# Global existence and non-existence of weak solutions for non-local
stochastic semilinear reaction-diffusion equations driven by a 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><EMAIL_ADDRESS>
###### Abstract

In the present paper, we study the existence and blow-up behavior to the
following stochastic non-local reaction-diffusion equation:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
du(t,x)&=\mathopen{}\mathclose{{}\left[(\Delta+\gamma)u(t,x)+\int_{D}u^{q}(t,y)dy-
ku^{p}(t,x)+\delta u^{m}(t,x)\int_{D}u^{n}(t,y)dy}\right]dt\\\ &\quad+\eta
u(t,x)dB^{H}(t),\\\ u(t,x)&=0,\ \ t>0,\ \ x\in\partial D,\\\ u(0,x)&=f(x)\geq
0,\ \ x\in D,\\\ \end{aligned}}\right.$

where $D\subset\mathbb{R}^{d}\ (d\geq 1)$ is a bounded domain with smooth
boundary $\partial D$. Here, $k>0,\gamma,\delta,\eta\geq 0$ and $p,q,n>1,\
m\geq 0$ with $m+n\geq q\geq p$. The initial data $f$ is a non-negative
bounded measurable function in class $C^{2}$ which is not identically zero.
Here, $\mathopen{}\mathclose{{}\left\\{B^{H}(t)}\right\\}_{t\geq 0}$ is a one-
dimensional fractional Brownian motion with Hurst parameter $\frac{1}{2}\leq
H<1$ defined on a filtered probability space
$\mathopen{}\mathclose{{}\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P}}\right)$. First, we estimate a lower bound for the finite-time
blow-up and by choosing a suitable initial data, we obtain the upper bound for
the finite-time blow-up of the above equation. Next, we provide a sufficient
condition for the global existence of a weak solution of the above equation.
Further, we obtain the bounds for the probability of blow-up solution.

Keywords: Semilinear SPDEs; non-local reaction-diffusion equations; blow-up
times; stopping times; lower and upper bounds.

MSC: 35R60; 60H15; 74H35; 35B50

##  1 Introduction

A number of physical phenomena, including chemical reactions, electrical
heating, and fluid flow, give rise to nonlinear diffusion problems that are
modeled as non-local reaction-diffusion equations [37]. In many applications
in engineering and biology, blow-up behaviour is associated with some
destructive behavior of the mathematical models. Thus from the mathematical
point of view, it is very interesting to investigate the conditions under
which such a finite-time blow-up occurs. In the past few decades, many authors
have discussed the global existence and blow-up of solutions for local and
non-local reaction-diffusion equations, and there have been many monographs as
well as review papers [4, 24, 39] etc. Kaplan [25] and Fujita [21, 22] and
followed by them, many researchers have developed the concept of blow-up to a
more general class of non-linear partial differential equations (PDEs). Over
the past few years, a considerable effort has been devoted to study the blow-
up properties of solutions to parabolic equations with local boundary
conditions, say Dirichlet, Neumann or Robin boundary conditions, which can be
used to describe heat propagation on the boundary, see the papers (cf.[12, 28]
and references therein).

Non-local mathematical processes appear in different physical forms (see [2,
10, 11]). Bebernes et al. in [3] obtained the blow-up results for a more
general class of non-local reaction diffusion equations of the form in a
bounded domain $D\subset\mathbb{R}^{d}\ (d\geq 1)$ with suffciently smooth
boundary $\partial D$

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} u_{t}-\Delta u&=f(u)+g(t),\ \
t>0,\ x\in D,\\\ u(x,t)&=0,\ \ t>0,\ x\in\partial{D},\\\ u(x,0)&=u_{0}(x),\ \
x\in D,\end{aligned}}\right.$ (1.1)

when $g^{\prime}(t)\geq 0$ or
$g(t)=\frac{k}{\mathopen{}\mathclose{{}\left|D}\right|}\int_{\Omega}u_{t}(x,t)dx$,
here $|D|$ is the volume of $D$ with $0<k<1$. Chadam et al. [5] proved that
under suitable conditions, the solution of (1.1) with $f(u)=0$ and
$g=\int_{D}F(u)dx$ blows-up in a finite-time. M. Wang and Y. Wang [43]
investigated the existence and non-existence of global positive solutions for
a non-local reaction-diffusion equation of the type

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} u_{t}-d\Delta
u&=\int_{D}u^{q}(t,x)dx-ku^{p}(t,x),\ \ x\in D,\ t>0,\\\ u(x,0)&=u_{0}(x),\ \
x\in D,\\\ \end{aligned}}\right.$ (1.2)

subject to the boundary (Dirichlet or Neumann type) conditions

$\displaystyle u(t,x)=0,\ \mbox{ or }\ \frac{\partial u}{\partial\upnu}=0,\
x\in\partial D,\ t>0,$ (1.3)

where $D\subset\mathbb{R}^{d}\ (d\geq 1)$ with suffciently smooth boundary
$\partial D$. Here $\upnu$ is the unit outside normal to the boundary
$\partial D$, the parameters $k>0,p,q\geq 1$ and $d>0$. Song in [41] obtained
the lower bounds for the finite-time blow-up of a non-local reaction-diffusion
equation for (1.2)-(1.3) with $d=1.$ Li et al. [31] proved that the solution
exists globally or blow-up in finite-time to the reaction-diffusion system of
non-local sources with appropriate hypotheses.

Since the above models are deterministic in nature, the dynamics is entirely
determined by the initial data. This is obviously not the case with natural
phenomena, where random environmental influences often play an important role.
Many researchers have studied the blow-up results for non-linear stochastic
partial differential equations (SPDEs) with the Dirichlet boundary conditions
(cf. [6, 13, 15, 16]) etc. Chow [7, 8] proved that the solution of the
stochastic reaction-diffusion equation is positive and explodes in finite-time
in the mean $L^{p}$-norm. The stochastic counterpart of non-local problems
have received less attention. Recently, Liang and Zhao [29] addressed the
existence and blow-up of positive solutions for a stochastic non-local
reaction-diffusion equation of the form:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} u_{t}-d_{1}\Delta
u&=\int_{D}u^{q}(t,x)dx-ku^{p}(t,x)+\epsilon u(t,x)dB_{t},\ \ x\in D,\ t>0,\\\
u(t,x)&=0,\ \ t>0,\ x\in\partial D,\\\ u(0,x)&=u_{0}(x),\ \ x\in D,\\\
\end{aligned}}\right.$

where $d_{1},k,\epsilon>0$ and $p,q\geq 1$ are constants.

Inspired by the above facts, in the present paper, our aim is to prove the
global existence of a weak solution for certain parameters and obtain lower
and upper bounds for the blow-up time for a more general class of stochastic
non-local reaction-diffusion equations driven by fractional Brownian motion
(fBm):

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
du(t,x)&=\mathopen{}\mathclose{{}\left[(\Delta+\gamma)u(t,x)+\int_{D}u^{q}(t,y)dy-
ku^{p}(t,x)+\delta u^{m}(t,x)\int_{D}u^{n}(t,y)dy}\right]dt\\\ &\quad+\eta
u(t,x)dB^{H}(t),\\\ u(t,x)&=0,\ \ t>0,\ \ x\in\partial D,\\\ u(0,x)&=f(x)\geq
0,\ \ x\in D,\\\ \end{aligned}}\right.$ (1.4)

where $D$ is a bounded domain in $\mathbb{R}^{d}\ (d\geq 1)$ with smooth
boundary $\partial D$. Here $k>0,\gamma\geq 0,\delta\geq 0,\eta\geq 0$ and
$p,q,n>1,\ m\geq 0$ with $m+n\geq q\geq p$. The initial data $f$ is non-
negative and bounded measurable in the class $C^{2}$ which is not identically
zero. Further, $\mathopen{}\mathclose{{}\left\\{B^{H}(t)}\right\\}_{t\geq 0}$
is a standard one-dimensional fBm defined on a filtered probability space
$\mathopen{}\mathclose{{}\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P}}\right)$ with Hurst parameter $\frac{1}{2}\leq H<1$. An fBm of
Hurst parameter $H\in(0,1)$ is a centered Gaussian process
$\mathopen{}\mathclose{{}\left\\{B^{H}(t)}\right\\}_{t\geq 0}$, with
covariance function

$\displaystyle
R_{H}(t,s):=E[B^{H}(t)B^{H}(s)]=\frac{1}{2}\mathopen{}\mathclose{{}\left(s^{2H}+t^{2H}-|t-s|^{2H}}\right).$

It is known that $B^{H}(\cdot)$ admits the so called _Volterra representation_
(Nualart [35]),

$\displaystyle B^{H}(t):=\int_{0}^{t}K_{H}(t,s)dW(s),$

where $\\{W(t)\\}_{t\geq 0}$ is a standard Brownian motion and the Volterra
kernel $K_{H}(t,s)$ is defined by

$\displaystyle
K_{H}(t,s)=C_{H}\mathopen{}\mathclose{{}\left[\frac{t^{H-\frac{1}{2}}}{s^{H-\frac{1}{2}}}(t-s)^{H-\frac{1}{2}}-\mathopen{}\mathclose{{}\left(H-\frac{1}{2}}\right)\int_{s}^{t}\frac{u^{H-\frac{3}{2}}}{s^{H-\frac{1}{2}}}(u-s)^{H-\frac{1}{2}}du}\right],\
\ s\leq t,$

where $C_{H}$ is a constant depending only on $H.$ When $H=\frac{1}{2}$, it is
the standard one-dimensional Brownian motion $W(\cdot)$. We closely follow the
paper [29] to obtain our results.

The main goal of this article is to obtain the existence and non-existence of
a solution $u(\cdot,\cdot)$ of the equation (1.4). First, we estimate a lower
bound for the finite-time blow-up of the solution of (1.4) and by choosing a
suitable initial data we obtain the upper bound for the finite-time blow-up of
the equation (1.4). Further, we provide a sufficient condition for a weak
solution $u$ of (1.4) to exist globally and obtain the bounds for the
probability of blow-up solution $u(\cdot,\cdot)$ of the equation (1.4).

The rest of the article is organized as follows: Section 2 covers the
preliminaries required for this work. Section 3 is devoted to obtain an
associated random PDE, by applying a random transformation (cf. (3.1)) to the
equation (1.4). The result established in Theorem 3.1 will be useful to
discuss our further sections. The existence of random PDE is based on a
transformation which converts a stochastic partial differential equation
(SPDE) into an equivalent random PDE. Such type of transformation is available
in the literature when the noise is either additive or linear multiplicative.
The results obtained in Sections 3, 4 and 5 are outlined in the following
table:

Parameters | Theorem | Result
---|---|---
$|D|\leq k$ & | Theorem 3.2 | The existence of a global solution $v(\cdot,\cdot)$ of the
$p=q=n,m=0$ |  | equation (3.2).
| Theorem 4.1 | Lower bounds for the finite-time blow-up $\tau$ of
|  | (1.4) with any non-negative initial value $f$.
$|D|>k$ & | Theorem 4.2 | The existence of a global solution $u(\cdot,\cdot)$ of the
$m+n\geq q\geq p$ |  | equation(1.4) under the condition (4.5).
| Theorem 5.1 | Upper bounds for $\tau$ with the initial data
|  |  $f(x)\geq b\varphi(x),$ $x\in D$, for any $b>1$ such that
|  | (5.2) holds.

Further, we estimate the blow-up probability to the solution of (1.4) for the
case $m+n=q>p$, the initial value $f(x)\geq b\varphi(x),\ b>1$ and
$\frac{1}{2}<H<1$ by using the Malliavin calculus and the method adopted in
[17]. We consider the case $H=\frac{1}{2}$ in the last section and estimate
the upper and lower bounds for the blow-up time of the solution of the
equation (1.4). Further, we establish the lower and upper bounds for the
probability of the blow-up solution $u(\cdot,\cdot)$ of the equation (1.4).

##  2 Preliminaries

In this section, we first recall some known results and we begin with some
necessary function spaces to be used in the subsequent sections. Let $D$ be a
bounded domain in $\mathbb{R}^{d},d\geq 1$ with smooth boundary $\partial D,$
$\overline{D}$ be the closure of $D$ and $|D|$ be the volume of $D$. We denote
$L^{2}(D)=L^{2}(D;\mathbb{R}),$ the usual $L^{2}$-real Hilbert space equipped
with the norm
$\|u\|=\mathopen{}\mathclose{{}\left(\int_{D}|u(x)|^{2}dx}\right)^{1/2}$ and
inner product $(u,v)=\int_{D}u(x)v(x)dx$ for all $u,v\in L^{2}(D)$.

