# On global solvability and regularity for generalized Rayleigh-Stokes
equations with history-dependent nonlinearities

Tran Dinh Ke & Nguyen Nhu Thang∗

###### Abstract.

We are concerned with the initial value problem governed by generalized
Rayleigh-Stokes equations, where the nonlinearity depends on history states
and takes values in Hilbert scales of negative order. The solvability and
Hölder regularity of solutions are proved by using fixed point arguments and
embeddings of fractional Sobolev spaces. An application to a related inverse
source problem is given.

###### Key words and phrases:

Rayleigh-Stokes problem; nonlocal PDE; well-posedness; Hölder continuity

###### 2010 Mathematics Subject Classification:

35B40,35R11,35C15,45D05,45K05

* Corresponding author. Email<EMAIL_ADDRESS>(N.N.Thang)

##  1\. Introduction

Let $\Omega\subset\mathbb{R}^{d}$ be a bounded domain with smooth boundary
$\partial\Omega$. We consider the following problem

$\displaystyle\partial_{t}u-(1+D_{t}^{\\{m\\}})\Delta u$
$\displaystyle=f(u,\mathcal{H}u)\text{ in }\Omega,t\in(0,T),$ (1.1)
$\displaystyle u$ $\displaystyle=0\text{ on }\partial\Omega,\;t\geq 0,$ (1.2)
$\displaystyle u(0)$ $\displaystyle=\xi\text{ in }\Omega,$ (1.3)

where $f$ is a given function, $\partial_{t}=\frac{\partial}{\partial t}$,
$D_{t}^{\\{m\\}}$ is the nonlocal derivative of Riemann-Liouville type defined
by

$D_{t}^{\\{m\\}}v(t)=\frac{d}{dt}\int_{0}^{t}m(t-s)v(s)ds,$

with kernel $m\in L^{1}_{loc}(\mathbb{R}^{+})$, $\mathcal{H}$ is the
convolution operator given by

$\mathcal{H}v(t)=\int_{0}^{t}\ell(t-\tau)v(\tau)d\tau,\;\ell\in L^{1}(0,T),$

which tells us that $f$ depends on the history state of the system.

The proposed system is a general model for some problems studied in
literature. Indeed, in the case $m$ is a constant, (1.1) is of classical
diffusion type. If $m$ is a regular function, e.g. $m\in
C^{1}(\mathbb{R}^{+})$ then one gets

$\partial_{t}u-(1+m_{0})\Delta u-\int_{0}^{t}m_{1}(t-s)\Delta u(s)ds=f,$

with $m_{0}=m(0)$ và $m_{1}(t)=m^{\prime}(t)$, which is a nonclassical
diffusion equation. Let $m(t)=m_{0}t^{-\alpha}/\Gamma(\alpha)$ with $m_{0}>0$,
then we see that (1.1) is a Rayleigh-Stokes equation, i.e.

$\partial_{t}u-(1+m_{0}\partial_{t}^{\alpha})\Delta u=f.$ (1.4)

The constitution of the last equation was formed in [10, 19] to describe the
behavior of second-grade fluids. It should be mentioned that, some numerical
schemes for Rayleigh-Stokes equations were developed in [1, 2, 5, 6]. On the
other hand, analytical representations for solution of (1.4) in linear case
were obtained in [11, 19], and recently, the regularity for nonlinear
Rayleigh-Stokes equations has been established in [13, 14, 21]. For more
studies related to (1.4), we refer the reader to [15, 17, 20], where the
terminal value problem was carried out.

In this work, we are interested in the solvability and regularity analysis for
problem (1.1)-(1.3) in the circumstance that the nonlinearity function
involves the history state expressed by $\mathcal{H}$. The appearance of
$\mathcal{H}u$ comes from, e.g. the inverse source problem as presented in the
last section. In another way, this factor may arise from control problems,
where the feedback requires some history information of the system. One will
find that, the term $\mathcal{H}u$ causes some difficulties for analyzing
regularity of solutions. As an important feature of our study, it is noted
that the nonlinearity function $f$ is allowed to take weak values, i.e.
$f(u,\mathcal{H}u)$ may belong to dual of fractional Sobolev spaces. This
enables us to consider the case when $f$ contains polymonial or gradient
terms, which have connections with practical applications. This also extends
the recent results established in [12, 13, 14].

The rest of this note is as follows. In the next section, we first recall some
notions and facts related to Hilbert scales and fractional Sobolev spaces.
Additionally, a representation for the solution of linear problem will be
shown together with some essential estimates for resolvent operator. Section 3
is devoted to proving the main results, including the global existence and
Hölder regularity of solutions. It is worth noting, in particular, that the
Hölder regularity result in our work can not be obtained by the technique used
in [12], since the differentiability of resolvent operator is unavailable on
dual spaces. In order to overcome this impediment, we construct a regular
closed subset of solution space, which is invariant under the solution
operator, and make use of fixed point arguments. The last section shows an
application of the obtained results, where we demonstrate that an inverse
source problem governed by Rayleigh-Stokes equations is solvable by
transforming it to a prototype of problem (1.1)-(1.3).

##  2\. Preliminaries

###  2.1. Functional spaces

Let $-\Delta$ be the Laplacian associated with the homogenuous Dirichlet
boundary condition. Then one has a sequence of eigenfunctions $\\{e_{n}\\}$ of
$-\Delta$ which forms an orthonormal basis of $L^{2}(\Omega)$, and we have the
following representation

$-\Delta
v=\sum_{n=1}^{\infty}\left(\lambda_{n}\int\limits_{\Omega}v(x)e_{n}(x)dx\right)e_{n},$

where $\lambda_{n}>0$ is the eigenvalue corresponding to the eigenfunction
$e_{n}$. For $\varrho\geq 0$, we define the following functional space

$\mathbb{H}^{\varrho}:=\left\\{\varphi\in
L^{2}(\Omega)\bigg{|}\|\varphi\|^{2}_{\mathbb{H}^{\varrho}}:=\sum_{n=1}^{\infty}\left|\lambda_{n}^{\frac{\varrho}{2}}\int\limits_{\Omega}\varphi(x)e_{n}(x)dx\right|^{2}<\infty\right\\}.$

Let $\mathbb{H}^{-\varrho}$ denote the dual space of $\mathbb{H}^{\varrho}$
with respect to the dual pair $\langle\cdot,\cdot\rangle_{-\varrho,\varrho}$
on $\mathbb{H}^{-\varrho}\times\mathbb{H}^{\varrho}$ with the norm

$\|\varphi\|^{2}_{\mathbb{H}^{-\varrho}}:=\sum_{n=1}^{\infty}\left|\lambda_{n}^{-\frac{\varrho}{2}}\langle\varphi,e_{n}\rangle_{-\varrho,\varrho}\right|^{2}<\infty.$

Clearly, $\mathbb{H}^{\varrho_{2}}\hookrightarrow\mathbb{H}^{\varrho_{1}}$ and
$\mathbb{H}^{-\varrho_{1}}\hookrightarrow\mathbb{H}^{-\varrho_{2}}$ with
$\varrho_{2}\geq\varrho_{1}\geq 0$. The family of Hilbert spaces
$\mathbb{H}^{\varrho},\varrho\in\mathbb{R}$, is said to be the Hilbert scales.

