambda$. Hence $T_{1}(t)=T_{2}(t)=T(t)\ (say)$ which
are defined in (2.4).

######  Theorem 7.1.

If $\beta_{1},\beta_{2}$ are positive constants and the initial values are of
the form $f_{1}=C_{1}\psi,f_{2}=C_{2}\psi$ where $C_{1}$ and $C_{2}$ are any
positive constants with $C_{1}\leq C_{2}$ and $\psi\in
C_{c}^{\infty}({\mathbb{R}}^{d}).$ Then $\tau_{\ast}\leq\tau,$ where
$\tau_{\ast}$ is given by

$\displaystyle\tau_{\ast}=\inf\Bigg{\\{}t\geq 0:$
$\displaystyle\int_{0}^{t}e^{(1+\beta_{1})k_{21}-k_{11}B^{H}_{1}(r)+(1+\beta_{1})k_{22}-k_{12}B^{H}_{2}(r)+\lambda\beta_{1}r}r^{-d\beta_{1}/\alpha}dr\geq\frac{1}{\beta_{1}p(1,0)^{\beta_{1}}C_{1}^{\beta_{1}}\mathopen{}\mathclose{{}\left\|\psi}\right\|_{\infty}^{\beta_{1}}},$
or
$\displaystyle\int_{0}^{t}e^{(1+\beta_{2})k_{11}-k_{21}B^{H}_{1}(r)+(1+\beta_{2})k_{12}-k_{22}B^{H}_{2}(r)+\lambda\beta_{2}r}r^{-d\beta_{2}/\alpha}dr\geq\frac{1}{\beta_{1}p(1,0)^{\beta_{1}}C_{1}^{\beta_{1}}\mathopen{}\mathclose{{}\left\|\psi}\right\|_{\infty}^{\beta_{1}}}\Bigg{\\}}.$
(7.1)

###### Proof.

For all $x\in\mathbb{R}^{d},\ t\geq 0,$ let us define the operators
$\mathcal{J}_{1},\ \mathcal{J}_{2}$ as

$\displaystyle\mathcal{J}_{1}v(t,x)$ $\displaystyle=T(t)f_{1}(x)$
$\displaystyle\quad+\int_{0}^{t}\exp\\{(1+\beta_{1})k_{21}-k_{11}B^{H}_{1}(r)+(1+\beta_{1})k_{22}-k_{12}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left(T(t-r)v}\right)^{1+\beta_{1}}dr,$
$\displaystyle\mathcal{J}_{2}v(t,x)$ $\displaystyle=T(t)f_{2}(x)$
$\displaystyle\quad+\int_{0}^{t}\exp\\{(1+\beta_{2})k_{11}-k_{21}B^{H}_{1}(r)+(1+\beta_{2})k_{12}-k_{22}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left(T(t-r)v}\right)^{1+\beta_{2}}dr,$

where $v$ is any non-negative, bounded and measurable function. Moreover, on
the set $t\leq\tau_{*},$ we set

$\displaystyle\mathscr{G}_{1}(t)$
$\displaystyle=\mathopen{}\mathclose{{}\left[1-\beta_{1}\int_{0}^{t}\exp\\{(1+\beta_{1})k_{21}-k_{11}B^{H}_{1}(r)+(1+\beta_{1})k_{22}-k_{12}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(t)f_{1}}\right\|_{\infty}^{\beta_{1}}dr}\right]^{\frac{-1}{\beta_{1}}},$
$\displaystyle\mathscr{G}_{2}(t)$
$\displaystyle=\mathopen{}\mathclose{{}\left[1-\beta_{2}\int_{0}^{t}\exp\\{(1+\beta_{2})k_{11}-k_{21}B^{H}_{1}(r)+(1+\beta_{2})k_{12}-k_{22}B^{H}_{2}(r)\\}\mathopen{}\mathclose{{}\left\|T(t)f_{2}}\right\|_{\infty}^{\beta_{2}}dr}\right]^{\frac{-1}{\beta_{2}}}.$

By using same way as in the proof of Theorem 3.1, we get

$\displaystyle v_{1}(t,x)=\mathcal{J}_{1}v_{2}(t,x),\
v_{2}(t,x)=\mathcal{J}_{2}v_{1}(t,x),\ \ x\in\mathbb{R}^{d},\ \ 0\leq
t<\tau_{*}.$