Let the semigroup $\mathopen{}\mathclose{{}\left\\{T_{t}}\right\\}_{t\geq 0}$
of bounded linear operators be defined by

$\displaystyle T_{t}f(x)=\mathbb{E}\mathopen{}\mathclose{{}\left[f(X_{t}),\
t<\tau_{D}|X_{0}=x}\right],\quad x\in D,$

for all bounded and measurable functions $f:D\rightarrow\mathbb{R},$ where
$\mathopen{}\mathclose{{}\left\\{X_{t}}\right\\}_{t\geq 0}$ is the
$d$-dimensional Brownian motion with variance parameter $2$, killed at the
time $\tau_{D}$ at which it strike the boundary $\partial D.$ Moreover,
$\lambda_{1}>0$ is the first eigenvalue of the Dirichlet Laplacian operator
$-\Delta$ on $D$ which satisfies

$\displaystyle-\Delta\varphi(x)=\lambda_{1}\varphi(x),\ \ x\in D,$ (2.1)

where $\varphi$ is the corresponding eigenfunction which is strictly positive
on $D$ (see Corollary 3.3.7 in [9]) and $\varphi|_{\partial D}=0.$ Remember
that

$T_{t}\varphi(x)=\exp\\{{-\lambda_{1}t}\\}\varphi(x),\ \ t\geq 0,\ x\in D.$

We assume that $\varphi$ is normalized so that
$\displaystyle\int_{D}\varphi(x)dx=1$. Let
$\mathopen{}\mathclose{{}\left\\{p_{t}(x,y)}\right\\}_{t>0}$ be the transition
kernel of $\mathopen{}\mathclose{{}\left\\{T_{t}}\right\\}_{t\geq 0}$ and one
can refer to [36] for its properties. For any $\varphi\in C^{\infty}_{0}(D)$,
we denote

$u(t,\varphi):=\int_{D}u(t,x)\varphi(x)dx,\ \ t\geq 0,\ x\in D.$

Let us recall the notion of weak solution of (1.4). Let $\tau<+\infty$ be a
stopping time. A continuous $\\{\mathcal{F}_{t}\\}_{t\geq 0}$-adapted random
field $u=\mathopen{}\mathclose{{}\left\\{u(t,x),\ t\geq 0,\ x\in D}\right\\}$
is a _weak solution_ of (1.4) on the interval $(0,\tau)$ provided

$\displaystyle u\mathopen{}\mathclose{{}\left(t,\varphi}\right)$
$\displaystyle=u\mathopen{}\mathclose{{}\left(0,\varphi}\right)+\int_{0}^{t}u\mathopen{}\mathclose{{}\left(s,(\Delta+\gamma)\varphi}\right)ds+\int_{0}^{t}\int_{D}u^{q}(s,y)dy\int_{D}\varphi(z)dzds-k\int_{0}^{t}u^{p}(s,\varphi)ds$
$\displaystyle\quad+\delta\int_{0}^{t}u^{m}\mathopen{}\mathclose{{}\left(s,x}\right)\int_{D}u^{n}(s,y)dy\int_{D}\varphi(z)dzds+\eta\int_{0}^{t}u(s,\varphi)dB^{H}(s),\
\mathbb{P}\text{-a.s.},$ (2.2)

holds for every $\varphi\in C^{\infty}_{0}(D).$ For any bounded measurable
initial data $f\geq 0,$ we say that
$u=\mathopen{}\mathclose{{}\left\\{u(t,x),\ t\geq 0,\ x\in D}\right\\}$ is a
local mild solution of the equation (1.4), if there exists a number
$0<\tau=\tau(\omega)\leq\infty$ such that $u$ satisfies the integral equation
([38, Chapter 4]),

$\displaystyle u(t,x)$
$\displaystyle=\exp\mathopen{}\mathclose{{}\left\\{\gamma
t}\right\\}T_{t}f(x)+\int_{0}^{t}e^{\gamma(t-r)}T_{t-r}\Bigg{[}\int_{D}u^{q}(r,y)dy-
ku^{p}(r,x)$ $\displaystyle\qquad+\delta
u^{m}(r,x)\int_{D}u^{n}(r,y)dy\Bigg{]}dr+\eta\int_{0}^{t}e^{\gamma(t-r)}T_{t-r}u(r,x)dB^{H}(r),\
\mathbb{P}\text{-a.s.},$

for each $0\leq t<\tau$ and $x\in D$. The equivalence between weak and mild
solutions of (1.4) can be established in a similar way as in [23, Proposition
3.7].

The following result will be used in the sequel.

######  Lemma 2.1.

([16, Lemma 1]) Let $0<H<1$ and let
$\mathopen{}\mathclose{{}\left\\{B^{H}(t)}\right\\}_{t\geq 0}$ be a one-
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

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

If $\nu<0,$ then

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

##  3 Random PDE

In this section, we obtain a random PDE by making use of the random
transformation

$\displaystyle v(t,x)=\exp\\{-\eta B^{H}(t)\\}u(t,x),$ (3.1)

for $\frac{1}{2}<H<1$ and $t\geq 0,\ x\in D$. The equation $\eqref{b1}$ is
transformed into the following random PDE:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
v(t,x)}{\partial
t}&=(\Delta+\gamma)v(t,x)+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\int_{D}v^{q}(t,y)dy-
ke^{\mathopen{}\mathclose{{}\left(p-1}\right)\eta B^{H}(t)}v^{p}(t,x)\\\
&\qquad+\delta e^{(m+n-1)\eta B^{H}(t)}v^{m}(t,x)\int_{D}v^{n}(t,y)dy,\\\
v(t,x)&=0,\ \ x\in\partial D,\ t>0,\\\ v(0,x)&=f(x),\ \ x\in
D.\end{aligned}}\right.$ (3.2)

The following result provides an equivalence between the weak solutions of the
stochastic PDE $\eqref{b1}$ and the random PDE (3.2).

######  Theorem 3.1.

If $u(\cdot,\cdot)$ is a weak solution of $\eqref{b1}$. Then the function
$v(\cdot,\cdot)$ defined by

$\displaystyle v(t,x)=\exp\\{-\eta B^{H}(t)\\}u(t,x),\ t\geq 0,\ x\in D,$

is a weak solution of the random PDE $\eqref{s1}$ and viceversa.

###### Proof.

By using Ito’s formula ([34, Lemma 2.7.1]), we have

$\displaystyle e^{-\eta B^{H}(t)}=1-\eta\int_{0}^{t}e^{-\eta
B^{H}(s)}dB^{H}(s).$

For any $\varphi\in C^{\infty}_{0}(D),$ the weak solution of $\eqref{b1}$ is
given by

$\displaystyle u\mathopen{}\mathclose{{}\left(t,\varphi}\right)$
$\displaystyle=u\mathopen{}\mathclose{{}\left(0,\varphi}\right)+\int_{0}^{t}u\mathopen{}\mathclose{{}\left(s,(\Delta+\gamma)\varphi}\right)ds+\int_{0}^{t}\int_{D}u^{q}(s,y)dy\int_{D}\varphi(z)dzds-k\int_{0}^{t}u^{p}(s,\varphi)ds$
$\displaystyle\quad+\delta\int_{0}^{t}u^{m}\mathopen{}\mathclose{{}\left(s,x}\right)\int_{D}u^{n}(s,y)dy\int_{D}\varphi(z)dzds+\eta\int_{0}^{t}u(s,\varphi)dB^{H}(s).$
(3.3)

By applying the integration by parts formula ([27, Chapter 8]), we obtain

$\displaystyle v(t,\varphi):$ $\displaystyle=\int_{D}v(t,x)\varphi(x)dx$
$\displaystyle=v(0,\varphi)+\int_{0}^{t}e^{-\eta
B^{H}(s)}du(s,\varphi)+\int_{0}^{t}u(s,\varphi)\mathopen{}\mathclose{{}\left(-\eta
e^{-\eta B^{H}(s)}dB^{H}(s)}\right).$

Therefore,

$\displaystyle v(t,\varphi)$
$\displaystyle=v(0,\varphi)+\int_{0}^{t}v\mathopen{}\mathclose{{}\left(s,(\Delta+\gamma)\varphi}\right)ds+\int_{0}^{t}e^{(q-1)\eta
B^{H}(s)}\int_{D}v^{q}(s,y)dy\int_{D}\varphi(z)dzds$
$\displaystyle-k\int_{0}^{t}e^{(p-1)\eta
B^{H}(s)}v^{p}(s,\varphi)ds+\delta\int_{0}^{t}e^{(m+n-1)\eta
B^{H}(s)}v^{m}(s,x)\int_{D}v^{n}(s,y)dy\int_{D}\varphi(z)dzds.$ (3.4)

The preceding equality means that $v(\cdot,\cdot)$ is a weak solution of
$\eqref{s1}.$ The converse part follows from the fact that the change of
variable is given by a homeomorphism which transforms one random dynamical
system into an another equivalent one. ∎

This equation (1.4) is understood in the pathwise (or trajectory wise) sense
and the existence and uniqueness of local solution is clear by a standard
theory of parabolic type. It is also clear that the solution can be extended
in $t$ direction, as long as the $L^{\infty}$ norm of the solution remains
finite. Moreover, the classical results for parabolic PDE can be applied to
show existence and uniqueness of solution
$v\in\mathrm{C}\mathopen{}\mathclose{{}\left([0,\tau);L^{p}(D;\mathbb{R})}\right)$
up to an eventual blowup (see [20, Chapter 7] and [29]).

Let us denote

$\displaystyle B^{H}_{\ast}(t)=\sup_{0\leq s\leq t}|B^{H}_{s}|,\ \mbox{for
each}\ t\geq 0.$

######  Lemma 3.1.

(see [18, Lemma 2.1]) Let $Z$ be a centered random variable in
$\mathbb{D}^{1,2}$ (cf. Sec. 5.1 for more details). Assume that there exists a
non-random constant $M_{0}$ such that

$\int_{0}^{T}(D_{r}Z)^{2}dr\leq M_{0}^{2},\ \mbox{a.s.}$

Then the following estimate for tail probabilities holds:

$\mathbb{P}(|Z|\geq x)\leq 2e^{-\frac{x^{2}}{2M_{0}^{2}}},\ x>0.$

Therefore (see (5.20) below)

$\mathbb{P}(B_{\ast}^{H}(t)<\infty)=1-\lim_{x\rightarrow\infty}\mathbb{P}(B_{\ast}^{H}(t)>x)=1,\
t>0.$

Let us denote $N_{t}=\\{\omega\in\Omega:B^{H}_{\ast}(t)=\infty\\},$ for every
$t>0.$ Then, we have $\mathbb{P}(N_{t})=0,$ for every $t>0.$ If we take
$t=1,2,\ldots,$ it is clear that $N_{t}\subset N_{m},$ for $t\geq m.$ Define
$N=\displaystyle\lim_{m\rightarrow\infty}N_{m},$ we have
$\mathbb{P}(N)=\displaystyle\lim_{m\to\infty}\mathbb{P}(N_{m})=0.$ Therefore,
for all $\omega\in N,B^{H}_{\ast}(t;\omega)<\infty,$ for all $t>0.$ From the
above observation, without loss of generality, we can assume that for all
$\omega\in\Omega,$

$\displaystyle B^{H}_{\ast}(t;\omega)<\infty,\ \mbox{for all}\ t>0.$ (3.5)

By Theorem 3.1 and (3.5), the global existence and finite-time blow-up of
$u(\cdot,\cdot)$ is guarenteed by considering the properties of
$v(\cdot,\cdot)$. So in this paper, we mainly consider the random partial
differential equation (3.2) for further investigation.

From classical results in semigroup theory, for any bounded measurable initial
data $f\geq 0,$ there exists a unique local mild solution $v(\cdot,\cdot)$ of
the random PDE (3.2), if there exists a number $0<\tau=\tau(\omega)\leq\infty$
such that $v(t,x)$ satisfies the integral equation [38, Chapter 4]

$\displaystyle v(t,x)$
$\displaystyle=\exp\mathopen{}\mathclose{{}\left\\{\gamma
t}\right\\}T_{t}f(x)+\int_{0}^{t}e^{\gamma(t-r)}T_{t-r}\Bigg{[}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\int_{D}v^{q}(r,y)dy$ $\displaystyle\qquad-
ke^{\mathopen{}\mathclose{{}\left(p-1}\right)\eta B^{H}(r)}v^{p}(r,x)+\delta
e^{(m+n-1)\eta B^{H}(r)}v^{m}(r,x)\int_{D}v^{n}(r,y)dy\Bigg{]}dr,$ (3.6)

for each $0\leq t<\tau$ and $x\in D.$

Let $\tau$ be the blow-up time of the equation (3.2) with the initial value
$f$. Due to Theorem 3.1 and a.s continuity of $B^{H}(\cdot)$, $\tau$ is also
the blow-up time for the equation (1.4) as well ([14, Corollary 1]).

The following comparison result is a consequence of [32, Lemma 2.1] and [43,
Lemma 2.1] (for more details see [37, Lemma 1]).

######  Lemma 3.2.

Suppose that $w(t,x)\in C^{1,2}(D_{T})\cap C(\overline{D}_{T}),$ where
$D_{T}=(0,T]\times D$ and satisfies

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} w_{t}(t,x)-d\Delta
w(t,x)&\geq c_{1}(t,x)w(t,x)+c_{3}(t,x)\int_{D}c_{2}(t,x)w(t,x)dx\\\
&\quad+\int_{D}c_{4}(t,x)w(t,x)dx,\ x\in D,\ 0<t\leq T,\\\ w(t,x)&\geq 0,\
x\in\partial D,\ 0\leq t\leq T,\\\ w(0,x)&\geq 0,\ x\in
D,\end{aligned}}\right.$ (3.7)

where $c_{1}(t,x),c_{2}(t,x),c_{3}(t,x)$ and $c_{4}(t,x)$ are bounded
functions and $c_{2}(t,x)\geq 0,\ c_{3}(t,x)\geq 0,\ c_{4}(t,x)\geq 0$ and
$d>0$ in $D_{T}$. Then $w(t,x)\geq 0$ on $\overline{D}_{T}.$ Moreover,
$w(t,x)>0$ in $\overline{D}_{T}$ if $w(0,x)$ is not identiaclly zero.