The fractional Laplacian $(-\Delta)^{\gamma}$, $\gamma\geq 0$, is defined as
follows

$\displaystyle(-\Delta)^{\gamma}:\;\mathbb{H}^{\gamma}$
$\displaystyle\to\mathbb{H}^{-\gamma};$
$\displaystyle(-\Delta)^{\gamma}\varphi$
$\displaystyle=\sum_{n=1}^{\infty}\lambda_{n}^{\gamma}\left(\int\limits_{\Omega}\varphi(x)e_{n}(x)dx\right)e_{n}.$

Then,
$\|(-\Delta)^{\gamma}\varphi\|_{\mathbb{H}^{-\gamma}}=\|\varphi\|_{\mathbb{H}^{\gamma}}$.

We now recall the notion of the fractional Sobolev spaces (see, e.g. [8, 9]
for details). For $r\in(0,1)$ and $p\in[1,+\infty)$, the functional space

$W^{r,p}(\Omega):=\left\\{u\in
L^{p}(\Omega):\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{d+pr}}dxdy<\infty\right\\}$

is called the fractional Sobolev space with the norm

$\|u\|_{W^{r,p}(\Omega)}=\left(\int_{\Omega}|u|^{p}dx+\int_{\Omega}\int_{\Omega}\frac{|u(x)-u(y)|^{p}}{|x-y|^{d+pr}}dxdy\right)^{1/p}$

For $r\geq 1$, denote by $[r]$ the integral part of $r$, we define

$W^{r,p}(\Omega):=\left\\{u\in W^{[r],p}(\Omega):D^{\eta}u\in
W^{r-[r],p}(\Omega),\forall|\eta|\leq[r]\right\\}$

which is a Banach space with the norm

$\|u\|_{W^{r,p}(\Omega)}=\left(\|u\|_{W^{[r],p}}^{p}(\Omega)+\sum_{|\eta|=[r]}\int_{\Omega}\int_{\Omega}\frac{\left|D^{\eta}u(x)-D^{\eta}u(y)\right|^{p}}{|x-y|^{d+p(r-[r])}}dxdy\right)^{1/p}.$

Denote

$W_{0}^{r,p}(\Omega):=\overline{C_{c}^{\infty}(\Omega)}^{W^{r,p}(\Omega)},H^{r}(\Omega):=W^{r,2}(\Omega),H_{0}^{r}(\Omega):=W_{0}^{r,2}(\Omega).$

Assume that $\Omega$ is a domain with sufficiently smooth boundary such that
$C_{c}^{\infty}(\Omega)$ is dense in $H^{r}(\Omega)$ with $0<r<1/2$, then
$H_{0}^{r}(\Omega)=H^{r}(\Omega)$ (see [3, Corollary 8.10.1]). It follows from
[4] that

$\mathbb{H}^{r}=\begin{cases}H_{0}^{r}(\Omega),&0\leq r<1/2,\\\
H_{00}^{1/2}(\Omega)\varsubsetneqq H_{0}^{1/2}(\Omega),&r=1/2,\\\
H_{0}^{r}(\Omega),&1/2<r\leq 1,\\\ H_{0}^{1}(\Omega)\cap
H^{r}(\Omega),&1<r\leq 2,\end{cases}$ (2.1)

where $H_{00}^{1/2}(\Omega)$ is the Lions-Magenes space determined by

$H_{00}^{1/2}(\Omega)=\left\\{u\in
H^{1/2}(\Omega):\int_{\Omega}\frac{|u(x)|^{2}}{(\operatorname{dist}(x,\partial\Omega))^{2}}dx<\infty\right\\}.$

Then the following lemma is a direct consequence of (2.1).

######  Lemma 2.1.

Denote by $H^{-r}(\Omega)$ the dual space of $H_{0}^{r}(\Omega)$ with $r\geq
0$. If $0\leq r\leq r^{\prime}\leq 2$, then

$\mathbb{H}^{r^{\prime}}\hookrightarrow\mathbb{H}^{r}\hookrightarrow
H^{r}(\Omega)\hookrightarrow L^{2}(\Omega)\hookrightarrow
H^{-r}(\Omega)\hookrightarrow\mathbb{H}^{-r}\hookrightarrow\mathbb{H}^{-r^{\prime}}.$

The following lemma represents embeddings between fractional Sobolev spaces.

######  Lemma 2.2.

[3, Theorem 8.12.6] Let $1\leq p,p^{\prime}\leq\infty,0\leq
r,r^{\prime}<\infty$ and $r^{\prime}-\frac{d}{p^{\prime}}\geq r-\frac{d}{p}$.
Then,

$W^{r^{\prime},p^{\prime}}(\Omega)\hookrightarrow W^{r,p}(\Omega).$

Using Lemma 2.1 and 2.2, we have the following embeddings.

######  Lemma 2.3.

We have

  1. a)

$L^{p}(\Omega)\hookrightarrow H^{r}(\Omega)\hookrightarrow\mathbb{H}^{r}$ if
$\left\\{-\frac{d}{2}<r\leq 0,p\geq\frac{2d}{d-2r}\right\\}$.

  2. b)

$\mathbb{H}^{r}\hookrightarrow H^{r}(\Omega)\hookrightarrow L^{p}(\Omega)$ if
$\left\\{0\leq r<\frac{d}{2},1\leq p\leq\frac{2d}{d-2r}\right\\}.$

###  2.2. Representation of solutions to linear problem

In order to investigate problem (1.1)-(1.3), we assume the following
hypothesis:

  * (M)

The function $m\in L^{1}_{loc}(\mathbb{R}^{+})$ is nonnegative such that
$a(t):=1+m(t)$ is completely positive.

Recall that the real valued function $a$ is said to be completely positive if
the solutions to the following integral equations

$\displaystyle s(t)+\theta\int_{0}^{t}a(t-\tau)s(\tau)d\tau=1,\;t\geq 0,$
(2.2) $\displaystyle r(t)+\theta\int_{0}^{t}a(t-\tau)r(\tau)d\tau=a(t),\;t>0,$
(2.3)

are nonnegative for each $\theta>0$. The theory of completely positive
functions can be found in [7, 18]. An equivalent condition for $a$ to be
completely positive is as follows:

  1. (PC)

There exists a nonincreasing and nonnegative function $k\in
L^{1}_{loc}(\mathbb{R}^{+})$ and $\epsilon\geq 0$ such that

$\epsilon a+k*a=1\text{ on }(0,\infty).$

In the present work, we assume that $m$ is unbounded on $\mathbb{R}^{+}$,
which implies $\epsilon=0$. The condition (PC) is satisfied if $m$ is
completely monotone, i.e. $(-1)^{n}m^{(n)}(t)\geq 0$ on $(0,\infty)$, for all
$n\in\mathbb{N}$ (see [16]).

First we consider the relaxation problem:

$\displaystyle\omega^{\prime}(t)+\lambda(1+D_{t}^{\\{m\\}})\omega(t)$
$\displaystyle=0,\;t>0,$ (2.4) $\displaystyle\omega(0)$ $\displaystyle=1,$
(2.5)

where the unknown $\omega$ is a scalar function, $\lambda$ is a positive
parameter. Integrating both sides of (2.4) over $(0,t)$, we obtain

$\displaystyle\omega(t)+\lambda\int_{0}^{t}(1+m(t-\tau))\omega(\tau)d\tau=1,$
(2.6)

which is just (2.2) with $\lambda=\theta$.

We denote by $\omega(t,\lambda)$ the solution of (2.6) to emphasize the
dependence of $\omega$ on the parameter $\lambda$. We list some properties of
$\omega$ in the following proposition.