Moreover, we have

$\displaystyle\mathcal{J}_{1}v(t,x)\leq T(t)f_{1}(x)\mathscr{G}_{1}(t)\ \text{
and }\ \mathcal{J}_{2}u(t,x)\leq T(t)f_{2}(x)\mathscr{G}_{2}(t),$

so that

$\displaystyle v_{1}(t,x)$
$\displaystyle\leq\frac{T(t)f_{1}(x)}{\mathopen{}\mathclose{{}\left[1-\beta_{1}p(1,0)^{\beta_{1}}\mathopen{}\mathclose{{}\left\|f_{1}}\right\|_{\infty}^{\beta_{1}}\int_{0}^{t}e^{(1+\beta_{1})k_{21}-k_{11}B^{H}_{1}(r)+(1+\beta_{1})k_{22}-k_{12}B^{H}_{2}(r)+\lambda\beta_{1}r}r^{-d\beta_{1}/\alpha}dr}\right]^{\frac{1}{\beta_{1}}}},$
$\displaystyle v_{2}(t,x)$
$\displaystyle\leq\frac{T(t)f_{2}(x)}{\mathopen{}\mathclose{{}\left[1-\beta_{2}p(1,0)^{\beta_{2}}\mathopen{}\mathclose{{}\left\|f_{1}}\right\|_{\infty}^{\beta_{2}}\int_{0}^{t}e^{(1+\beta_{2})k_{11}-k_{21}B^{H}_{1}(r)+(1+\beta_{2})k_{12}-k_{22}B^{H}_{2}(r)+\lambda\beta_{2}r}r^{-d\beta_{2}/\alpha}dr}\right]^{\frac{1}{\beta_{2}}}},$

and one can complete the proof of (7.1). ∎

The following theorem gives an upper bounds for the finite-time blow-up of
solution $u=(u_{1},u_{2})^{\top}$ of the system (1.3) for $1/2\leq H<1$ with
$\beta_{1}\geq\beta_{2}>0.$

######  Theorem 7.2.

Let $\frac{1}{2}\leq H<1$ and let $u=(u_{1},u_{2})^{\top}$ be a weak solution
of (1.3) with nonnegative initial data $f_{1}$ and $f_{2}$ which are bounded
measurable functions.

If $\beta_{1}=\beta_{2}=\beta>0.$ Then $u=(u_{1},u_{2})^{\top}$ blow-up in
finite time which is given by

$\displaystyle\theta_{1}=\inf\Bigg{\\{}$ $\displaystyle
t>r_{0}:\int_{r_{0}}^{t}e^{k_{1,2}s+((1+\beta)k_{21}-k_{11})B^{H}_{1}(s)+((1+\beta)k_{22}-k_{12})B^{H}_{2}(s)}s^{\frac{-d\beta}{\alpha}}$
$\displaystyle\wedge
e^{k_{2,1}s+((1+\beta)k_{21}-k_{11})B^{H}_{1}(s)+((1+\beta)k_{22}-k_{12})B^{H}_{2}(s)}s^{\frac{-d\beta}{\alpha}}ds\geq\frac{2^{(1+\beta)}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}$
(or)
$\displaystyle\int_{r_{0}}^{t}e^{k_{1,2}s+((1+\beta)k_{11}-k_{21})B^{H}_{1}(s)+((1+\beta)k_{12}-k_{22})B^{H}_{2}(s)}s^{\frac{-d\beta}{\alpha}}$
$\displaystyle\wedge
e^{k_{2,1}s+((1+\beta)k_{11}-k_{21})B^{H}_{1}(s)+((1+\beta)k_{12}-k_{22})B^{H}_{2}(s)}s^{\frac{-d\beta}{\alpha}}ds\geq\frac{2^{(1+\beta)}}{\beta\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta}}\Bigg{\\}}.$

Moreover, if $\tau$ is the blow-up time of (1.3), then

$\displaystyle\tau$
$\displaystyle\leq\min\mathopen{}\mathclose{{}\left\\{(r_{0}+\tau_{m_{1}})(1+2^{\alpha}),(r_{0}+\tau_{m_{2}})(1+2^{\alpha})}\right\\}\leq(r_{0}+\theta_{1})(1+2^{\alpha}),$

where
$r_{i}=p(1,0)2^{-2d}\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right],$
and $k_{i,j}=-\mathit{N}_{i}+(1+\beta_{i})\mathit{N}_{j}$ for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$

If $\beta_{1}>\beta_{2}>0$ and let
$D_{1}=\mathopen{}\mathclose{{}\left(\frac{\beta_{1}-\beta_{2}}{1+\beta_{1}}}\right)\mathopen{}\mathclose{{}\left(\frac{1+\beta_{1}}{1+\beta_{2}}}\right)^{\frac{1+\beta_{2}}{\beta_{1}-\beta_{2}}},$