###### Proof.

Let $\widetilde{c}_{i}$ be the least upper bound of $c_{i}(t,x)$ in
$D_{T},i=1,2,3,4.$ Let $v(t,x)=e^{-\gamma_{1}t}w(t,x)$ for some
$\gamma_{1}>\widetilde{c}_{1}+(\widetilde{c}_{2}\widetilde{c}_{3}+\widetilde{c}_{4})|D|.$
Therefore from (3.7), we have

$\displaystyle Lv:$ $\displaystyle=v_{t}(t,x)-d\Delta
v(t,x)+(\gamma_{1}-c_{1}(t,x))v(t,x)$ $\displaystyle\geq
e^{-\gamma_{1}t}\mathopen{}\mathclose{{}\left[c_{3}(t,x)\int_{D}c_{2}(t,x)v(t,x)dx+\int_{D}c_{4}(t,x)v(t,x)dx}\right],\
x\in D,\ 0<t\leq T,$ (3.8) $\displaystyle v(t,x)$ $\displaystyle\geq 0,\
x\in\partial D,\ 0<t\leq T,$ $\displaystyle v(0,x)$ $\displaystyle\geq 0,\
x\in D.$

We need to show that $v(t,x)\geq 0$ on $\overline{D}_{T}.$ Suppose that $v$
has a negative minimum for some $(t_{0},x_{0})\in\overline{D}_{T}$. If
$x_{0}\in\partial D,$ then $v(t_{0},x_{0})\geq 0$ for $0<t_{0}\leq T,$ which
is a contradiction, since $v(t_{0},x_{0})<0$. Hence $(t_{0},x_{0})\in D_{T},$
and it is immediate that $v_{t}(t_{0},x_{0})=0,$ for $0<t_{0}<T$ and
$v_{t}(t_{0},x_{0})\leq 0,$ for $t_{0}=T$, $\Delta v(t_{0},x_{0})\geq 0$ and

$\displaystyle
Lv(t_{0},x_{0})\leq(\gamma_{1}-c_{1}(t_{0},x_{0}))v(t_{0},x_{0}).$

From (3), we have

$\displaystyle
e^{-\gamma_{1}t_{0}}\mathopen{}\mathclose{{}\left[c_{3}(t_{0},x_{0})\int_{D}c_{2}(t_{0},x)v(t_{0},x)dx+\int_{D}c_{4}(t_{0},x)v(t_{0},x)dx}\right]\leq(\gamma_{1}-c_{1}(t_{0},x_{0}))v(t_{0},x_{0}),$
$\displaystyle\Rightarrow(\widetilde{c}_{3}\widetilde{c}_{2}|D|+\widetilde{c}_{4}|D|)v(t_{0},x_{0})\leq(\gamma_{1}-\widetilde{c}_{1})v(t_{0},x_{0}),$
$\displaystyle\Rightarrow\gamma_{1}\leq\widetilde{c}_{1}+(\widetilde{c}_{3}\widetilde{c}_{2}+\widetilde{c}_{4})|D|,$

which is a contradiction, since
$\gamma_{1}>\widetilde{c}_{1}+(\widetilde{c}_{2}\widetilde{c}_{3}+\widetilde{c}_{4})|D|.$
Hence $v(t,x)\geq 0$ on $\overline{D}_{T}$ implies that
$w(t,x)=e^{\gamma_{1}t}v(t,x)\geq 0$ in $\overline{D}_{T}.$ ∎

Let $N>0$ be a constant and define the following stopping time:

$\tau_{N}(\omega)=\inf\mathopen{}\mathclose{{}\left\\{t>0:\
|B^{H}_{t}(\omega)|\geq N}\right\\}.$

Clearly,

$\mathopen{}\mathclose{{}\left\\{\omega\in\Omega:\ \tau_{N}(\omega)\leq
t}\right\\}=\mathopen{}\mathclose{{}\left\\{\omega\in\Omega:\
B^{H}_{\ast}(t)\geq N}\right\\}.$

Let $z(t,x)$ be the solution of

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
z(t,x)}{\partial
t}&=(\Delta+\gamma)z(t,x)+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\int_{D}|z(t,y)|^{q-1}z(t,y)dy-
ke^{\mathopen{}\mathclose{{}\left(p-1}\right)\eta
B^{H}(t)}|z(t,x)|^{p-1}z(t,x)\\\ &\qquad+\delta e^{(m+n-1)\eta
B^{H}(t)}|z(t,x)|^{m}\int_{D}|z(t,y)|^{n-1}z(t,y)dy,\ (x,t)\in
D\times(0,T\wedge\tau_{N}],\\\ z(t,x)&=0,\ \ (x,t)\in\partial
D\times(0,T\wedge\tau_{N}],\\\ z(0,x)&=f(x),\ x\in D.\end{aligned}}\right.$

Here $f(x)\geq 0,\ c_{1}(t,x)=\gamma-ke^{(p-1)\eta B^{H}(t)}|z(t,x)|^{p-1},\
c_{2}(t,x)=\delta e^{(m+n-1)\eta B^{H}(t)}|z(t,x)|^{n-1}\geq 0,\
c_{3}(t,x)=|z(t,x)|^{m}\geq 0,\
c_{4}(t,x)=e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}|z(t,y)|^{q-1}\geq 0$ and $c_{1}(t,x),c_{2}(t,x),c_{3}(t,x)$ and
$c_{4}(t,x)$ are bounded in $D\times(0,T\wedge\tau_{N}].$ By using Lemma 3.2,
we have $z(t,x)\geq 0$ and hence, $z(t,x)$ is the solution of (3.2). Moreover,
by uniqueness, we have $v(t,x)=z(t,x)\geq 0.$ Again by using Lemma 3.2, we
obtain that if $f(x)$ is not identically zero, then $v(t,x)>0$ in
$D\times(0,T\wedge\tau_{N}]$.

######  Remark 3.1.

If the initial data $f$ is non-negative, then the weak solution
$u(\cdot,\cdot)$ and $v(\cdot,\cdot)$, of (1.4) and (3.2) respectively are
non-negative as well.

Using the increasing property of the function $x^{i}$ for $x\geq 0$,
$i\in\mathopen{}\mathclose{{}\left\\{p,q,m,n}\right\\}$ with $p,q,n>1$ and
$m\geq 0,$ as a consequence of Lemma 3.2, we have the following comparison
principle:

######  Proposition 3.1.

Let $v(\cdot,\cdot)$ be the solution of poblem (3.2) and the positive function
$V\in C^{2,1}(D_{T})\cap C(\overline{D}_{T}),$ where
$D_{T}=D\times(0,T\wedge\tau_{N}]$ satisfy

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
V(t,x)}{\partial
t}&\geq(\leq)(\Delta+\gamma)V(t,x)+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\int_{D}V^{q}(t,y)dy-
ke^{\mathopen{}\mathclose{{}\left(p-1}\right)\eta B^{H}(t)}V^{p}(t,x)\\\
&\qquad+\delta e^{(m+n-1)\eta B^{H}(t)}V^{m}(t,x)\int_{D}V^{n}(t,y)dy,\
(x,t)\in D\times(0,T\wedge\tau_{N}],\\\ V(t,x)&\geq(\leq)0,\ \
(x,t)\in\partial D\times(0,T\wedge\tau_{N}],\\\ V(0,x)&\geq(\leq)f(x),\ x\in
D.\end{aligned}}\right.$

Then

$\displaystyle V(t,x)\geq(\leq)v(t,x)\ \mbox{on}\
\overline{D}\times[0,T\wedge\tau_{N}].$

The following theorem establishes the existence of a global solution
$v(\cdot,\cdot)$ of (3.2) with $|D|\leq k.$

######  Theorem 3.2.

Let $v(\cdot,\cdot)$ be a solution of (3.2) and $\frac{1}{2}<H<1$. If
$p=q=n=\beta\ (say),\ m=0$ and $|D|\leq k,$ then there exist $\sigma_{0}>0$
and $M_{1}>0$ such that

$v(t,x)\leq M_{1}e^{-\sigma_{0}t+(\beta-1)\eta B^{H}_{\ast}(t)},\ \mbox{for
all}\ (t,x)\in[0,\infty)\times\overline{D}.$

The proof follows from [29, Theorem 3.3].

Our next aim is to obtain random times $\tau_{*}$ and $\tau^{*}$ such that
$0\leq\tau_{*}\leq\tau\leq\tau^{*}.$

##  4 A Lower Bound for $\tau$

This section is devoted to obtain the lower bounds $\tau_{\ast}$ to the blow-
up time $\tau$ such that $\tau_{\ast}\leq\tau$ of (1.4) with
$\frac{1}{2}<H<1$. Also, we provide a sufficient condition for the existence
of a global mild solution of the equation (1.4) for $|D|>k$.

Let us first consider the problem:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
w(t,x)}{\partial
t}&=(\Delta+\gamma)w(t,x)+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\int_{D}w^{q}(t,y)dy+\delta e^{(m+n-1)\eta
B^{H}(t)}w^{m}(t,x)\int_{D}w^{n}(t,y)dy,\\\ w(t,x)&=0,\ \ x\in\partial D,\
t>0,\\\ w(0,x)&=f(x),\ \ x\in D.\end{aligned}}\right.$ (4.1)

Then the mild solution (see [38, Chapter 4]), $w(\cdot,\cdot)$ of (4.1)
satisfies the following integral equation:

$\displaystyle w(t,x)=e^{\gamma t}T_{t}f(x)+$
$\displaystyle\int_{0}^{t}e^{\gamma(t-r)}T_{t-r}\Bigg{[}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\int_{D}w^{q}(r,y)dy$ $\displaystyle\qquad+\delta e^{(m+n-1)\eta
B^{H}(r)}w^{m}(r,x)\int_{D}w^{n}(r,y)dy\Bigg{]}dr,$ (4.2)

for each $x\in D$ and $0\leq t<\tau.$ The following result provides a lower
bound for the finite-time blow-up of the solution of (1.4).

######  Theorem 4.1.

Assume that $n,p,q>1,\ m\geq 0$ with $m+n\geq q\geq p>1$, $f$ is a non-
negative bounded function and $|D|>k$. Let $\tau_{\ast}$ be given by

$\displaystyle\tau_{\ast}=\inf\Bigg{\\{}t\geq 0:$
$\displaystyle\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}dr$
$\displaystyle\qquad\geq\frac{1}{2M(m+n+q-1)\|f\|_{\infty}^{m+n-1}}\Bigg{\\}},$
(4.3)

where $M=\max\mathopen{}\mathclose{{}\left\\{|D|,\delta|D|}\right\\}\
\mbox{and}\ \|f\|_{\infty}:=\displaystyle\sup_{x\in D}f(x).$ Then
$\tau_{\ast}\leq\tau$.

###### Proof.

Let $w(\cdot,\cdot)$ solve the equation (4.1). Then, we have

$\displaystyle w(t,x)$ $\displaystyle=e^{\gamma
t}T_{t}f(x)+\int_{0}^{t}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(\int_{D}w^{q}(r,y)dy}\right)dr$
$\displaystyle\qquad+\delta\int_{0}^{t}e^{(m+n-1)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(w^{m}(r,x)\int_{D}w^{n}(r,y)dy}\right)dr.$

For all $x\in D,\ t\geq 0,$ let us define the operator $\mathcal{H}$ as
follows:

$\displaystyle\mathcal{H}v(t,x)$ $\displaystyle=e^{\gamma
t}T_{t}f(x)+\int_{0}^{t}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(\int_{D}v^{q}(r,y)dy}\right)dr$
$\displaystyle\qquad+\delta\int_{0}^{t}e^{(m+n-1)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(v^{m}(r,x)\int_{D}v^{n}(r,y)dy}\right)dr,$

where $v$ is any non-negative, bounded and measurable function. First, we
shall prove that

$\displaystyle\mathcal{H}w(t,x)=w(t,x),\ \ x\in D,\ \ 0\leq t<\tau_{*},$

for some non-negative bounded and measurable function $w$. Moreover, on the
set $t<\tau_{*},$ we set

$\displaystyle\mathscr{G}(t)$
$\displaystyle=\Bigg{[}1-2(m+n+q-1)M\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)$
$\displaystyle\qquad\qquad\qquad\times\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}dr\Bigg{]}^{-\frac{1}{m+n+q-1}}.$

Then, it can be easily seen that

$\mathopen{}\mathclose{{}\left\\{\begin{aligned}
\frac{d\mathscr{G}(t)}{dt}&=2M\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\vee e^{(m+n-1)\eta B^{H}(t)}}\right)\|e^{\gamma
t}T_{t}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}\mathscr{G}^{m+n+q}(t),\\\
\mathscr{G}(0)&=1,\end{aligned}}\right.$

so that

$\displaystyle\mathscr{G}(t)$
$\displaystyle=1+2M\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}\mathscr{G}^{m+n+q}(r)dr.$

Let us choose $v\geq 0$ such that

$v(t,x)\leq e^{\gamma t}T_{t}\|f\|_{\infty}\mathscr{G}(t),$

for $x\in D$ and $t<\tau_{*}.$ Then $e^{\gamma
t}T_{t}\|f\|_{\infty}\leq\mathcal{H}v(t,x)$ and

$\displaystyle\mathcal{H}v(t,x)$ $\displaystyle=e^{\gamma
t}T_{t}f(x)+\int_{0}^{t}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(\int_{D}v^{q}(r,y)dy}\right)dr$
$\displaystyle\quad+\delta\int_{0}^{t}e^{(m+n-1)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(v^{m}(r,x)\int_{D}v^{n}(r,y)dy}\right)dr$
$\displaystyle\leq e^{\gamma
t}T_{t}f(x)+\int_{0}^{t}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(\int_{D}\mathopen{}\mathclose{{}\left(e^{\gamma
r}T_{r}\|f\|_{\infty}\mathscr{G}(r)}\right)^{q}dy}\right)dr$
$\displaystyle\quad+\delta\int_{0}^{t}e^{(m+n-1)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(e^{\gamma
r}T_{r}\|f\|_{\infty}\mathscr{G}(r)}\right)^{m}\int_{D}\mathopen{}\mathclose{{}\left(e^{\gamma
r}T_{r}\|f\|_{\infty}\mathscr{G}(r)}\right)^{n}dy}\right)dr$
$\displaystyle\leq e^{\gamma
t}T_{t}\|f\|_{\infty}+\int_{0}^{t}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)+\gamma(t-r)}\|e^{\gamma
r}T_{r}\|_{\infty}^{q-1}\|f\|_{\infty}^{q-1}T_{t-r}(e^{\gamma
r}T_{r}\|f\|_{\infty})\mathscr{G}^{q}(r)|D|dr$
$\displaystyle\quad+\delta\int_{0}^{t}e^{(m+n-1)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(e^{\gamma
r}T_{r}\|f\|_{\infty}}\right)\|e^{\gamma
r}T_{r}\|_{\infty}^{n+m-1}\|f\|_{\infty}^{m+n-1}\mathscr{G}^{m+n}(r)|D|dr$
$\displaystyle=e^{\gamma
t}T_{t}\|f\|_{\infty}\Bigg{[}1+|D|\int_{0}^{t}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\|e^{\gamma
r}T_{r}\|_{\infty}^{q-1}\|f\|_{\infty}^{q-1}\mathscr{G}^{q}(r)dr$
$\displaystyle\quad+\delta|D|\int_{0}^{t}e^{(m+n-1)\eta B^{H}(r)}\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}\mathscr{G}^{m+n}(r)dr\Bigg{]}$
$\displaystyle\leq e^{\gamma
t}T_{t}\|f\|_{\infty}\Bigg{[}1+M\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)$
$\displaystyle\quad\times\mathopen{}\mathclose{{}\left(\|e^{\gamma
r}T_{r}\|_{\infty}^{q-1}\|f\|_{\infty}^{q-1}+\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}}\right)\mathopen{}\mathclose{{}\left(\mathscr{G}^{q}(r)+\mathscr{G}^{m+n}(r)}\right)dr\Bigg{]}$
$\displaystyle\leq e^{\gamma
t}T_{t}\|f\|_{\infty}\Bigg{[}1+2M\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)$
$\displaystyle\quad\times\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}\mathscr{G}^{m+n+q}(r)dr\Bigg{]}$
$\displaystyle=e^{\gamma t}T_{t}\|f\|_{\infty}\mathscr{G}(t),$

where $M=\max\mathopen{}\mathclose{{}\left\\{|D|,\delta|D|}\right\\}$. Thus,
we have