######  Proposition 2.4.

Let $\omega$ be the solution to (2.4)-(2.5). Then

  1. (a)

$\omega$ is nonincreasing on $\mathbb{R}^{+}$ and

$\displaystyle
0<\omega(t,\lambda)\leq\frac{1}{1+\lambda\int_{0}^{t}(1+m(\tau))d\tau},\;\forall
t\geq 0,\;\lambda>0.$

  2. (b)

The following estimate holds

$\displaystyle\int_{0}^{t}\omega(\tau,\lambda)d\tau\leq\lambda^{-1}(1-\omega(t,\lambda)),\;\forall
t\geq 0,\lambda>0.$

  3. (c)

For each $t>0$, the function $\lambda\mapsto\omega(t,\lambda)$ is
nonincreasing.

  4. (d)

The function
$v(t)=\omega(t,\lambda)v_{0}+\int_{0}^{t}\omega(t-\tau,\lambda)g(\tau)d\tau$
is a solution to the problem

$\displaystyle v^{\prime}(t)+\lambda(1+D^{\\{m\\}}_{t})v(t)$
$\displaystyle=g(t),$ $\displaystyle v(0)$ $\displaystyle=v_{0}.$

###### Proof.

The properties (a) and (b) are implied from (2.6). The properties (c) and (d)
were proved in [12]. ∎

Now we look for a representation of the solution to the following initial
value linear problem

$\displaystyle\partial_{t}u-(1+D_{t}^{\\{m\\}})\Delta u$
$\displaystyle=F\;\text{ in }\Omega,t\in(0,T],$ (2.7) $\displaystyle u$
$\displaystyle=0\;\text{ on }\partial\Omega,\;t\in[0,T],$ (2.8) $\displaystyle
u(0)$ $\displaystyle=\xi\;\text{ in }\Omega,$ (2.9)

where $F\in C([0,T];L^{2}(\Omega))$.

Assume that

$\displaystyle
u(t)=\sum_{n=1}^{\infty}u_{n}(t)e_{n},\;F(t)=\sum_{n=1}^{\infty}F_{n}(t)e_{n}.$

Using these expansions in (2.7), we obtain

$\displaystyle
u_{n}^{\prime}(t)+\lambda_{n}(1+D_{t}^{\\{m\\}})u_{n}(t)=F_{n}(t),$
$\displaystyle u_{n}(0)=\xi_{n}:=(\xi,e_{n}).$

Applying Proposition 2.4(d), we get

$u_{n}(t)=\omega(t,\lambda_{n})\xi_{n}+\int_{0}^{t}\omega(t-\tau,\lambda_{n})F_{n}(\tau)d\tau.$

Therefore,

$\displaystyle u(t)=S(t)\xi+\int_{0}^{t}S(t-\tau)F(\tau)d\tau,$ (2.10)

where $S(t)$ is the resolvent operator determined by

$\displaystyle
S(t)\xi=\sum_{n=1}^{\infty}\omega(t,\lambda_{n})\xi_{n}e_{n},\;\xi\in
L^{2}(\Omega).$ (2.11)

Obviously, $S(t)$ is a bounded linear operator on $L^{2}(\Omega)$ for all
$t\geq 0$. Moreover, we have the following statements.

######  Lemma 2.5.

Let $\\{S(t)\\}_{t\geq 0}$ be the resolvent family defined by (2.11), $v\in
L^{2}(\Omega)$ and $T>0$. Then,

  1. (a)

$S(\cdot)v\in C([0,T];L^{2}(\Omega))$ and $\|S(t)\|\leq\omega(t,\lambda_{1})$
for all $t\geq 0$.

  2. (b)

For $g\in C([0,T];\mathbb{H}^{\mu-1})$, $\mu>0$, we have $S*g\in
C([0,T];\mathbb{H}^{\mu})$. Furthermore,

$\|S*g(t)\|^{2}_{\mathbb{H}^{\mu}}\leq\int_{0}^{t}\omega(t-\tau,\lambda_{1})\|g(\tau)\|^{2}_{\mathbb{H}^{\mu-1}}d\tau,\text{
for all }t\geq 0.$ (2.12)

  3. (c)

If $m$ is nonincreasing, then $S(\cdot)v\in C^{1}((0,T];L^{2}(\Omega))$ and it
holds that

$\|S^{\prime}(t)\|\leq t^{-1}\;\text{ for all }t>0.$ (2.13)

  4. (d)

For $\delta\in(0,1)$, $g\in C([0,T];\mathbb{H}^{\mu-1-\delta})$, we have

$\displaystyle\|S*g(t)\|^{2}_{\mathbb{H}^{\mu}}\leq\int_{0}^{t}(t-\tau)^{-\delta}\|g(\tau)\|^{2}_{\mathbb{H}^{\mu-1-\delta}}d\tau.$

  5. (e)

If $(1*m)^{-1}\in L^{1}(0,T)$, then for $g\in C([0,T];\mathbb{H}^{\mu-2})$, we
have

$\displaystyle\|S*g(t)\|^{2}_{\mathbb{H}^{\mu}}\leq\int_{0}^{t}\frac{\|g(\tau)\|^{2}_{\mathbb{H}^{\mu-2}}}{(1*m)(t-\tau)}d\tau.$

###### Proof.

The properties (a) and (c) were proved in [12, Lemma 1]. Now we show (b). For
$g\in C([0,T];\mathbb{H}^{\mu-1})$, we have

$\displaystyle\|S*g(t)\|^{2}_{\mathbb{H}^{\mu}}=\sum_{n=1}^{\infty}\lambda^{\mu}_{n}\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})g_{n}(\tau)d\tau\right)^{2},\;g_{n}(\tau)=(g(\tau),e_{n}).$

Moreover, by the Hölder inequality, we can estimate

$\displaystyle\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})g_{n}(\tau)d\tau\right)^{2}$
$\displaystyle\leq\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})d\tau\right)\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})|g_{n}(\tau)|^{2}d\tau\right)$
$\displaystyle\leq\lambda_{n}^{-1}\int_{0}^{t}\omega(t-\tau,\lambda_{1})|g_{n}(\tau)|^{2}d\tau.$

Hence,

$\displaystyle\|S*g(t)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\leq\sum_{n=1}^{\infty}\int_{0}^{t}\omega(t-\tau,\lambda_{1})\lambda_{n}^{\mu-1}|g_{n}(\tau)|^{2}d\tau$
$\displaystyle=\int_{0}^{t}\omega(t-\tau,\lambda_{1})\|g(\tau)\|^{2}_{\mathbb{H}^{\mu-1}}d\tau.$

Next, we prove (d). Assume that $g\in C([0,T];\mathbb{H}^{\mu-1-\delta})$.
Then

$\displaystyle\|S*g(t)\|^{2}_{\mathbb{H}^{\mu}}=\sum_{n=1}^{\infty}\lambda_{n}^{\mu}\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})g_{n}(\tau)d\tau\right)^{2},\;g_{n}(\tau)=(g(\tau),e_{n}).$

Using the Hölder inequality and Proposition 2.4, we obtain

$\displaystyle\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})g_{n}(\tau)d\tau\right)^{2}$
$\displaystyle\leq\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})d\tau\right)\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})|g_{n}(\tau)|^{2}d\tau\right)$
$\displaystyle\leq\lambda_{n}^{-1}\int_{0}^{t}\frac{|g_{n}(\tau)|^{2}}{1+\lambda_{n}(t-\tau)}d\tau$
$\displaystyle\leq\lambda_{n}^{-1}\int_{0}^{t}\frac{|g_{n}(\tau)|^{2}}{\lambda_{n}^{\delta}(t-\tau)^{\delta}}d\tau,$

here, we used the inequality $1+b\geq b^{\delta}$ for $b\geq
0,\delta\in(0,1)$.