$\displaystyle\epsilon_{0}$
$\displaystyle=\min\Bigg{\\{}1,\mathopen{}\mathclose{{}\left(h_{1}(r_{0})/D_{1}^{1/1+\beta_{2}}}\right)^{\beta_{1}-\beta_{2}},\mathopen{}\mathclose{{}\left(2^{-(1+\beta_{2})}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}/D_{1}}\right)^{\frac{\beta_{1}-\beta_{2}}{1+\beta_{1}}}\Bigg{\\}}.$

Assume that

$\displaystyle
2^{-(1+\beta_{2})}\epsilon_{0}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}\geq\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}.$

Then $u=(u_{1},u_{2})^{\top}$ blow-up in finite time which is given by

$\displaystyle\theta_{2}=\inf\Bigg{\\{}$ $\displaystyle
t>r_{0}:\int_{r_{0}}^{t}2^{-d(1+\beta_{1})/\alpha}e^{k_{1,2}s+((1+\beta_{1})k_{21}-k_{11})B^{H}_{1}(s)+((1+\beta_{2})k_{22}-k_{12})B^{H}_{2}(s)}s^{\frac{-d\beta_{1}}{\alpha}}$
$\displaystyle\wedge
2^{-d(1+\beta_{2})/\alpha}e^{k_{2,1}s+((1+\beta_{1})k_{21}-k_{11})B^{H}_{1}(s)+((1+\beta_{2})k_{22}-k_{12})B^{H}_{2}(s)}s^{\frac{-d\beta_{2}}{\alpha}}ds$
$\displaystyle\qquad\geq\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1}$
or
$\displaystyle\quad\int_{r_{0}}^{t}2^{-d(1+\beta_{1})/\alpha}e^{k_{1,2}s+((1+\beta_{2})k_{11}-k_{21})B^{H}_{1}(s)+((1+\beta_{2})k_{12}-k_{22})B^{H}_{2}(s)}s^{\frac{-d\beta_{1}}{\alpha}}$
$\displaystyle\wedge
2^{-d(1+\beta_{2})/\alpha}e^{k_{2,1}s+((1+\beta_{2})k_{11}-k_{21})B^{H}_{1}(s)+((1+\beta_{2})k_{12}-k_{22})B^{H}_{2}(s)}s^{\frac{-d\beta_{2}}{\alpha}}ds$
$\displaystyle\qquad\geq\mathopen{}\mathclose{{}\left[\beta_{2}\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{\beta_{2}}\mathopen{}\mathclose{{}\left(\frac{\epsilon_{0}}{2^{1+\beta_{2}}}-\frac{\epsilon_{0}^{\frac{1+\beta_{1}}{\beta_{1}-\beta_{2}}}D_{1}}{\mathopen{}\mathclose{{}\left(r_{1}e^{-\gamma_{1}r_{0}}+r_{2}e^{-\gamma_{2}r_{0}}}\right)^{1+\beta_{2}}}}\right)}\right]^{-1}\Bigg{\\}}.$

Moreover, if $\tau$ is the blow-up time of (1.3), then

$\displaystyle\tau$
$\displaystyle\leq\min\mathopen{}\mathclose{{}\left\\{(r_{0}+\tau_{m_{1}})(1+2^{\alpha}),(r_{0}+\tau_{m_{2}})(1+2^{\alpha})}\right\\}\leq(r_{0}+\theta_{2})(1+2^{\alpha}),$

where
$r_{i}=p(1,0)2^{-2d}\exp\\{\gamma_{i}r_{0}\\}\mathbb{E}\mathopen{}\mathclose{{}\left[f_{i}(X_{2^{-\alpha}r_{0}})}\right],$
and $k_{i,j}=-\mathit{N}_{i}+(1+\beta_{i})\mathit{N}_{j}$ for $i=1,2,\
j\in\mathopen{}\mathclose{{}\left\\{1,2}\right\\}/\mathopen{}\mathclose{{}\left\\{i}\right\\}.$

Proof of Theorem 7.2 can be obtained by proceeding in the same way as in the
proof of Theorem 4.1.

Acknowledgements: The first author is supported by the University Research
Fellowship of Periyar University, India. M. T. Mohan would like to thank the
Department of Science and Technology (DST), India for Innovation in Science
Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110). The third
author is supported by the Fund for Improvement of Science and Technology
Infrastructure (FIST) of DST (SR/FST/MSI-115/2016).

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