$e^{\gamma t}T_{t}f(x)\leq\mathcal{H}v(t,x)\leq e^{\gamma
t}T_{t}\|f\|_{\infty}\mathscr{G}(t).$

For $x\in D,\ 0\leq t\leq\tau_{*},$ let us take,

$u^{(0)}(t,x)=e^{\gamma t}T_{t}f(x)\mbox{ and
}u^{(n)}(t,x)=\mathcal{H}u^{(n-1)}(t,x).$

Our aim is to show that the sequence $\\{u^{(n)}\\}_{n\in\mathbb{N}}$ of
functions is monotonically increasing. Note that

$\displaystyle u^{(0)}(t,x)$ $\displaystyle\leq e^{\gamma
t}T_{t}f(x)+\int_{0}^{t}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left(\int_{D}(u^{(0)})^{q}(r,y)dy}\right)dr$
$\displaystyle\quad+\delta\int_{0}^{t}e^{(m+n-1)\eta
B^{H}(r)+\gamma(t-r)}T_{t-r}\mathopen{}\mathclose{{}\left((u^{(0)})^{m}(r,x)\int_{D}(u^{(0)})^{n}(r,y)dy}\right)dr$
$\displaystyle=\mathcal{H}u^{(0)}(t,x)=u^{(1)}(t,x).$

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

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

and therefore the limit

$\displaystyle\lim_{n\rightarrow\infty}u^{(n)}(t,x)=w(t,x),$

exists for $x\in D$ and $0\leq t<\tau_{*}.$ As a result of the monotone
convergence theorem, we obtain

$\displaystyle w(t,x)=\mathcal{H}w(t,x).$

Moreover, we have

$\mathcal{H}w(t,x)\leq e^{\gamma t}T_{t}\|f\|_{\infty}\mathscr{G}(t),$

so that

$\displaystyle 0\leq w(t,x)\leq$ $\displaystyle e^{\gamma
t}T_{t}\|f\|_{\infty}\Bigg{[}1-2(m+n+q-1)M\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)$
$\displaystyle\qquad\times\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}dr\Bigg{]}^{-\frac{1}{m+n+q-1}}.$
(4.4)

From (4.1), we infer that

$\displaystyle\frac{\partial w(t,x)}{\partial t}$
$\displaystyle\geq(\Delta+\gamma)w(t,x)+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\int_{D}w^{q}(t,y)dy$ $\displaystyle\quad-
ke^{\mathopen{}\mathclose{{}\left(p-1}\right)\eta B^{H}(t)}w^{p}(t,x)+\delta
e^{(m+n-1)\eta B^{H}(t)}w^{m}(t,x)\int_{D}w^{n}(t,y)dy.$

By Proposition 3.1, we deduce that $0\leq v(t,x)\leq w(t,x)$, for each $x\in
D$ and $t\geq 0$, where $v(\cdot,\cdot)$ is the unique solution of (3.2).
Therefore from (4), we get

$\displaystyle 0\leq v(t,x)$ $\displaystyle\leq e^{\gamma
t}T_{t}\|f\|_{\infty}\Bigg{[}1-2(m+n+q-1)M\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)$
$\displaystyle\qquad\times\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}dr\Bigg{]}^{-\frac{1}{m+n+q-1}},$

which completes the proof of the theorem. ∎

The following results show that the solution $u(\cdot,\cdot)$ of the equation
(1.4) exists globally.

######  Theorem 4.2.

Assume that $n,p,q>1,\ m\geq 0$ with $m+n\geq q\geq p.$ If $f$ is a non-
negaive bounded function, $|D|>k$ and the inequality

$\displaystyle\Bigg{\\{}2(m+n+q-1)M$
$\displaystyle\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}dr\Bigg{\\}}<1,$ (4.5)

holds, then the weak solution $v(t,x)$ of (3.2) exists globally for all
$(t,x)\in[0,\infty)\times D.$ Moreover, for all $(t,x)\in[0,\infty)\times D$

$\displaystyle 0\leq v(t,x)$ $\displaystyle\leq e^{\gamma
t}T_{t}\|f\|_{\infty}\Bigg{[}1-2(m+n+q-1)M\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta B^{H}(r)}}\right)$
$\displaystyle\qquad\times\|e^{\gamma
r}T_{r}\|_{\infty}^{m+n-1}\|f\|_{\infty}^{m+n-1}dr\Bigg{]}^{-\frac{1}{m+n+q-1}},$

where $M=\max\mathopen{}\mathclose{{}\left\\{|D|,\delta|D|}\right\\}.$

The proof of Theorem 4.2 immediately follows from Theorem 4.1.

The following thoerem gives sharp bounds for
$\mathopen{}\mathclose{{}\left\\{p_{t}(x,y),\ t>0}\right\\}$ (see [36, Theorem
1.1]).

######  Theorem 4.3.

([36]). Let $\varphi$ be the first (normalized) Dirichlet eigenfunction of
(2.1) on a connected bounded $C^{1,\alpha}$-domain $D\subset\mathbb{R}^{d}$,
where $\alpha>0$ and $d\geq 1$, and let $\\{p_{t}(x,y),\ t>0\\}$ be the
corresponding Dirichlet heat kernel. Then, there exists a constant $c>0$ such
that

$\displaystyle\max\mathopen{}\mathclose{{}\left\\{1,\frac{1}{c}t^{-\frac{(d+2)}{2}}}\right\\}\leq
e^{\lambda_{1}t}\sup_{x,y\in D}\frac{p_{t}(x,y)}{\varphi(x)\varphi(y)}\leq
1+c(1\wedge t)^{-\frac{(d+2)}{2}}e^{-(\lambda_{2}-\lambda_{1})t},\ t>0,$ (4.6)

where $\lambda_{1},\lambda_{2}$ are the first two Dirichlet eigenvalues of the
Laplacian with $\lambda_{2}>\lambda_{1}$. This estimate is sharp for both
short and long times.

The following result establishes the existence of a global weak solution to
the problem (1.4) under the assumption that $f(x)\leq C_{0}\varphi(x)$, for
each $x\in D$ and $C_{0}>0$ by using Theorem 4.3.

######  Theorem 4.4.

Let $D$ be a bounded domain $D\subset\mathbb{R}^{d}\ (d\geq 1)$ with smooth
boundary. Assume that $n,q>1,\ m\geq 0$ along with the conditions $m+n\geq q$
and $|D|>k$. If the initial data $f(x)\leq C_{0}\varphi(x)$ satisfies

$\displaystyle\mathcal{K}\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta
B^{H}(r)}}\right)e^{-\mathopen{}\mathclose{{}\left(\lambda_{1}-\gamma}\right)(m+n-1)r}dr<1,$

where $C_{0}>0$,
$\mathcal{K}=2M(m+n+q-1)\mathopen{}\mathclose{{}\left(C_{0}\mathopen{}\mathclose{{}\left\|\varphi}\right\|_{\infty}^{2}(1+c)}\right)^{m+n-1}$
and $M=\max\mathopen{}\mathclose{{}\left\\{|D|,\delta|D|}\right\\},$ then the
solution $v(\cdot,\cdot)$ of (3.2) exists globally.

###### Proof.

For any $f(x)\geq 0,x\in D$ and any $t>0$

$\displaystyle T_{t}f(x)$
$\displaystyle=\int_{D}p_{t}(x,y)f(y)dy=\int_{D}e^{\lambda_{1}t}\frac{p_{t}(x,y)}{\varphi(x)\varphi(y)}e^{-\lambda_{1}t}\varphi(x)\varphi(y)f(y)dy$
$\displaystyle\leq\mathopen{}\mathclose{{}\left\|f}\right\|_{\infty}\mathopen{}\mathclose{{}\left\|\varphi}\right\|_{\infty}\int_{D}e^{\lambda_{1}t}\frac{p_{t}(x,y)}{\varphi(x)\varphi(y)}e^{-\lambda_{1}t}\varphi(y)dy.$

By using (4.6), we obtain

$\displaystyle T_{t}f(x)$
$\displaystyle\leq\mathopen{}\mathclose{{}\left\|f}\right\|_{\infty}\mathopen{}\mathclose{{}\left\|\varphi}\right\|_{\infty}\int_{D}\mathopen{}\mathclose{{}\left(1+c(1\wedge
t)^{-\frac{(d+2)}{2}}e^{-(\lambda_{2}-\lambda_{1})t}}\right)e^{-\lambda_{1}t}\varphi(y)dy$
$\displaystyle\leq\mathopen{}\mathclose{{}\left\|f}\right\|_{\infty}\mathopen{}\mathclose{{}\left\|\varphi}\right\|_{\infty}\mathopen{}\mathclose{{}\left(e^{-\lambda_{1}t}+ce^{-\lambda_{2}t}}\right)\int_{D}\varphi(y)dy$
$\displaystyle\leq\mathopen{}\mathclose{{}\left\|f}\right\|_{\infty}\mathopen{}\mathclose{{}\left\|\varphi}\right\|_{\infty}\mathopen{}\mathclose{{}\left(1+c}\right)e^{-\lambda_{1}t}.$

Therefore, we get

$\displaystyle\mathopen{}\mathclose{{}\left\|e^{\gamma
t}T_{t}f}\right\|_{\infty}\leq
C_{0}\mathopen{}\mathclose{{}\left\|\varphi}\right\|_{\infty}^{2}\mathopen{}\mathclose{{}\left(1+c}\right)e^{-\mathopen{}\mathclose{{}\left(\lambda_{1}-\gamma}\right)t},$
(4.7)

and the condition (4.5) in Theorem 4.2 is satisfied, provided

$\displaystyle\mathcal{K}\int_{0}^{\infty}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(r)}\vee e^{(m+n-1)\eta
B^{H}(r)}}\right)e^{-\mathopen{}\mathclose{{}\left(\lambda_{1}-\gamma}\right)(m+n-1)r}dr<1,$

where
$\mathcal{K}=2M(m+n+q-1)\mathopen{}\mathclose{{}\left(C_{0}\mathopen{}\mathclose{{}\left\|\varphi}\right\|_{\infty}^{2}(1+c)}\right)^{m+n-1}$
and $M=\max\mathopen{}\mathclose{{}\left\\{|D|,\delta|D|}\right\\}.$ Then the
remaining proof follows by Theorem 4.2. ∎

##  5 Upper bound for $\tau$

In this section, we obtain an upper bound $\tau^{*}$ for the explosion time
$\tau$ under suitable assumptions for the case $\frac{1}{2}<H<1$. Using the
fact that $\varphi(x)=0$ for $x\in\partial D,$ we have

$\displaystyle
v(s,\Delta\varphi)=\int_{D}v(s,x)\Delta\varphi(x)dx=\int_{D}\Delta
v(s,x)\varphi(x)dx=\Delta v(s,\varphi)=-\lambda_{1}v(s,\varphi).$ (5.1)

We recall that $\lambda_{1}$ is the first eigenvalue and $\varphi$ is the
coresponding eigenfunction of $-\Delta$ on $D$ and
$\displaystyle\int_{D}\varphi(x)dx=1.$ Let $t\in(0,\infty)$ be some random
number and $b>1$ be chosen so that

$\mathopen{}\mathclose{{}\left.\begin{aligned}
b^{q-p}e^{-\eta(m+n-1)B_{\ast}^{H}(t)}\geq\frac{kC_{1}^{p}+\lambda_{1}C_{1}}{|D|^{1-q}},\
\ &b^{q-p}e^{-\eta(q-p)B_{\ast}^{H}(t)}\geq\frac{kC_{1}^{p}}{|D|^{1-q}}\\\
\mbox{ and }\
b^{q-p}e^{-\eta(q-1)B_{\ast}^{H}(t)}&\geq\frac{2k\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{q-p}{p}}}{\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{p+1}(x)dx}\right)^{\frac{q-p}{p}}}\end{aligned}}\right\\},$
(5.2)

where $C_{1}=\displaystyle\sup_{x\in D}\varphi(x)$.