Therefore,

$\displaystyle\|S*g(t)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\leq\sum_{n=1}^{\infty}\lambda_{n}^{\mu-1-\delta}\int_{0}^{t}(t-\tau)^{-\delta}|g_{n}(\tau)|^{2}d\tau$
$\displaystyle=\int_{0}^{t}(t-\tau)^{-\delta}\|g(\tau)\|^{2}_{\mathbb{H}^{\mu-1-\delta}}d\tau.$

The last property is proved similarly by noting that

$\displaystyle\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})g_{n}(\tau)d\tau\right)^{2}$
$\displaystyle\leq\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})d\tau\right)\left(\int_{0}^{t}\omega(t-\tau,\lambda_{n})|g_{n}(\tau)|^{2}d\tau\right)$
$\displaystyle\leq\lambda_{n}^{-1}\int_{0}^{t}\frac{|g_{n}(\tau)|^{2}}{1+\lambda_{n}(1*m)(t-\tau)}d\tau$
$\displaystyle\leq\lambda_{n}^{-2}\int_{0}^{t}\frac{|g_{n}(\tau)|^{2}}{(1*m)(t-\tau)}d\tau.$

The proof is complete. ∎

##  3\. Global solvability and Hölder regularity

In order to solve problem (1.1)-(1.3), we require the following assumption on
the nonlinearity:

  1. (F)

The function $f:\mathbb{H}^{\mu}\times\mathbb{H}^{\mu}\to\mathbb{H}^{-\theta}$
satisfies $f(0,0)=0$ and for $\rho,\rho^{\prime}>0$, we have

$\|f(v_{1},w_{1})-f(v_{2},w_{2})\|_{\mathbb{H}^{-\theta}}\leq
L_{f}(\rho)\|v_{1}-v_{2}\|_{\mathbb{H}^{\mu}}+K_{f}(\rho^{\prime})\|w_{1}-w_{2}\|_{\mathbb{H}^{\mu}},$

where $\|v_{1}\|_{\mathbb{H}^{\mu}},\|v_{2}\|_{\mathbb{H}^{\mu}}\leq\rho$,
$\|w_{1}\|_{\mathbb{H}^{\mu}},\|w_{2}\|_{\mathbb{H}^{\mu}}\leq\rho^{\prime}$,
$0<\mu<2$, $\theta>0$ , $L_{f}$ and $K_{f}$ are nonnegative functions.

By the representation of the solution to the linear problem given by (2.10),
we have the following definition of mild solution to (1.1)-(1.3).

######  Definition 3.1.

Let $\xi\in\mathbb{H}^{\mu}$. The function $u\in C([0,T];\mathbb{H}^{\mu})$ is
called a mild solution to (1.1)-(1.3) on the interval $[0,T]$ if the following
identity holds

$u(t)=S(t)\xi+\int_{0}^{t}S(t-\tau)f(u(\tau),\mathcal{H}u(\tau))d\tau,\;t\in[0,T].$

###  3.1. Global solvability

######  Theorem 3.1.

Assume that the condtion (F) is satisfied with $\theta=1+\delta-\mu$,
$\delta\in(0,1)$ and

$\limsup\limits_{\rho\to
0}L_{f}(\rho)=L_{f}^{*},\;\limsup\limits_{\rho^{\prime}\to
0}K_{f}(\rho^{\prime})=K_{f}^{*},$

such that

$8T^{1-\delta}(1-\delta)^{-1}({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|\ell\|^{2}_{L^{1}})<1.$

Then there exists $\rho^{*}>0$ such that for
$\|\xi\|_{\mathbb{H}^{\mu}}\leq\frac{1}{2}\rho^{*}$, problem (1.1)-(1.3)
possesses a unique mild solution $u$ on $[0,T]$ obeying
$\|u(t)\|_{\mathbb{H}^{\mu}}\leq\rho^{*}$, for $t\in[0,T]$.

###### Proof.

We will show that the operator $\Phi$ determined by

$\Phi(u)(t)=S(t)\xi+\int_{0}^{t}S(t-\tau)f(u(\tau),\mathcal{H}u(\tau))d\tau,\;t\in[0,T],$

has a fixed point in the space $C([0,T];\mathbb{H}^{\mu})$ furnished by the
norm $\|u\|_{\infty}=\sup\limits_{t\in[0,T]}\|u(t)\|_{\mathbb{H}^{\mu}}$.

Denote by $B_{\rho}$ the closed ball of radius $\rho$ in
$C([0,T];\mathbb{H}^{\mu})$ which centers at the origin. For $u\in B_{\rho}$,
we have $\mathcal{H}u\in B_{\rho^{\prime}}$ with
$\rho^{\prime}=\rho\|\ell\|_{L^{1}}$. Then

$\displaystyle\|\Phi(u)(t)\|^{2}_{\mathbb{H}^{\mu}}$ $\displaystyle\leq
2\|S(t)\xi\|^{2}_{\mathbb{H}^{\mu}}+2\|S*f(u(\cdot),\mathcal{H}u(\cdot))(t)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\leq
2\|S(t)\xi\|^{2}_{\mathbb{H}^{\mu}}+2\int_{0}^{t}(t-\tau)^{-\delta}\|f(u(\tau),\mathcal{H}u(\tau))\|^{2}_{\mathbb{H}^{\mu-1-\delta}}d\tau,$

by Lemma 2.5(d). Using the condition (F), we have

$\displaystyle\|\Phi(u)(t)\|^{2}_{\mathbb{H}^{\mu}}$ $\displaystyle\leq
2\|\xi\|^{2}_{\mathbb{H}^{\mu}}+4\int_{0}^{t}(t-\tau)^{-\delta}[L_{f}(\rho)^{2}\|u(\tau)\|^{2}_{\mathbb{H}^{\mu}}+K_{f}(\rho^{\prime})^{2}\|\mathcal{H}u(\tau)\|^{2}_{\mathbb{H}^{\mu}}]d\tau$
$\displaystyle\leq
2\|\xi\|^{2}_{\mathbb{H}^{\mu}}+4\int_{0}^{t}(t-\tau)^{-\delta}({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon)\rho^{2}d\tau$
$\displaystyle\leq
2\|\xi\|^{2}_{\mathbb{H}^{\mu}}+4T^{1-\delta}(1-\delta)^{-1}({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon)\rho^{2},$

where $\epsilon>0$ and $\rho^{*}>0$ is chosen such that

$8T^{1-\delta}(1-\delta)^{-1}({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon)\leq
1,\text{ for all }\rho\leq\rho^{*}.$

Then, for $u\in B_{\rho^{*}}$ and $\|\xi\|\leq\frac{1}{2}\rho^{*}$, we can
estimate

$\displaystyle\|\Phi(u)(t)\|_{\mathbb{H}^{\mu}}\leq\rho^{*},\text{ for all
}t\in[0,T].$

Hence $\Phi(B_{\rho^{*}})\subset B_{\rho^{*}}$. It remains to verify that
$\Phi$ is a contraction on $B_{\rho^{*}}$. Indeed, for $u_{1},u_{2}\in
B_{\rho^{*}}$, we have

$\displaystyle\|f(u_{1}(\tau),\mathcal{H}u_{1}(\tau))-f(u_{2}(\tau),\mathcal{H}u_{2}(\tau))\|^{2}_{\mathbb{H}^{\mu-1-\delta}}$
$\displaystyle\qquad\leq
2L_{f}(\rho^{*})^{2}\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}+2K_{f}(\rho^{*}\|h\|_{L^{1}})^{2}\|\mathcal{H}u_{1}(\tau)-\mathcal{H}u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\qquad\leq
2({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|h\|^{2}_{L^{1}}+\epsilon)\sup_{\tau\in[0,T]}\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}.$

This yields that

$\displaystyle\|\Phi(u_{1})(t)-\Phi(u_{2})(t)\|^{2}_{\mathbb{H}^{\mu}}\leq\|S*[f(u_{1}(\cdot),\mathcal{H}u_{1}(\cdot))-f(u_{2}(\cdot),\mathcal{H}u_{2}(\cdot))](t)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\qquad\leq\int_{0}^{t}(t-\tau)^{-\delta}\|f(u_{1}(\tau),\mathcal{H}u_{1}(\tau))-f(u_{2}(\tau),\mathcal{H}u_{2}(\tau))\|^{2}_{\mathbb{H}^{\mu-1-\delta}}d\tau$
$\displaystyle\qquad\leq
2T^{1-\delta}(1-\delta)^{-1}({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|h\|^{2}_{L^{1}}+\epsilon)\sup_{\tau\in[0,T]}\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\qquad\leq\frac{1}{4}\sup_{\tau\in[0,T]}\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}.$

Finally, we arrive at

$\|\Phi(u_{1})-\Phi(u_{2})\|_{\infty}\leq\frac{1}{2}\|u_{1}-u_{2}\|_{\infty}.$

The proof is complete. ∎

In the case the function $f(\cdot,\cdot)$ satisfies global Lipschitz
condition, we have the following result.

######  Theorem 3.2.

Assume that the nonlinearity
$f:\mathbb{H}^{\mu}\times\mathbb{H}^{\mu}\to\mathbb{H}^{\mu-1-\delta}$ is
continuous and satisfies the condition

$\|f(v_{1},w_{1})-f(v_{2},w_{2})\|_{\mathbb{H}^{\mu-1-\delta}}\leq
L^{*}_{f}\|v_{1}-v_{2}\|_{\mathbb{H}^{\mu}}+K^{*}_{f}\|w_{1}-w_{2}\|_{\mathbb{H}^{\mu}},$

for all $v_{1},v_{2},w_{1},w_{2}\in\mathbb{H}^{\mu}$, where
$L^{*}_{f},K^{*}_{f}\geq 0$. Then, Problem (1.1)-(1.3) possesses a unique mild
solution in the space $C([0,T];\mathbb{H}^{\mu})$.

###### Proof.

In $C([0,T];\mathbb{H}^{\mu})$, we use an equivalent norm

$\|u\|_{\beta,\infty}=\sup_{t\in[0,T]}e^{-\beta
t}\|u(t)\|_{\mathbb{H}^{\mu}},$

where $\beta>0$ satisfies

$L^{*}:=2({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|\ell\|^{2}_{L^{1}})\int_{0}^{T}e^{-2\beta\tau}\tau^{-\delta}d\tau<1.$

Consider the solution operator $\Phi$ as in the proof of Theorem 3.1, where
$u_{1},u_{2}\in C([0,T];\mathbb{H}^{\mu})$, we have

$\displaystyle\|\Phi(u_{1})(t)-\Phi(u_{2})(t)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\leq
2\int_{0}^{t}(t-\tau)^{-\delta}{L^{*}_{f}}^{2}\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}d\tau$
$\displaystyle\qquad+2\int_{0}^{t}(t-\tau)^{-\delta}{K^{*}_{f}}^{2}\|\mathcal{H}u_{1}(\tau)-\mathcal{H}u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}d\tau,$

according to the assumption of the theorem and Lemma 2.5(d). Then, it yields

$\displaystyle e^{-2\beta t}\|\Phi(u_{1})(t)$
$\displaystyle-\Phi(u_{2})(t)\|^{2}_{\mathbb{H}^{\mu}}$ $\displaystyle\leq
2\int_{0}^{t}e^{-2\beta(t-\tau)}(t-\tau)^{-\delta}{L^{*}_{f}}^{2}[e^{-2\beta\tau}\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}]d\tau$
$\displaystyle\qquad+2\int_{0}^{t}e^{-2\beta(t-\tau)}(t-\tau)^{-\delta}{K^{*}_{f}}^{2}[e^{-2\beta\tau}\|\mathcal{H}u_{1}(\tau)-\mathcal{H}u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}]d\tau$
$\displaystyle\leq
2({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|\ell\|^{2}_{L^{1}})\left(\int_{0}^{t}e^{-2\beta\tau}\tau^{-\delta}d\tau\right)\|u_{1}-u_{2}\|^{2}_{\beta,\infty},$

here, we have used the following estimate

$\displaystyle
e^{-2\beta\tau}\|\mathcal{H}u_{1}(\tau)-\mathcal{H}u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\leq\left(\int_{0}^{\tau}e^{-\beta(\tau-z)}\ell(\tau-z)[e^{-\beta
z}\|u_{1}(z)-u_{2}(z)\|_{\mathbb{H}^{\mu}}]dz\right)^{2}$
$\displaystyle\leq\left(\int_{0}^{\tau}e^{-\beta
z}|\ell(z)|dz\right)^{2}\|u_{1}-u_{2}\|^{2}_{\beta,\infty}$
$\displaystyle\leq\|\ell\|^{2}_{L^{1}}\|u_{1}-u_{2}\|^{2}_{\beta,\infty}.$

Therefore,

$\|\Phi(u_{1})-\Phi(u_{2})\|_{\beta,\infty}\leq\sqrt{L^{*}}\|u_{1}-u_{2}\|_{\beta,\infty}.$

In other words, $\Phi$ is a contraction on $C([0,T];\mathbb{H}^{\mu})$. This
completes the proof. ∎

######  Remark 3.1.

(i) In Theorem 3.2, we do not require that $\|\xi\|_{\mathbb{H}^{\mu}}$ is
small. Besides, we also relax the condition on $L^{*}_{f}$ and $K^{*}_{f}$ in
comparison with the assumptions of Theorem 3.1. Moreover, the result obtained
in Theorem 3.2 still holds if we add to the right-hand side of equation (1.1)
an external force, that means the right-hand side of (1.1) has the form
$f(u,\mathcal{H}u)+g(t,x)$.

(ii) In the case $(1*m)^{-1}\in L^{1}(0,T)$, we achieve a similar result to
that in Theorem 3.2 assumming that the nonlinearity $f$ can take ‘weaker’
values:

$\|f(v_{1},w_{1})-f(v_{2},w_{2})\|_{\mathbb{H}^{\mu-2}}\leq
L^{*}_{f}\|v_{1}-v_{2}\|_{\mathbb{H}^{\mu}}+K^{*}_{f}\|w_{1}-w_{2}\|_{\mathbb{H}^{\mu}},$

for all $v_{1},v_{2},w_{1},w_{2}\in\mathbb{H}^{\mu}$, where
$L^{*}_{f},K^{*}_{f}\geq 0$. In this situation, we use the estimate in Lemma
2.5(e).

###  3.2. Hölder regularity

In this part, we assume an additional contition on the kernel $m$ as follows.

  1. (M*)

The hypothesis (M) is satisfied with a nonincreasing function $m$.

Note that, under the assumption (M*), the resolvent $S(\cdot)$ is
differentiable in $(0,\infty)$ and $\|S(t)\|\leq t^{-1}$ for $t>0$ by Lemma
2.5.