######  Theorem 5.1.

Suppose that $n,p,q>1,\ m\geq 0$ with $q\geq p$. For the given initial data
$f(x)\geq b\varphi(x),$ $x\in D$ and for each $b>1$ such that (5.2) holds.
Then, we have the following results:

If $m+n=q=\mu\ (say),$ then $\tau\leq\tau^{\ast}_{1},$ where $\tau^{\ast}_{1}$
is given by

$\displaystyle\tau^{\ast}_{1}=$ $\displaystyle\inf\Biggl{\\{}t\geq
0:\int_{0}^{t}e^{\eta(\mu-1)B^{H}(s)+(-\lambda_{1}+\gamma)(\mu-1)s}ds$
$\displaystyle\qquad\geq
J^{1-\mu}(0)\Bigg{[}(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)\Bigg{]}^{-1}\Biggr{\\}}.$
(5.3)

If $m+n>q,$ let
$A_{0}=\mathopen{}\mathclose{{}\left(\frac{m+n-q}{m+n}}\right)\mathopen{}\mathclose{{}\left(\frac{m+n}{q}}\right)^{\frac{q}{m+n-q}}$
and
$\epsilon_{0}\leq\mathopen{}\mathclose{{}\left(J^{q}(0)/A_{0}}\right)^{\frac{q}{m+n-q}}$,
then $\tau\leq\tau^{\ast}_{2},$ where $\tau^{\ast}_{2}$ is given by

$\displaystyle\tau^{\ast}_{2}=\inf\Bigg{\\{}t\geq
0:\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\eta(q-1)B^{H}(s)}\wedge
e^{\eta(m+n-1)B^{H}(s)}}\right)e^{-(-\lambda_{1}+\gamma)(q-1)s}ds\geq
2{J^{1-q}(0)}$
$\displaystyle\times\Bigg{[}((q-1)(-\lambda_{1}+\gamma))\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}+\delta\mathopen{}\mathclose{{}\left(\epsilon_{0}-\frac{A_{0}\epsilon_{0}^{\frac{m+n}{m+n-q}}}{J^{q}(0)}}\right)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)\Bigg{]}^{-1}\Bigg{\\}},$
(5.4)

where $J(0)=\displaystyle\int_{D}f(x)\varphi(x)dx.$

###### Proof.

Let $\bar{v}(t,x)=b\varphi(x),$ for $t>0,$ $x\in D$, $v(0,x)\geq b\varphi(x)$,
and $v(t,x)=\bar{v}(t,x)=0$ on $\partial D.$ Firstly, we show that
$v(t,x)\geq\bar{v}(t,x)=b\varphi(x),$ for each $x\in D$ and $t\geq 0.$ Let us
take

$\displaystyle
I_{1}(t,x):=e^{\eta\mathopen{}\mathclose{{}\left(q-1}\right)B^{H}(t)}\int_{D}\bar{v}^{q}(t,y)dy-
ke^{\eta(p-1)B^{H}(t)}\bar{v}^{p}(t,x)+\delta
e^{\eta(m+n-1)B^{H}(t)}\bar{v}^{m}(t,x)\int_{D}\bar{v}^{n}(t,y)dy$
$\displaystyle=e^{\eta\mathopen{}\mathclose{{}\left(q-1}\right)B^{H}(t)}b^{q}\int_{D}\varphi^{q}(y)dy-
ke^{\eta(p-1)B^{H}(t)}b^{p}\varphi^{p}(x)+\delta
e^{\eta(m+n-1)B^{H}(t)}b^{m+n}\varphi^{m}(x)\int_{D}\varphi^{n}(y)dy.$ (5.5)

Using Hölder’s inequality, we get

$\displaystyle\int_{D}\varphi(y)dy$
$\displaystyle\leq\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{q}(y)dy}\right)^{\frac{1}{q}}\mathopen{}\mathclose{{}\left(\int_{D}1^{\frac{q}{q-1}}dx}\right)^{\frac{q-1}{q}}\Rightarrow\int_{D}\varphi^{q}(y)dy\geq\mathopen{}\mathclose{{}\left(\int_{D}\varphi(y)dy}\right)^{q}|D|^{1-q}=|D|^{1-q}.$

Therefore from (5), we have

$\displaystyle I_{1}(t,x)$ $\displaystyle\geq
e^{\eta\mathopen{}\mathclose{{}\left(q-1}\right)B^{H}(t)}b^{q}|D|^{1-q}-ke^{\eta(p-1)B^{H}(t)}b^{p}\varphi^{p}(x)$
$\displaystyle\geq
e^{\eta\mathopen{}\mathclose{{}\left(p-1}\right)B^{H}(t)}b^{p}\Big{[}e^{\eta\mathopen{}\mathclose{{}\left(q-p}\right)B^{H}(t)}b^{q-p}|D|^{1-q}-kC_{1}^{p}\Big{]}$
$\displaystyle\geq
e^{-\eta\mathopen{}\mathclose{{}\left(p-1}\right)B^{H}_{\ast}(t)}b^{p}\Big{[}e^{-\eta\mathopen{}\mathclose{{}\left(q-p}\right)B^{H}_{\ast}(t)}b^{q-p}|D|^{1-q}-kC_{1}^{p}\Big{]}.$
(5.6)

If $m+n\geq q$, the inequality $\eqref{e5}$ reduces to

$\displaystyle I_{1}(t,x)$ $\displaystyle\geq
b\Big{[}e^{-\eta\mathopen{}\mathclose{{}\left(m+n-1}\right)B^{H}_{\ast}(t)}b^{q-p}|D|^{1-q}-kC_{1}^{p}\Big{]}.$
(5.7)

and the conditions (5.2) leads to $I_{1}(t,x)\geq\lambda_{1}b\varphi(x),$ for
each $x\in D$ and $t\geq 0.$ Note that

$\displaystyle\frac{\partial\bar{v}(t,x)}{\partial
t}-(\Delta+\gamma)\bar{v}(t,x)=(\lambda_{1}-\gamma)b\varphi(x)\leq
I_{1}(t,x)-\gamma\bar{v}(t,x)\leq I_{1}(t,x).$

Hence from (3.2) and by comparison principle in Proposition 3.1, we obtain
that $v(t,x)\geq\bar{v}(t,x)=b\varphi(x),$ for each $x\in D$ and $t\geq 0.$

Let $J(t):=\displaystyle\int_{D}v(t,x)\varphi(x)dx.$ Then,

$\displaystyle J{{}^{\prime}}(t)=\int_{D}\frac{\partial v(t,x)}{\partial
t}\varphi(x)dx,$

and by using (3.2) and (5.1), we have

$\displaystyle J{{}^{\prime}}(t)$
$\displaystyle=\int_{D}(\Delta+\gamma)v(t,x)\varphi(x)dx+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\int_{D}\varphi(z)dz\int_{D}v^{q}(t,y)dy$ $\displaystyle\quad-
ke^{(p-1)\eta B^{H}(t)}\int_{D}v^{p}(t,x)\varphi(x)dx+\delta e^{(m+n-1)\eta
B^{H}(t)}\int_{D}v^{m}(t,z)\varphi(z)dz\int_{D}v^{n}(t,y)dy$
$\displaystyle=\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)\int_{D}v(t,x)\varphi(x)dx+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\int_{D}v^{q}(t,x)dx-ke^{(p-1)\eta
B^{H}(t)}\int_{D}v^{p}(t,x)\varphi(x)dx$ $\displaystyle\qquad+\delta
e^{(m+n-1)\eta B^{H}(t)}\int_{D}v^{m}(t,z)\varphi(z)dz\int_{D}v^{n}(t,y)dy.$

Using Hölder’s inequality, we deduce

$\displaystyle\int_{D}v^{p}(t,x)\varphi(x)dx$
$\displaystyle\leq\mathopen{}\mathclose{{}\left(\int_{D}v^{q}(t,x)dx}\right)^{\frac{p}{q}}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{q-p}{q}},$
$\displaystyle\Rightarrow\int_{D}v^{q}(t,x)dx$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{q}{p}}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{p-q}{p}}$

and

$\displaystyle\int_{D}v^{n}(t,y)dy\geq\mathopen{}\mathclose{{}\left(\int_{D}v(t,y)\varphi(y)dy}\right)^{n}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(y)dy}\right)^{1-n}.$

By using Jensen’s inequality, we obtain

$\displaystyle\int_{D}v^{m}(t,z)\varphi(z)dz\geq\mathopen{}\mathclose{{}\left(\int_{D}v(t,z)\varphi(z)dz}\right)^{m}.$
(5.8)

Thus, it is immediate that

$\displaystyle J{{}^{\prime}}(t)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{q}{p}}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{p-q}{p}}$
$\displaystyle\quad-ke^{(p-1)\eta
B^{H}(t)}\int_{D}v^{p}(t,x)\varphi(x)dx+\delta e^{(m+n-1)\eta
B^{H}(t)}J^{m+n}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}$
$\displaystyle=\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{q}{p}}\Bigg{[}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{p-q}{p}}$
$\displaystyle\quad-ke^{(p-q)\eta
B^{H}(t)}\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{p-q}{p}}\Bigg{]}+\delta
e^{(m+n-1)\eta
B^{H}(t)}J^{m+n}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}.$
(5.9)

Using assumption (5.2), we find

$\displaystyle b^{q-p}e^{(q-p)\eta B^{H}(t)}\geq b^{q-p}e^{-(q-p)\eta
B^{H}_{\ast}(t)}$ $\displaystyle=b^{q-p}e^{-(q-1)\eta
B^{H}_{\ast}(t)}e^{(p-1)\eta B^{H}_{\ast}(t)}$ $\displaystyle\geq
b^{q-p}e^{-(q-1)\eta
B^{H}_{\ast}(t)}\geq\frac{2k\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{q-p}{p}}}{\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{p+1}(x)dx}\right)^{\frac{q-p}{p}}}.$

We know that $v(t,x)\geq b\varphi(x),$ for each $x\in D$ and $t\geq 0,$ and so

$\displaystyle\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{q-p}{p}}\geq
b^{p-q}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{p+1}(x)dx}\right)^{\frac{q-p}{p}}.$

Therefore from the above inequalities, we have

$\displaystyle b^{q-p}e^{(q-p)\eta
B^{H}(t)}\geq\frac{2k\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{q-p}{p}}}{\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{p+1}(x)dx}\right)^{\frac{q-p}{p}}}\geq\frac{2b^{q-p}k\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{q-p}{p}}}{\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{q-p}{p}}}$
$\displaystyle\Rightarrow ke^{(p-q)\eta
B^{H}(t)}\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{p-q}{p}}\leq\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{p-q}{p}}$
$\displaystyle\Rightarrow\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{p-q}{p}}-ke^{(p-q)\eta
B^{H}(t)}\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{p-q}{p}}\geq\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{p-q}{p}}.$

Therefore from (5), we deduce

$\displaystyle J{{}^{\prime}}(t)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\frac{1}{2}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\mathopen{}\mathclose{{}\left(\int_{D}v^{p}(t,x)\varphi(x)dx}\right)^{\frac{q}{p}}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}$
$\displaystyle\quad+\delta e^{(m+n-1)\eta
B^{H}(t)}J^{m+n}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}.$

By using Jensen’s inequality (cf. (5.8)), we obtain

$\displaystyle J{{}^{\prime}}(t)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\frac{1}{2}e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}J^{q}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}$
$\displaystyle\quad+\delta e^{(m+n-1)\eta
B^{H}(t)}J^{m+n}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}.$
(5.10)

Case 1: If $m+n=q=\mu\ (say),$ then we have

$\displaystyle J{{}^{\prime}}(t)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\frac{1}{2}e^{\mathopen{}\mathclose{{}\left(\mu-1}\right)\eta
B^{H}(t)}J^{\mu}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)}\right)^{\frac{p-\mu}{p}}$
$\displaystyle\quad+\delta e^{(\mu-1)\eta
B^{H}(t)}J^{\mu}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}$
$\displaystyle=\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\mathit{\widetilde{N}}e^{\mathopen{}\mathclose{{}\left(\mu-1}\right)\eta
B^{H}(t)}J^{\mu}(t),$ (5.11)

where
$\mathit{\widetilde{N}}=\frac{1}{2}\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}.$
Here, by a comparison argument (see Theorem 1.3 of [42] and Appendix 7 below),
we have $J(t)\geq I(t)$ for all $t\geq 0$, where $I(t)$ solves the
differential equation

$\displaystyle\frac{dI(t)}{dt}=\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)I(t)+\mathit{\widetilde{N}}e^{\mathopen{}\mathclose{{}\left(\mu-1}\right)\eta
B^{H}(t)}I^{\mu}(t),\ I(0)=J(0).$

Solving the above equation, we have

$\displaystyle
I(t)=e^{(-\lambda_{1}+\gamma)t}\mathopen{}\mathclose{{}\left\\{I^{1-\mu}(0)-\mathit{\widetilde{N}}(\mu-1)\int_{0}^{t}e^{\eta(\mu-1)B^{H}(t)+(-\lambda_{1}+\gamma)(\mu-1)s}ds}\right\\}^{-\frac{1}{\mu-1}}.$

For the above equation, the blow-up time is given by (5.1). Case 2: If
$m+n>q$, then we infer

$\displaystyle J^{\prime}(t)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\wedge e^{(m+n-1)\eta
B^{H}(t)}}\right)\Bigg{[}\frac{1}{2}J^{q}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}$
$\displaystyle\quad+\delta
J^{m+n}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}\Bigg{]}.$
(5.12)

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

$\displaystyle
xy\leq\frac{\delta^{a_{0}}x^{a_{0}}}{a_{0}}+\frac{\delta^{-b_{1}}y^{b_{1}}}{b_{1}},\
x,y\geq 0.$ (5.13)

By setting $b_{1}=\frac{m+n}{q},\ y=J^{q}(t),\ x=\epsilon,\
\delta=\mathopen{}\mathclose{{}\left(\frac{m+n}{q}}\right)^{\frac{q}{m+n}}$
and using the fact that $q<m+n$ in (5.13), it follows that for any
$\epsilon>0,$