For $\gamma\in(0,1)$, denote

$V^{\mu,\gamma}_{\rho,\rho^{*}}=B_{\rho^{*}}\cap\\{u\in
C([0,T];\mathbb{H}^{\mu}):\sup_{\begin{subarray}{c}h>0\\\
t\in(0,T-h]\end{subarray}}\frac{t^{\gamma}\|u(t+h)-u(t)\|_{\mathbb{H}^{\mu}}}{h^{\gamma}}\leq\rho\\}.$

We will show that the mild solution to problem (1.1)-(1.3) obtained by Theorem
3.1 is Hölder continuous in $(0,T]$ by proving that the solution mapping
$\Phi$ is contractive on $V^{\mu,\gamma}_{\rho,\rho^{*}}$.

######  Theorem 3.3.

Assume that (M*) and all assumptions in Theorem 3.1 are satisfied. Moreover,
with $\gamma\in(\frac{1}{2}\delta,\frac{1}{2})$, we have

$\displaystyle\ell^{*}_{1}=\sup_{\begin{subarray}{c}h>0\\\
t\in(0,T-h]\end{subarray}}\left(\frac{t}{h}\right)^{\gamma}\int_{t}^{t+h}|\ell(\tau)|d\tau<\infty,$
$\displaystyle
16B(1-\delta,1-2\gamma)T^{1-\delta}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}{\ell^{*}_{2}}^{2})<1,$

where $B(\cdot,\cdot)$ is the Beta function and

$\ell^{*}_{2}=\sup_{t\in(0,T]}t^{\gamma}\int_{0}^{t}\frac{|\ell(\tau)|}{(t-\tau)^{\gamma}}d\tau.$

Then, the solution to problem (1.1)-(1.3) is Hölder continuous on $(0,T]$.

###### Proof.

It suffices to show that $\Phi(V^{\mu,\gamma}_{\rho,\rho^{*}})\subset
V^{\mu,\gamma}_{\rho,\rho^{*}}$ for a certain $\rho>0$.

Since $S(\cdot)$ is differentiable in $(0,\infty)$, using the Mean value
theorem, we have

$\displaystyle\|[S(t+h)-S(t)]\xi\|_{\mathbb{H}^{\mu}}$ $\displaystyle\leq
h\int_{0}^{1}\|S^{\prime}(t+\zeta h)\xi\|_{\mathbb{H}^{\mu}}d\zeta$
$\displaystyle\leq
h\|\xi\|_{\mathbb{H}^{\mu}}\int_{0}^{1}\frac{d\zeta}{t+\zeta
h}=\|\xi\|_{\mathbb{H}^{\mu}}\ln\left(1+\frac{h}{t}\right)$
$\displaystyle\leq\|\xi\|_{\mathbb{H}^{\mu}}\gamma^{-1}t^{-\gamma}h^{\gamma}.$
(3.1)

On the other hand, for $u\in V^{\mu,\gamma}_{\rho,\rho^{*}}$, we have

$\displaystyle\|\mathcal{H}u(t+h)-\mathcal{H}u(t)\|_{\mathbb{H}^{\mu}}\leq\int_{t}^{t+h}|\ell(\tau)|\|u(t+h-\tau)\|_{\mathbb{H}^{\mu}}d\tau$
$\displaystyle\quad+\int_{0}^{t}|\ell(\tau)|\|u(t+h-\tau)-u(t-\tau)\|_{\mathbb{H}^{\mu}}d\tau$
$\displaystyle\leq
t^{-\gamma}h^{\gamma}\rho^{*}\left(\frac{t}{h}\right)^{\gamma}\int_{t}^{t+h}|\ell(\tau)|d\tau$
$\displaystyle\quad+t^{-\gamma}h^{\gamma}t^{\gamma}\int_{0}^{t}\frac{|\ell(\tau)|}{(t-\tau)^{\gamma}}[(t-\tau)^{\gamma}h^{-\gamma}\|u(t+h-\tau)-u(t-\tau)\|_{\mathbb{H}^{\mu}}]d\tau$
$\displaystyle\leq
t^{-\gamma}h^{\gamma}\rho^{*}\left[\left(\frac{t}{h}\right)^{\gamma}\int_{t}^{t+h}|\ell(\tau)|d\tau\right]+t^{-\gamma}h^{\gamma}\rho\left[t^{\gamma}\int_{0}^{t}\frac{|\ell(\tau)|}{(t-\tau)^{\gamma}}d\tau\right].$

Hence,

$\|\mathcal{H}u(t+h)-\mathcal{H}u(t)\|_{\mathbb{H}^{\mu}}\leq
t^{-\gamma}h^{\gamma}(\rho^{*}\ell^{*}_{1}+\rho\ell^{*}_{2}).$ (3.2)

Denote

$D_{h}f(u)(t)=f(u(t+h),\mathcal{H}u(t+h))-f(u(t),\mathcal{H}u(t)).$ (3.3)

We have

$\displaystyle\|D_{h}f(u)(t)\|_{\mathbb{H}^{\mu-1-\delta}}\leq\|f(u(t+h),\mathcal{H}u(t+h))-f(u(t),\mathcal{H}u(t))\|_{\mathbb{H}^{\mu-1-\delta}}$
$\displaystyle\quad\leq
L_{f}(\rho^{*})\|u(t+h)-u(t)\|_{\mathbb{H}^{\mu}}+K_{f}(\rho^{*}\|\ell\|_{L^{1}})\|\mathcal{H}u(t+h)-\mathcal{H}u(t)\|_{\mathbb{H}^{\mu}}$
$\displaystyle\quad\leq
t^{-\gamma}h^{\gamma}L_{f}(\rho^{*})\rho+t^{-\gamma}h^{\gamma}K_{f}(\rho^{*}\|\ell\|_{L^{1}})(\rho^{*}\ell^{*}_{1}+\rho\ell^{*}_{2}),$
(3.4)

here, we employed the estimate (3.2).

Finally, it holds that

$\displaystyle\|\Phi(u)(t+h)$
$\displaystyle-\Phi(u)(t)\|^{2}_{\mathbb{H}^{\mu}}\leq
2\|[S(t+h)-S(t)]\xi\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle+4\int_{0}^{t}\tau^{-\delta}\|D_{h}f(u)(t-\tau)\|^{2}_{\mathbb{H}^{\mu-1-\delta}}d\tau$
$\displaystyle+4\int_{t}^{t+h}\tau^{-\delta}\|f(u(t+h-\tau),\mathcal{H}u(t+h-\tau))\|^{2}_{\mathbb{H}^{\mu-1-\delta}}d\tau$
$\displaystyle=E_{1}(t)+E_{2}(t)+E_{3}(t).$

By (3.1), we get

$E_{1}(t)\leq
2\gamma^{-2}t^{-2\gamma}h^{2\gamma}\|\xi\|^{2}_{\mathbb{H}^{\mu}}.$

Using (3.4), we gain

$\displaystyle E_{2}(t)$ $\displaystyle\leq
8h^{2\gamma}[\rho^{2}L_{f}(\rho^{*})^{2}+K_{f}(\rho^{*}\|\ell\|^{2}_{L^{1}})^{2}(\rho^{*}\ell^{*}_{1}+\rho\ell^{*}_{2})^{2}]\int_{0}^{t}\tau^{-\delta}(t-\tau)^{-2\gamma}d\tau$
$\displaystyle\leq
8h^{2\gamma}t^{-2\gamma}[\rho^{2}L_{f}(\rho^{*})^{2}+K_{f}(\rho^{*}\|\ell\|^{2}_{L^{1}})^{2}(\rho^{*}\ell^{*}_{1}+\rho\ell^{*}_{2})^{2}]B(1-\delta,1-2\gamma)T^{1-\delta}$
$\displaystyle\leq 16h^{2\gamma}t^{-2\gamma}B(1-\delta,1-2\gamma)T^{1-\delta}$
$\displaystyle\quad\times[(L_{f}(\rho^{*})^{2}+K_{f}(\rho^{*}\|\ell\|^{2}_{L^{1}})^{2}{\ell^{*}_{2}}^{2})\rho^{2}+K_{f}(\rho^{*}\|\ell\|^{2}_{L^{1}})^{2}{\rho^{*}}^{2}{\ell^{*}_{1}}^{2}]$

here we utilized the identity

$\displaystyle\int_{0}^{t}\tau^{-\delta}(t-\tau)^{-2\gamma}d\tau=B(1-\delta,1-2\gamma)t^{1-\delta-2\gamma}.$

We can estimate $E_{3}(t)$ as follows

$\displaystyle E_{3}(t)$
$\displaystyle=4\int_{0}^{h}(t+h-\tau)^{-\delta}\|f(u(\tau),\mathcal{H}u(\tau))\|^{2}_{\mathbb{H}^{\mu-1-\delta}}d\tau$
$\displaystyle\leq
8{\rho^{*}}^{2}[L_{f}(\rho^{*})^{2}+K_{f}(\rho^{*}\|\ell\|_{L^{1}})^{2}\|\ell\|^{2}_{L^{1}}]\int_{0}^{h}(t+h-\tau)^{-\delta}d\tau.$

Noting that

$\displaystyle\int_{0}^{h}(t+h-\tau)^{-\delta}d\tau$
$\displaystyle=(1-\delta)^{-1}[(t+h)^{1-\delta}-t^{1-\delta}]$
$\displaystyle\leq ht^{-\delta}\leq
h^{2\gamma}t^{-2\gamma}h^{1-2\gamma}t^{2\gamma-\delta}\leq
h^{2\gamma}t^{-2\gamma}T^{1-\delta},$

we obtain

$E_{3}(t)\leq
8h^{2\gamma}t^{-2\gamma}(\rho^{*})^{2}T^{1-\delta}[L_{f}(\rho^{*})^{2}+K_{f}(\rho^{*}\|\ell\|_{L^{1}})^{2}\|\ell\|^{2}_{L^{1}}].$

Combining the estimates of $E_{1}(t),E_{2}(t)$ and $E_{3}(t)$ above, we
conclude that

$\displaystyle\left(\frac{t}{h}\right)^{2\gamma}$
$\displaystyle\|\Phi(u)(t+h)-\Phi(u)(t)\|^{2}_{\mathbb{H}^{\mu}}\leq
2\gamma^{-2}\|\xi\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle+16B(1-\delta,1-2\gamma)T^{1-\delta}[({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}{\ell^{*}_{2}}^{2}+\epsilon)\rho^{2}+K_{f}(\rho^{*}\|\ell\|^{2}_{L^{1}})^{2}{\rho^{*}}^{2}{\ell^{*}_{1}}^{2}]$
$\displaystyle+8{\rho^{*}}^{2}T^{1-\delta}[L_{f}(\rho^{*})^{2}+K_{f}(\rho^{*}\|\ell\|_{L^{1}})^{2}\|\ell\|^{2}_{L^{1}}],$

here, $\epsilon>0$ is chosen such that

$16T^{1-\delta}[L_{f}(\rho^{*})^{2}+K_{f}(\rho^{*}\|\ell\|^{2}_{L^{1}})^{2}{\ell^{*}_{2}}^{2}]\leq
16T^{1-\delta}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}{\ell^{*}_{2}}^{2}+\epsilon)<\frac{1}{B(1-\delta,1-2\gamma)}.$

Now, we take a sufficiently large $\rho>0$ such that

$\left(\frac{t}{h}\right)^{2\gamma}\|\Phi(u)(t+h)-\Phi(u)(t)\|^{2}_{\mathbb{H}^{\mu}}\leq\rho^{2},\text{
for all }h>0,t\in(0,T-h].$

This implies that $\Phi(u)\in W^{\mu,\gamma}_{\rho,\rho^{*}}$. The proof is
complete. ∎

We now consider the case when the nonlinearity $f(u(t),\mathcal{H}u(t))$ is
more regular, i.e. it takes values in $\mathbb{H}^{\mu-1}$. We will show the
Hölder continuity of the solution without the assumptions on the coefficients
as in Theorem 3.3.

######  Theorem 3.4.

Assume that the condition (M*) and (F) are satisfied with $\theta=1-\mu$, and

$\limsup\limits_{\rho\to
0}L_{f}(\rho)=L_{f}^{*},\;\limsup\limits_{\rho^{\prime}\to
0}K_{f}(\rho^{\prime})=K_{f}^{*},$

such that

$4({L^{*}_{f}}^{2}+{K^{*}_{f}}^{2}\|\ell\|^{2}_{L^{1}})<\lambda_{1}.$

Then, there exists $\eta>0$ such that for
$\|\xi\|_{\mathbb{H}^{\mu}}\leq\eta$, problem (1.1)-(1.3) has a unique mild
solution $u$ in $[0,T]$. Moreover, $u(\cdot)$ is Hölder continuous on $(0,T]$.

###### Proof.

Similar to the proof of Theorem 3.1, first we look for $\rho^{*}>0$ such that
$\Phi(B_{\rho^{*}})\subset B_{\rho^{*}}$. Let $u\in B_{\rho}$, $\rho>0$, and
$f(u(t),\mathcal{H}u(t))\in\mathbb{H}^{\mu-1}$.

Then using Lemma 2.5, we obtain

$\displaystyle\|S*f(u(\cdot),$
$\displaystyle\mathcal{H}u(\cdot))\|^{2}_{\mathbb{H}^{\mu}}\leq\int_{0}^{t}\omega(t-\tau,\lambda_{1})\|f(u(\tau),\mathcal{H}u(\tau))\|^{2}_{\mathbb{H}^{\mu-1}}d\tau$
$\displaystyle\leq
2\int_{0}^{t}\omega(t-\tau,\lambda_{1})[L_{f}(\rho)^{2}\|u(\tau)\|^{2}_{\mathbb{H}^{\mu}}+K_{f}(\rho\|\ell\|_{L^{1}})^{2}\|\mathcal{H}u(\tau)\|^{2}_{\mathbb{H}^{\mu}}]d\tau$
$\displaystyle\leq
2\int_{0}^{t}\omega(t-\tau,\lambda_{1})[L_{f}(\rho)^{2}+K_{f}(\rho\|\ell\|_{L^{1}})^{2}\|\ell\|^{2}_{L^{1}}]\|u(\tau)\|^{2}_{\mathbb{H}^{\mu}}d\tau$
$\displaystyle\leq
2\int_{0}^{t}\omega(t-\tau,\lambda_{1})({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|_{L^{1}}^{2}+\epsilon)\|u(\tau)\|^{2}_{\mathbb{H}^{\mu}}d\tau,$

for $\epsilon>0$ and $\rho^{*}>0$ is chosen such that

$4[L_{f}(\rho)^{2}+K_{f}(\rho\|\ell\|_{L^{1}})^{2}\|\ell\|^{2}_{L^{1}}]\leq
4({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|_{L^{1}}^{2}+\epsilon)\leq\lambda_{1},$

for all $\rho\leq\rho^{*}$. Then,

$\displaystyle\|S*f(u(\cdot),$
$\displaystyle\mathcal{H}u(\cdot))\|^{2}_{\mathbb{H}^{\mu}}\leq
2\lambda_{1}^{-1}(1-\omega(t,\lambda_{1}))({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|_{L^{1}}^{2}+\epsilon){\rho^{*}}^{2}.$