$\displaystyle\epsilon J^{q}(t)$
$\displaystyle\leq\frac{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{m+n}{q}}\right)^{\frac{q}{m+n}}}\right)^{\frac{m+n}{m+n-q}}\epsilon^{\frac{m+n}{m+n-q}}}{\frac{m+n}{m+n-q}}+\frac{\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\frac{m+n}{q}}\right)^{\frac{q}{m+n}}}\right)^{-\frac{m+n}{q}}\mathopen{}\mathclose{{}\left(J^{q}(t)}\right)^{\frac{m+n}{q}}}{\frac{m+n}{q}}$
$\displaystyle\Rightarrow J^{m+n}(t)$ $\displaystyle\geq\epsilon
J^{q}(t)-A_{0}\epsilon^{\frac{m+n}{m+n-q}}.$

Therefore from (5), we have

$\displaystyle J^{\prime}(t)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\wedge e^{(m+n-1)\eta
B^{H}(t)}}\right)\Bigg{[}\frac{1}{2}J^{q}(t)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}$
$\displaystyle\quad+\delta\mathopen{}\mathclose{{}\left(\epsilon_{0}J^{q}(t)-A_{0}\epsilon_{0}^{\frac{m+n}{m+n-q}}}\right)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}\Bigg{]}$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\wedge e^{(m+n-1)\eta
B^{H}(t)}}\right)J^{q}(t)\Bigg{[}\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}$
$\displaystyle\quad+\delta\mathopen{}\mathclose{{}\left(\epsilon_{0}-\frac{A_{0}\epsilon_{0}^{\frac{m+n}{m+n-q}}}{J^{q}(0)}}\right)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}\Bigg{]}.$

Here, by a comparison argument (see Theorem 1.3 of [42] and Appendix), we have
$J(t)\geq I(t)$ for all $t\geq 0$, where $I(t)$ solves the differential
equation

$\displaystyle\frac{dI(t)}{dt}$
$\displaystyle=\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(t)}\wedge e^{(m+n-1)\eta
B^{H}(t)}}\right)J^{q}(t)\Bigg{[}\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}$
$\displaystyle\quad+\delta\mathopen{}\mathclose{{}\left(\epsilon_{0}-\frac{A_{0}\epsilon_{0}^{\frac{m+n}{m+n-q}}}{J^{q}(0)}}\right)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}\Bigg{]},\
I(0)=J(0).$

Solving the above equation, we have

$\displaystyle I(t)$
$\displaystyle=e^{(q-1)(-\lambda_{1}+\gamma)t}\Bigg{\\{}I^{1-q}(0)-(q-1)(-\lambda_{1}+\gamma)D$
$\displaystyle\qquad\times\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
B^{H}(s)}\wedge e^{(m+n-1)\eta
B^{H}(s)}}\right)e^{-(q-1)(-\lambda_{1}+\gamma)s}ds\Bigg{\\}}^{-\frac{1}{p-1}},$
(5.14)

where
$D=\displaystyle\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}+\delta\mathopen{}\mathclose{{}\left(\epsilon_{0}-\frac{A_{0}\epsilon_{0}^{\frac{m+n}{m+n-q}}}{J^{q}(0)}}\right)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}.$
For the above inequality, the blow-up time is given by (5.1), which completes
the proof of the theorem. ∎

###  5.1 Probability of blow-up when $\frac{1}{2}<H<1$

In this section, we estimate the blow-up probability of the solution to (1.4)
for $\frac{1}{2}<H<1$.

First let us recall some basic concepts of Malliavin calculus (for more
details see [17, 35], etc.). Let $W=\\{W(t)\\}_{t\geq 0}$ be defined on a
complete probability space
$\mathopen{}\mathclose{{}\left(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq
0},\mathbb{P}}\right).$ For $h\in\mathrm{L}^{2}(\mathbb{R^{+}}),$ the Wiener
integral $\mathcal{W}(h)$ is denoted by

$\mathcal{W}(h)=\int_{0}^{\infty}h(s)dW(s).$

Let $\mathcal{S}$ denote the dense subset of
$\mathrm{L}^{2}(\Omega,\mathcal{F},\mathbb{P})$ consisting of smooth random
variables of the form

$\displaystyle
F=f(\mathcal{W}(h_{1}),\mathcal{W}(h_{2}),\ldots,\mathcal{W}(h_{n})),$ (5.15)

where $n\in\mathbb{N},\ f\in\mathrm{C}_{0}^{\infty}(\mathbb{R}^{n})$ and
$h_{1},h_{2},\ldots,h_{n}\in\mathrm{L}^{2}(\mathbb{R}^{+}).$ If $F$ is of the
form (5.15), we define its Malliavin derivative as the process
$DF:=\\{D_{t}F\\}_{t\geq 0}$ given by

$\displaystyle D_{t}F=\sum_{k=1}^{n}\frac{\partial f}{\partial
x_{k}}\mathopen{}\mathclose{{}\left(\mathcal{W}(h_{1}),\mathcal{W}(h_{2}),...,\mathcal{W}(h_{n})}\right)h_{k}(t).$

For any $1\leq p<\infty,$ we shall denote by $\mathbb{D}^{1,p}$, the closure
of $\mathcal{S}$ with respect to the norm

$\displaystyle\|F\|_{1,p}^{p}:=\mathbb{E}\mathopen{}\mathclose{{}\left[|F|^{p}}\right]+\mathbb{E}\mathopen{}\mathclose{{}\left[\int_{0}^{\infty}|\mathrm{D}_{t}F|^{p}dt}\right].$

A random variable $F$ is said to be _Malliavin differentiable_ if it belongs
to $\mathbb{D}^{1,2}.$

We establish an upper bound for probabilities of the form

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{0}^{\infty}\exp\\{-as+\sigma
X_{s}\\}ds<x}\right],$

where $x>0,\ \sigma>0$ and $a$ are real numbers, $\\{X_{s}\\}_{s\geq 0}$ is a
continuous stochastic processes, which also contains fBm $B^{H}$.

######  Assumption 1.

The stochastic process $\\{X_{t}\\}_{t\geq 0}$ is $\mathcal{F}_{t}$-adapted
and satisfies the following properties:

  * 1.

$\displaystyle\int_{0}^{\infty}\exp\\{-as\\}\mathbb{E}\mathopen{}\mathclose{{}\left[\exp\\{\sigma
X_{s}\\}}\right]ds<+\infty.$

  * 2.

For each $t\geq 0,\ X_{t}\in\mathbb{D}^{1,2}.$

  * 3.

There exists a function $f:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}$ such that
$\displaystyle\lim_{t\rightarrow\infty}f(t)=+\infty$ and for each $x>0,$

$\displaystyle\sup_{t\geq
0}\frac{\displaystyle\sup_{s\in[0,t]}\displaystyle\int_{0}^{s}|D_{\theta}X_{s}|^{2}d\theta}{\mathopen{}\mathclose{{}\left(\ln\mathopen{}\mathclose{{}\left(x+1}\right)+f(t)}\right)^{2}}\leq
M_{x}<+\infty\ \ \mathbb{P}\text{-a.s.}$

######  Theorem 5.2.

[17, Theorem 3.1] Suppose that Assumption 1 holds. We have

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left[\int_{0}^{\infty}\exp\\{-as+\sigma
X_{s}\\}ds<x}\right]\leq\exp\mathopen{}\mathclose{{}\left\\{-\frac{(m_{x}-1)^{2}}{2\sigma^{2}M_{x}}}\right\\},$
(5.16)

where

$\displaystyle m_{x}=\mathbb{E}\mathopen{}\mathclose{{}\left[\sup_{t\geq
0}\frac{\ln\mathopen{}\mathclose{{}\left(\int_{0}^{t}\exp\\{-as+\sigma
X_{s}\\}ds+1}\right)+f(t)}{\ln\mathopen{}\mathclose{{}\left(x+1}\right)+f(t)}}\right]\geq
1.$

The following theorem is useful to obtain a lower bound for the probability of
blow-up solution of the equation (1.4) with $\frac{1}{2}<H<1.$

######  Theorem 5.3.

Suppose that $\alpha>H$ with $q>p>1,m+n=q$. For each positive initial value
$f(x)\geq b\varphi(x),$ $x\in D$ and for each $b>1$ such that (5.2) holds, and
the blow-up times $\tau^{\ast}_{1}$ and $\tau^{\ast}_{2}$ of (1.4) as defined
in Theorem 5.1 with $J(0)=\int_{D}f(x)\varphi(x)dx,$ we have the following
results:

  * 1.

If $\lambda_{1}<\gamma,$ then $\mathbb{P}(\tau^{\ast}_{1}=\infty)=0$ for any
non-trivial positive solution $u(\cdot,\cdot)$ of the equation (1.4), that is
the solution $u(\cdot,\cdot)$ blows-up in finite-time $\mathbb{P}\text{-a.s.
}$

  * 2.

If $\lambda_{1}>\gamma$ and $m+n=q=\mu\ (say)$, then a lower bound for the
probability of blow-up solution of (1.4) is given by

$\displaystyle\mathbb{P}(\tau<\infty)$ $\displaystyle\geq
1-\exp\Bigg{\\{}-\frac{1}{2\rho^{2}_{1}}\mathopen{}\mathclose{{}\left(\log\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}+1}\right)}\right)^{\frac{2H}{\alpha}-2}$
$\displaystyle\qquad\qquad\times\mathopen{}\mathclose{{}\left(\frac{\alpha-H}{\alpha}}\right)^{2-\frac{2H}{\alpha}}\mathopen{}\mathclose{{}\left(N(H)-1}\right)^{2}\Bigg{\\}},$

where $\rho_{1}=\eta(\mu-1)$ and

$\displaystyle N(H):=\mathbb{E}\mathopen{}\mathclose{{}\left[\sup_{t\geq
0}\frac{\ln\mathopen{}\mathclose{{}\left(\displaystyle
1+\int_{0}^{t}\exp\\{(-\lambda_{1}+\gamma)(\mu-1)s-\frac{\rho^{2}_{1}}{2}s^{2H}+\rho_{1}B^{H}(s)\\}ds}\right)+t^{\alpha}}{\ln\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}+1}\right)+t^{\alpha}}}\right].$
(5.17)

###### Proof.

Case 1: Assume that $\lambda_{1}<\gamma,$ and recall the law of iterated
logarithm for $B^{H}$ which is given by (see more details in [1])

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

If $m+n=q,$ then by Lemma 2.1 and (5.1), we have
$\mathbb{P}(\tau^{\ast}_{1}=\infty)=0$, and any non-trivial positive solution
of the equation (1.4) blows-up in finite-time $\mathbb{P}\text{-a.s. }$

Case 2 : If $m+n=q=\mu\ (say)$ then by using the definition of
$\tau^{\ast}_{1}$ in (5.1), we have

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left(\tau^{\ast}_{1}=\infty}\right)$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{\infty}e^{\rho_{1}B^{H}(s)+(-\lambda_{1}+\gamma)(\mu-1)s}ds<\frac{J^{1-\mu}(0)}{(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{p-q}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\Bigg{)}$
$\displaystyle\leq\mathbb{P}\Bigg{(}\int_{0}^{\infty}e^{\rho_{1}B^{H}(s)-\frac{\rho_{1}^{2}}{2}s^{2H}+(-\lambda_{1}+\gamma)(\mu-1)s}ds<\frac{J^{1-\mu}(0)}{(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{p-q}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\Bigg{)}.$

We choose $X_{t}=-\frac{\rho_{1}}{2}t^{2H}+B^{H}(t),\ \rho_{1}=\eta(\mu-1).$
First we show that the stochastic process $\\{X_{t}\\}_{t\geq 0}$ satisfies
the conditions of Theorem 5.2. Consider

$\displaystyle\int_{0}^{\infty}\mathbb{E}\mathopen{}\mathclose{{}\left(e^{(-\lambda_{1}+\gamma)(\mu-1)s-\frac{\rho^{2}_{1}}{2}s^{2H}+\rho_{1}B^{H}(s)}}\right)ds$
$\displaystyle=\int_{0}^{\infty}e^{(-\lambda_{1}+\gamma)(\beta-1)s-\frac{\rho^{2}_{1}}{2}s^{2H}}\mathbb{E}\mathopen{}\mathclose{{}\left(e^{\rho_{1}B^{H}(s)}}\right)ds$
$\displaystyle=\int_{0}^{\infty}e^{(-\lambda_{1}+\gamma)(\mu-1)s}ds=\frac{1}{(-\lambda_{1}+\gamma)(\mu-1)}<\infty,$
(5.18)

and

$\displaystyle
D_{\theta}X_{t}=D_{\theta}\mathopen{}\mathclose{{}\left(-\frac{\rho_{1}}{2}t^{2H}+B^{H}(t)}\right)=D_{\theta}B^{H}(t)=K_{H}(t,\theta),\
\theta\leq t.$ (5.19)

Moreover, we have

$\displaystyle\sup_{s\in[0,t]}\int_{0}^{s}\mathopen{}\mathclose{{}\left|D_{\theta}X_{s}}\right|^{2}d\theta$
$\displaystyle=\sup_{s\in[0,t]}\int_{0}^{s}K^{2}_{H}(s,\theta)d\theta=\sup_{s\in[0,t]}\mathbb{E}\mathopen{}\mathclose{{}\left|B^{H}(s)}\right|^{2}=t^{2H}.$
(5.20)

Now we take $f(t)=t^{\alpha}$ with $\alpha>H.$ Therefore the stochastic
process $\\{X_{t}\\}_{t\geq 0}$ satisfies the conditions of Theorem 5.2.
Moreover,

$\displaystyle M_{H}:=\sup_{t\geq
0}t^{2H}\mathopen{}\mathclose{{}\left\\{\log\mathopen{}\mathclose{{}\left(\frac{J^{1-\mu}(0)}{(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}+1}\right)+t^{\alpha}}\right\\}^{-2},$

and by a simple calculation, we get

$\displaystyle
M_{H}=\mathopen{}\mathclose{{}\left(\frac{\alpha-H}{\alpha}}\right)^{2-\frac{2H}{\alpha}}\mathopen{}\mathclose{{}\left(\log\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}+1}\right)}\right)^{\frac{2H}{\alpha}-2}.$

Then by Theorem 5.2, we have

$\displaystyle\mathbb{P}\Bigg{(}\int_{0}^{\infty}e^{\rho_{1}B^{H}(s)-\frac{\rho^{2}_{1}}{2}s^{2H}+(-\lambda_{1}+\gamma)(\mu-1)s}ds<\frac{J^{1-\mu}(0)}{(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\Bigg{)}$
$\displaystyle\leq\exp\Bigg{\\{}-\frac{1}{2\rho^{2}_{1}}\mathopen{}\mathclose{{}\left(\log\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}+1}\right)}\right)^{\frac{2H}{\alpha}-2}$
$\displaystyle\qquad\qquad\times\mathopen{}\mathclose{{}\left(\frac{\alpha-H}{\alpha}}\right)^{2-\frac{2H}{\alpha}}\mathopen{}\mathclose{{}\left(N(H)-1}\right)^{2}\Bigg{\\}}.$

Therefore, we obtain

$\displaystyle\mathbb{P}(\tau<\infty)=1-\mathbb{P}(\tau=\infty)$
$\displaystyle\geq
1-\exp\Bigg{\\{}-\frac{1}{2\rho^{2}_{1}}\mathopen{}\mathclose{{}\left(\log\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}+1}\right)}\right)^{\frac{2H}{\alpha}-2}$
$\displaystyle\qquad\qquad\times\mathopen{}\mathclose{{}\left(\frac{\alpha-H}{\alpha}}\right)^{2-\frac{2H}{\alpha}}\mathopen{}\mathclose{{}\left(N(H)-1}\right)^{2}\Bigg{\\}},$

where $\rho_{1}=\eta(\mu-1)$ and $N(H)$ is defined in (5.17), which completes
the proof of theorem. ∎

######  Remark 5.1.

For the case $m+n>q$, we have the blow-up time given by (5.1) and the
probability distribution for estimating the probability of blow-up solution
for the case of fractional Brownian motion is not known and it will be a part
of our future work.

##  6 For the case $H=\frac{1}{2}$

In this section, we consider the case $H=\frac{1}{2}$ and
$B^{\frac{1}{2}}(\cdot)=W(\cdot),$ where $W(\cdot)$ is a standard one-
dimensional Brownian motion. We are devoted to establish the lower and upper
bounds for the probability of the blow-up solution to the equation (1.4) by
using the methods developed in [17] and [19]. By using the random
transformation

$v(t,x)=\exp\\{-\eta W(t)\\}u(t,x),$

for $t\geq 0,\ x\in D$, the equation (1.4) is transformed into the following
random PDE:

$\mathopen{}\mathclose{{}\left\\{\begin{aligned} \frac{\partial
v(t,x)}{\partial
t}&=\mathopen{}\mathclose{{}\left(\Delta+\gamma-\frac{\eta^{2}}{2}}\right)v(t,x)+e^{\mathopen{}\mathclose{{}\left(q-1}\right)\eta
W(t)}\int_{D}v^{q}(t,x)dx-ke^{\mathopen{}\mathclose{{}\left(p-1}\right)\eta
W(t)}v^{p}(t,x)\\\ &\qquad+\delta e^{(m+n-1)\eta
W(t)}v^{m}(t,x)\int_{D}v^{n}(t,x)dx,\\\ v(t,x)&=0,\ \ x\in D,\\\
v(0,x)&=f(x),\ \ x\in D.\end{aligned}}\right.$ (6.1)

Taking $\Lambda=\frac{\eta^{2}}{2}-\gamma,$ the results obtained in previous
sections are valid by replacing the constant $\gamma$ by $-\Lambda.$ Next, we
recall that $X(\alpha,\beta_{1})$ is said to be a _Gamma random variable_ with
parameters $\alpha>0,\beta_{1}>0$ if its density function
$\widetilde{f}_{\alpha,\beta_{1}}(x)$ is given by (cf. [30])

$\widetilde{f}(x)=\mathopen{}\mathclose{{}\left\\{\begin{aligned}
&\frac{x^{\alpha-1}}{\beta_{1}^{\alpha}\varGamma(\alpha)}\exp\mathopen{}\mathclose{{}\left\\{-\frac{x}{\beta_{1}}}\right\\},\
x\geq 0,\\\ &\hskip 21.68121pt0,\hskip 65.04256pt\ x<0.\end{aligned}}\right.$
(6.2)

The following lemma is very useful to estimate the probability of blow-up of
positive solutions $v(\cdot,\cdot)$ of the equation (6.1).

######  Lemma 6.1.

([33], [40], [44, Chapter 6, Corollary 1.2]) For any $\alpha>0,$ the
exponential functional $\displaystyle\int_{0}^{\infty}e^{2(W(t)-\alpha t)}dt$
is distributed as $(2X(\alpha,1))^{-1},$ where $W(\cdot)$ is a one-dimensional
standard Brownian motion.

The following theorem gives the lower bounds for the finite-time blow-up of
the solution $u(\cdot,\cdot)$ of the equation (6.1).

######  Theorem 6.1.

Assume that $p,q,n>1,$ $m\geq 0$ with $m+n\geq q$. If $f$ is a non-negative
bounded function and $|D|>k$, then $\sigma_{\ast}\leq\tau$, where
$\sigma_{\ast}$ is given by

$\displaystyle\sigma_{\ast}=\inf\Bigg{\\{}t\geq
0:\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\eta\mathopen{}\mathclose{{}\left(q-1}\right)W(r)}\vee
e^{\eta(m+n-1)W(r)}}\right)e^{-\Lambda(m+n-1)r}dr\geq\frac{1}{2M(m+n-1)\|f\|_{\infty}^{m+n-1}}\Bigg{\\}},$

where $M=\max\mathopen{}\mathclose{{}\left\\{|D|,\delta|D|}\right\\}.$

The proof of Theorem 6.1 follows from Theorem 4.1.
Let us denote

$\displaystyle W_{\ast}(t)=\sup_{0\leq s\leq t}|W(s)|,\ \mbox{for each}\ t>0.$

From [26, p.96], we infer that for any $A>0$ and $t>0$

$\displaystyle\mathbb{P}(W_{\ast}(t)\geq
A)\leq\frac{4\sqrt{t}}{A\sqrt{2\pi}}e^{-\frac{A^{2}}{2t}}.$

Hence

$\mathbb{P}(W_{\ast}(t)<\infty)=1-\lim_{A\rightarrow\infty}\mathbb{P}(W_{\ast}(t)>A)=1,\
t>0.$

Let us denote $U_{t}=\\{\omega\in\Omega:W_{\ast}(t)=\infty\\},$ for every
$t>0.$ Then, we have $\mathbb{P}(U_{t})=0,$ for every $t>0.$ If we take
$t=1,2,\ldots,$ it is clear that $U_{t}\subset U_{m},$ for $t\geq m.$ Define
$U=\displaystyle\lim_{m\rightarrow\infty}U_{m},$ we have
$P(U)=\displaystyle\lim_{m\to\infty}\mathbb{P}(U_{m})=0.$ Therefore, for all
$\omega\in U,$ $W_{\ast}(t;\omega)<\infty,$ for all $t>0.$ From the above
observation, without loss of generality, we can assume that for all
$\omega\in\Omega,$

$\displaystyle W_{\ast}(t;\omega)<\infty,\ \mbox{for all}\ t>0.$

Let $N>0$ be a constant and define the stopping time as follows:

$\tau_{N}(\omega)=\inf\mathopen{}\mathclose{{}\left\\{t>0:\ |W(t;\omega)|\geq
N}\right\\}.$

Clearly,

$\mathopen{}\mathclose{{}\left\\{\omega\in\Omega:\ \tau_{N}(\omega)\leq
t}\right\\}=\mathopen{}\mathclose{{}\left\\{\omega\in\Omega:\ W_{\ast}(t)\geq
N}\right\\}.$

Let $t\in(0,\infty)$ be some random number and $b>1$ be chosen so that

$\mathopen{}\mathclose{{}\left.\begin{aligned}
b^{q-p}e^{-\eta(m+n-1)W_{\ast}(t)}\geq\frac{kC_{1}^{p}+(\lambda_{1}+\Lambda)C_{1}}{|D|^{1-q}},\
\ &b^{q-p}e^{-\eta(q-p)W_{\ast}(t)}\geq\frac{kC_{1}^{p}}{|D|^{1-q}}\\\ \mbox{
and }\
b^{q-p}e^{-\eta(q-1)W_{\ast}(t)}&\geq\frac{2k\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)dx}\right)^{\frac{q-p}{p}}}{\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{p+1}(x)dx}\right)^{\frac{q-p}{p}}}\end{aligned}}\right\\},$
(6.3)

where $C_{1}=\displaystyle\sup_{x\in D}\varphi(x)$. From (5.7), we have

$\displaystyle I_{1}(t,x)$ $\displaystyle\geq
b\Big{[}e^{-\eta\mathopen{}\mathclose{{}\left(m+n-1}\right)W_{\ast}(t)}b^{q-p}|D|^{1-q}-kC_{1}^{p}\Big{]}$

and the conditions (6.3) leads to $I_{1}(t,x)\geq
b(\lambda_{1}+\Lambda)\varphi(x),$ for each $x\in D$ and $t\geq 0.$ Note that

$\displaystyle\frac{\partial\bar{v}(t,x)}{\partial
t}-\Delta\bar{v}(t,x)=\lambda_{1}b\varphi(x)\leq
I_{1}(t,x)-\Lambda\bar{v}(t,x).$

Hence from (6.1) and by comparison principle in Proposition 3.1, we obtain
$v(t,x)\geq\bar{v}(t,x)=b\varphi(x),$ for each $x\in D$ and $t\geq 0.$

The following theorem provides upper bounds for the finite-time blow-up.
Further, we estimate a lower bound for the probability of blow-up solution of
the random PDE (1.4) when $H=\frac{1}{2}$ and
$\Lambda=\frac{\eta^{2}}{2}-\gamma.$

######  Theorem 6.2.

Suppose that $n,p,q>1,\ m\geq 0$ with $q\geq p$ and for given initial value
$f(x)\geq b\varphi(x),\ x\in D$ and choose $b>1$ such that (6.3) holds. Then,
we have the following results:

If $m+n=q=\mu\ (say).$ Then $\tau\leq\sigma^{\ast}_{1},$ where
$\sigma^{\ast}_{1}$ is given by

$\displaystyle\sigma^{\ast}_{1}=$ $\displaystyle\inf\Biggl{\\{}t\geq
0:\int_{0}^{t}e^{\rho_{1}W(s)-(\lambda_{1}+\Lambda)(\mu-1)s}ds$
$\displaystyle\qquad\quad\geq
J^{1-\mu}(0)\mathopen{}\mathclose{{}\left[(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\right]^{-1}\Biggr{\\}}.$

Moreover, there exists $\theta_{1}>0$ such that a lower bound for the
probability of blow-up solution of the equation (6.1) is given by

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left(\tau<\infty}\right)\geq\mathbb{P}\mathopen{}\mathclose{{}\left(\sigma^{\ast}_{1}<\infty}\right)$
$\displaystyle=1-\mathbb{P}\Bigg{(}2(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)<X(\theta_{1},1)\rho^{2}_{1}J^{1-\mu}(0)\Bigg{)},$

where
$\theta_{1}=\displaystyle\frac{2(\lambda_{1}+\Lambda)(\mu-1)}{\rho^{2}_{1}}$
and $\rho_{1}=\eta(\mu-1).$

If $m+n>q>1$, let
$A_{0}=\mathopen{}\mathclose{{}\left(\frac{m+n-q}{m+n}}\right)\mathopen{}\mathclose{{}\left(\frac{m+n}{q}}\right)^{\frac{q}{m+n-q}}$
and
$\epsilon_{0}\leq\mathopen{}\mathclose{{}\left(J^{q}(0)/A_{0}}\right)^{\frac{q}{m+n-q}}$.
Then $\tau\leq\sigma^{\ast\ast}_{2},$ where $\sigma^{\ast\ast}_{2}$ is given
by

$\displaystyle\sigma^{\ast\ast}_{2}=\inf\Biggl{\\{}t\geq
0:\int_{0}^{t}e^{-(\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s)}\mathbf{1}_{\\{W(s)\geq
0\\}}ds\geq a_{1}\Biggr{\\}}.$

Moreover, a lower bound for the probability of blow-up solution of the
equation (6.1) is given by

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left(\tau<\infty}\right)\geq\mathbb{P}\mathopen{}\mathclose{{}\left(\sigma^{\ast\ast}_{2}<\infty}\right)=\frac{8(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}\sum_{n\geq
1}\frac{\exp\Bigg{\\{}-\frac{(\eta(q-1))^{2}a_{1}}{8}j^{2}_{\frac{2(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}-1,n}\Bigg{\\}}}{j^{2}_{\frac{2(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}-1,n}},$

where
$a_{1}=2{J^{1-q}(0)}\Bigg{[}(q-1)(-\lambda_{1}+\gamma)\mathopen{}\mathclose{{}\left(\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{q}{q-p}}(x)}\right)^{\frac{p-q}{p}}+\delta\mathopen{}\mathclose{{}\left(\epsilon_{0}-\frac{A_{0}\epsilon_{0}^{\frac{m+n}{m+n-q}}}{J^{q}(0)}}\right)\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)\Bigg{]}^{-1}$,
$\Big{\\{}j_{\frac{2(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}-1,n}\Big{\\}}_{n\geq
1}$ is an increasing sequence of all positive zeros of the Bessel function of
the first kind of order
$\frac{2(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}-1>-1$ (see more details
in [19]) and $J(0)=\displaystyle\int_{D}f(x)\varphi(x)dx.$

###### Proof.

Case 1: If $m+n=q=\mu\ \mbox{(say)}$ by proceeding in the same way as in the
case 1 of Theorem 5.1 and by using (6.1), we obtain

$\displaystyle\sigma^{\ast}_{1}=$ $\displaystyle\inf\Biggl{\\{}t\geq
0:\int_{0}^{t}e^{\rho_{1}W(s)-(\lambda_{1}+\Lambda)(\mu-1)s}ds$
$\displaystyle\qquad\quad\geq
J^{1-\mu}(0)\mathopen{}\mathclose{{}\left[(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\right]^{-1}\Biggr{\\}},$

where $\rho_{1}=\eta(\mu-1).$ From the definition of $\sigma^{\ast}_{1}$, we
infer

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left(\sigma^{\ast}_{1}=\infty}\right)$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{\infty}e^{\rho_{1}W(s)-(\lambda_{1}+\Lambda)(\mu-1)s}ds<\frac{J^{1-\mu}(0)}{(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{\infty}e^{2W\mathopen{}\mathclose{{}\left(\frac{\rho^{2}_{1}}{4}s}\right)-(\lambda_{1}+\Lambda)(\mu-1)s}ds<\frac{J^{1-\mu}(0)}{(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\Bigg{)}.$

By performing the transformation $s\mapsto\frac{4t}{\rho^{2}_{1}},$ we get

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left(\sigma^{\ast}_{1}=\infty}\right)$
$\displaystyle=\mathbb{P}\Bigg{(}\frac{4}{\rho^{2}_{1}}\int_{0}^{\infty}e^{2W(t)-(\lambda_{1}+\Lambda)(\mu-1)\frac{4t}{\rho^{2}_{1}}}dt<\frac{J^{1-\mu}(0)}{(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}\frac{4}{\rho^{2}_{1}}\int_{0}^{\infty}e^{2(W(t)-\theta_{1}t)}dt<\frac{J^{1-\mu}(0)}{(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\displaystyle\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\Bigg{)},$

where
$\theta_{1}=\displaystyle\frac{2(\lambda_{1}+\Lambda)(\mu-1)}{\rho^{2}_{1}}.$
By Lemma 6.1, we have

$\displaystyle\mathbb{P}\mathopen{}\mathclose{{}\left(\sigma^{\ast}_{1}=\infty}\right)$
$\displaystyle=\mathbb{P}\Bigg{(}\frac{1}{X(\theta_{1},1)}<\frac{\rho^{2}_{1}J^{1-\mu}(0)}{2(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)}\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}2(\mu-1)\mathopen{}\mathclose{{}\left(\frac{1}{2}\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{\mu}{\mu-p}}(x)dx}\right)^{\frac{p-\mu}{p}}+\delta\mathopen{}\mathclose{{}\left(\int_{D}\varphi^{\frac{n}{n-1}}(x)dx}\right)^{1-n}}\right)<X(\theta_{1},1)\rho^{2}_{1}J^{1-\mu}(0)\Bigg{)}.$

Case 2: If $m+n>q>1,$ by proceeding in the same way as in the case 2 of
Theorem 5.1 and by using (6.1), we obtain

$\displaystyle\sigma^{\ast}_{2}=\inf\Bigg{\\{}t\geq
0:\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\eta(q-1)W(s)}\wedge
e^{\eta(m+n-1)W(s)}}\right)e^{-(\lambda_{1}+\Lambda)(q-1)s}ds\geq
a_{1}\Bigg{\\}}.$

Note that $m+n-1>q-1>0$ and so

$\displaystyle\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\eta(q-1)W(s)}\wedge
e^{\eta(m+n-1)W(s)}}\right)e^{-(\lambda_{1}+\Lambda)(q-1)s}ds$
$\displaystyle=\int_{0}^{t}e^{\eta(m+n-1)W(s)-(\lambda+\Lambda)(q-1)s}\mathbf{1}_{\\{W(s)<0\\}}ds+\int_{0}^{t}e^{\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s}\mathbf{1}_{\\{W(s)\geq
0\\}}ds$
$\displaystyle\geq\int_{0}^{t}e^{\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s}\mathbf{1}_{\\{W(s)\geq
0\\}}ds,$

and

$\mathopen{}\mathclose{{}\left\\{\eta(q-1)W(t)-(\lambda_{1}+\Lambda)(q-1)t\geq
0}\right\\}\subseteq\mathopen{}\mathclose{{}\left\\{W(t)\geq 0}\right\\},$

for all $t\geq 0$. We conclude that

$\displaystyle\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\eta(q-1)W(s)}\wedge
e^{\eta(m+n-1)W(s)}}\right)e^{-(\lambda_{1}+\Lambda)(q-1)s}ds$
$\displaystyle\geq\int_{0}^{t}e^{-(\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s)}\mathbf{1}_{\\{\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s\geq
0\\}}ds.$

Therefore, we define

$\displaystyle\sigma^{\ast\ast}_{2}=\inf\Biggl{\\{}t\geq
0:\int_{0}^{t}e^{-(\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s)}\mathbf{1}_{\\{\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s\geq
0\\}}ds\geq a_{1}\Biggr{\\}}.$

It can be easily seen that
$\tau\leq\sigma^{\ast}_{2}\leq\sigma^{\ast\ast}_{2}.$ From the definition of
$\sigma^{\ast\ast}_{2}$ and [19, Theorem 2.6], we have

$\displaystyle\mathbb{P}\\{\tau<\infty\\}$
$\displaystyle\geq\mathbb{P}\\{\sigma^{\ast\ast}_{2}<\infty\\}$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{\infty}e^{-(\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s)}\mathbf{1}_{\\{\eta(q-1)W(s)-(\lambda_{1}+\Lambda)(q-1)s\geq
0\\}}ds\geq a_{1}\Bigg{)}$
$\displaystyle=\int^{\infty}_{a_{1}}(\lambda_{1}+\Lambda)(q-1)\sum_{n\geq
1}\exp\Bigg{\\{}-\mathopen{}\mathclose{{}\left(\frac{(\eta(q-1))^{2}}{8}j^{2}_{\frac{2(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}-1,n}}\right)y\Bigg{\\}}dy$
$\displaystyle=\frac{8(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}\sum_{n\geq
1}\frac{\exp\Bigg{\\{}-\frac{(\eta(q-1))^{2}a_{1}}{8}j^{2}_{\frac{2(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}-1,n}\Bigg{\\}}}{j^{2}_{\frac{2(\lambda_{1}+\Lambda)(q-1)}{(\eta(q-1))^{2}}-1,n}},$

where we have used the Monotone Convergence Theorem to obtain the final
equality, which completes the proof of the theorem. ∎

The following theorem provides an upper bound for the probability of blow-up
solution $v(\cdot,\cdot)$ of the equation (6.1).

######  Theorem 6.3.

Assume that $p,q,n>1,$ $m\geq 0$ with $m+n\geq q$. If $f$ is a non-negative
bounded function and $|D|>k$, then an upper bound for the probability of blow-
up solution of the equation (6.1) is given by

$\displaystyle\mathbb{P}\\{\tau<\infty\\}\leq\int_{\widetilde{N}_{1}}^{\infty}h_{3}(y)dy,$

where

$\displaystyle\widetilde{N}_{1}$
$\displaystyle=\frac{1}{2M(m+n-1)\|f\|_{\infty}^{m+n-1}}-\frac{1}{(\lambda+\Lambda)(q-1)},\
\mbox{here}\ M=\max\mathopen{}\mathclose{{}\left\\{|D|,\delta|D|}\right\\}\
\mbox{and}\ $ $\displaystyle h_{3}(y)$
$\displaystyle=\frac{(2/(\eta(q-1))^{2}y)^{(2\Lambda(m+n-1)/(\eta(q-1))^{2})}}{y\Gamma(2\Lambda(m+n-1)/(\eta(q-1))^{2})}\exp\mathopen{}\mathclose{{}\left(-\frac{2}{(\eta(q-1))^{2}y}}\right).\
$

###### Proof.

By using the same procedure as in Theorem 6.1, we have
$\sigma_{\ast}\leq\tau,$ where $\sigma_{\ast}$ is given by

$\displaystyle\sigma_{\ast}=\inf\Bigg{\\{}t\geq
0:\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\eta\mathopen{}\mathclose{{}\left(q-1}\right)W(r)}\vee
e^{\eta(m+n-1)W(r)}}\right)e^{-\Lambda(m+n-1)r}dr\geq\frac{1}{2M(m+n-1)\|f\|_{\infty}^{m+n-1}}\Bigg{\\}},$

Note that $1<q\leq m+n$ and so

$\displaystyle\int_{0}^{t}\mathopen{}\mathclose{{}\left(e^{\eta(q-1)W(s)}\wedge
e^{\eta(m+n-1)W(s)}}\right)e^{-\Lambda(m+n-1)s}ds$
$\displaystyle=\int_{0}^{t}e^{\eta(m+n-1)W(s)-\Lambda(m+n-1)s}\mathbf{1}_{\\{W(s)<0\\}}ds+\int_{0}^{t}e^{\eta(q-1)W(s)-\Lambda(m+n-1)s}\mathbf{1}_{\\{W(s)\geq
0\\}}ds$
$\displaystyle\leq\int_{0}^{\infty}e^{-\Lambda(m+n-1)s}ds+\int_{0}^{t}e^{\eta(q-1)W(s)-\Lambda(m+n-1)s}ds$
$\displaystyle=\frac{1}{\Lambda(m+n-1)}+\int_{0}^{t}e^{\eta(q-1)W(s)-\Lambda(m+n-1)s}ds,$
(6.4)

for all $t\geq 0$. Therefore, we define

$\displaystyle\sigma_{\ast\ast}=\inf\Bigg{\\{}t\geq
0:\int_{0}^{t}e^{\eta(q-1)W(s)-\Lambda(m+n-1)s}ds$
$\displaystyle\geq\frac{1}{2M(m+n-1)\|f\|_{\infty}^{m+n-1}}-\frac{1}{\Lambda(m+n-1)}\Bigg{\\}},$

and $\sigma_{\ast\ast}\leq\sigma_{\ast}\leq\tau$. Using the definition of
$\sigma_{\ast\ast}$, we have

$\displaystyle\mathbb{P}\\{\tau<\infty\\}\leq\mathbb{P}(\sigma_{\ast\ast}<\infty)$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{\infty}e^{\eta(q-1)W(s)-\Lambda(m+n-1)s}ds\geq\widetilde{N}_{1}\Bigg{)}$
$\displaystyle=\mathbb{P}\Bigg{(}\int_{0}^{\infty}\exp\\{2W_{s}^{(\hat{\alpha})}\\}ds\geq\widetilde{N}_{1}\Bigg{)},$

where $W_{s}^{(\hat{\alpha})}:=W(s)-\hat{\alpha}s$ and
$\hat{\alpha}=\frac{\Lambda(m+n-1)}{\eta(q-1)}.$ By performing the
transformation $s\mapsto\frac{4t}{(\eta(q-1))^{2}}$ and setting
$\nu_{1}=\frac{2\hat{\alpha}}{\eta(q-1)}$, we get

$\displaystyle\mathbb{P}\\{\tau<\infty\\}\leq\mathbb{P}\Bigg{(}\frac{4}{(\eta(q-1))^{2}}\int_{0}^{\infty}\exp\\{2W_{t}^{(\nu_{1})}\\}dt\geq\widetilde{N}_{1}\Bigg{)}.$

Therefore, we deduce

$\displaystyle\mathbb{P}\\{\tau<\infty\\}\leq\int_{\widetilde{N}_{1}}^{\infty}h_{3}(y)dy,$

which completes the proof. ∎

##  7 Appendix

In this appendix, we provide a comparison result which is used in Section 5
(see [42, Theorem 1.3] also). From (5), we have

$\displaystyle J^{\prime}(t)$
$\displaystyle\geq\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)J(t)+\mathit{\widetilde{N}}e^{\eta\mathopen{}\mathclose{{}\left(\mu-1}\right)B^{H}(t)}J^{\mu}(t),$

and

$\displaystyle\frac{dI(t)}{dt}=\mathopen{}\mathclose{{}\left(-\lambda_{1}+\gamma}\right)I(t)+\mathit{\widetilde{N}}e^{\eta\mathopen{}\mathclose{{}\left(\mu-1}\right)B^{H}(t)}I^{\mu}(t),\
I(0)=J(0).$

We have to show that $J(t)\geq I(t),$ for $t\in[0,\tau).$ Suppose that
$J(t)<I(t),$ for some $t\in[t_{0},t_{0}+\varepsilon),\ \varepsilon>0.$ Let
$\triangle(t)=I(t)-J(t).$ Then an application of Taylor’s formula yields

$\displaystyle\triangle^{\prime}(t)={I}^{\prime}(t)-{J}^{\prime}(t)$
$\displaystyle\leq(-\lambda_{1}+\gamma)\triangle(t)+\mathit{\widetilde{N}}e^{\mathopen{}\mathclose{{}\left(\mu-1}\right)\eta
B^{H}(t)}\mathopen{}\mathclose{{}\left(I^{\mu}(t)-J^{\mu}(t)}\right)$
$\displaystyle\leq(-\lambda_{1}+\gamma)\triangle(t)+\mu\kappa\mathit{\widetilde{N}}\triangle(t)\leq\kappa_{1}\triangle(t),$

where
$\kappa=\sup\limits_{t\in[0,\tau]}e^{\mathopen{}\mathclose{{}\left(\mu-1}\right)\eta
B^{H}(t)}\sup\limits_{t\in[0,\tau]}(|J(t)|+|I(t)|)^{\mu-1}$ and
$\kappa_{1}=\max\\{(-\lambda+\gamma),\mu\kappa\mathit{\widetilde{N}}\\}.$
Therefore, we obtain $\dot{\triangle}(t)-\kappa_{1}\triangle(t)\leq 0$ and
this implies that $e^{-\kappa_{1}t}\triangle(t)\leq 0,$ so that
$\triangle(t)\leq 0$ which is a contradiction, since $\triangle(t)>0,$ for all
$t\in[t_{0},t_{0}+\varepsilon)$. Hence $J(t)\geq I(t),$ for all
$t\in[0,\tau).$

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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