We can estimate the solution operator $\Phi$ as follows

$\displaystyle\|\Phi(u)(t)\|_{\mathbb{H}^{\mu}}^{2}$ $\displaystyle\leq
2\omega(t,\lambda_{1})^{2}\|\xi\|^{2}_{\mathbb{H}^{\mu}}+2\|S*f(u(\cdot),\mathcal{H}u(\cdot))\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\leq
2\omega(t,\lambda_{1})\|\xi\|^{2}_{\mathbb{H}^{\mu}}+4\lambda_{1}^{-1}(1-\omega(t,\lambda_{1}))({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|_{L^{1}}^{2}+\epsilon){\rho^{*}}^{2}$
$\displaystyle\leq
2\omega(t,\lambda_{1})[\|\xi\|^{2}_{\mathbb{H}^{\mu}}-2\lambda_{1}^{-1}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|_{L^{1}}^{2}+\epsilon){\rho^{*}}^{2}]$
$\displaystyle\qquad+4\lambda_{1}^{-1}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|_{L^{1}}^{2}+\epsilon){\rho^{*}}^{2}.$

Take
$\eta=2\lambda_{1}^{-1}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|_{L^{1}}^{2}+\epsilon){\rho^{*}}^{2}$,
with $\|\xi\|_{\mathbb{H}^{\mu}}\leq\eta$ we have

$\displaystyle\|\Phi(u)(t)\|_{\mathbb{H}^{\mu}}^{2}$ $\displaystyle\leq
4\lambda_{1}^{-1}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|_{L^{1}}^{2}+\epsilon){\rho^{*}}^{2}\leq{\rho^{*}}^{2}.$

Hence $\Phi(u)\in B_{\rho^{*}}$. Then, we show that $\Phi$ is contractive on
$B_{\rho^{*}}$. For $u_{1},u_{1}\in B_{\rho^{*}}$, we have

$\displaystyle\|\Phi(u_{1})(t)$
$\displaystyle-\Phi(u_{2})(t)\|_{\mathbb{H}^{\mu}}^{2}$
$\displaystyle\leq\int_{0}^{t}\omega(t-\tau,\lambda_{1})\|f(u_{1}(\tau),\mathcal{H}u_{1}(\tau))-f(u_{2}(\tau),\mathcal{H}u_{2}(\tau))\|^{2}_{\mathbb{H}^{\mu-1}}d\tau$
$\displaystyle\leq\int_{0}^{t}\omega(t-\tau,\lambda_{1})({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon)\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}d\tau$
$\displaystyle\leq\lambda_{1}^{-1}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon)\sup_{\tau\in[0,T]}\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle\leq\frac{1}{4}\sup_{\tau\in[0,T]}\|u_{1}(\tau)-u_{2}(\tau)\|^{2}_{\mathbb{H}^{\mu}}.$

Therefore

$\|\Phi(u_{1})-\Phi(u_{2})|_{\infty}\leq\frac{1}{2}\|u_{1}-u_{2}\|_{\infty},$

which implies that $\Phi$ is contractive on $B_{\rho^{*}}$, and problem
(1.1)-(1.3) possesses a unique mild solution $u\in B_{\rho^{*}}$.

It remains to prove that this solution is Hölder continuous on $(0,T]$.
Observe that

$\displaystyle u(t+h)-u(t)$
$\displaystyle=[S(t+h)-S(t)]\xi+\int_{0}^{t}S(\tau)D_{h}f(u)(t-\tau)d\tau$
$\displaystyle+\int_{t}^{t+h}S(\tau)f(u(t+h-\tau),\mathcal{H}u(t+h-\tau))d\tau.$

Then

$\displaystyle\|u(t+h)-u(t)\|^{2}_{\mathbb{H}^{\mu}}$ $\displaystyle\leq
3\|[S(t+h)-S(t)]\xi\|^{2}_{\mathbb{H}^{\mu}}$
$\displaystyle+3\int_{0}^{t}\|D_{h}f(u)(t-\tau)\|^{2}_{\mathbb{H}^{\mu-1}}d\tau$
$\displaystyle+3\int_{t}^{t+h}\|f(u(t+h-\tau),\mathcal{H}u(t+h-\tau))\|_{\mathbb{H}^{\mu-1}}^{2}d\tau$
$\displaystyle=F_{1}(t)+F_{2}(t)+F_{3}(t),$

here we employed Lemma 2.5(b) and the fact that $\omega(t,\lambda_{1})\leq 1$.

By an analoguous argument as in the proof of Theorem 3.3, we get

$\displaystyle\|[S(t+h)-S(t)]\xi\|_{\mathbb{H}^{\mu}}$
$\displaystyle\leq\gamma^{-1}h^{\gamma}t^{-\gamma}\|\xi\|_{\mathbb{H}^{\mu}},\gamma\in(0,\frac{1}{2}).$

So it is clear that

$F_{1}(t)\leq
3\gamma^{-2}h^{2\gamma}t^{-2\gamma}\|\xi\|_{\mathbb{H}^{\mu}}^{2}.$

Moreover,

$\displaystyle F_{2}(t)$ $\displaystyle\leq
6\int_{0}^{t}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon)\|u(t+h-\tau)-u(t-\tau)\|^{2}_{\mathbb{H}^{\mu}}d\tau$
$\displaystyle=6\int_{0}^{t}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon)\|u(\tau+h)-u(\tau)\|^{2}_{\mathbb{H}^{\mu}}d\tau,$
$\displaystyle F_{3}(t)$ $\displaystyle\leq
6\int_{t}^{t+h}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon){\rho^{*}}^{2}d\tau$
$\displaystyle\leq
6h({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon){\rho^{*}}^{2}$
$\displaystyle\leq
6Th^{2\gamma}t^{-2\gamma}({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon){\rho^{*}}^{2},$

thanks to the fact that $0<h,t<T$. Putting the estimates of
$F_{1}(t),F_{2}(t)$ and $F_{3}(t)$ together, we get

$\displaystyle\|u(t+h)$ $\displaystyle-u(t)\|^{2}_{\mathbb{H}^{\mu}}\leq
C_{0}h^{2\gamma}t^{-2\gamma}+\int_{0}^{t}C_{1}\tau^{-2\gamma}[\tau^{2\gamma}\|u(\tau+h)-u(\tau)\|^{2}_{\mathbb{H}^{\mu}}]d\tau$

with

$\displaystyle C_{0}$
$\displaystyle=3\gamma^{-2}\|\xi\|^{2}_{\mathbb{H}^{\mu}}+6T({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon){\rho^{*}}^{2},$
$\displaystyle C_{1}$
$\displaystyle=6({L_{f}^{*}}^{2}+{K_{f}^{*}}^{2}\|\ell\|^{2}_{L^{1}}+\epsilon).$

Applying the Gronwall inequality, we obtain

$\displaystyle t^{2\gamma}\|u(t+h)$
$\displaystyle-u(t)\|^{2}_{\mathbb{H}^{\mu}}\leq
e^{C_{1}(1-2\gamma)^{-1}t^{1-2\gamma}}C_{0}h^{2\gamma}\text{ for }t>0.$

In other words, $u(\cdot)$ is Hölder continuous on $(0,T]$. ∎

######  Example 3.1.

Consider the case $d\geq 3$ and

$f(u(t),\mathcal{H}u(t))=|u(t)|^{p}+\int_{0}^{t}\ell(t-\tau)(\chi\cdot\nabla)u(\tau)d\tau,p>1,$

with $\chi\in[L^{\infty}(\Omega)]^{d}$. The nonlinear function contains the
advection term which depends on the history of states. Noting that, for
$\theta\in(0,1)$, $q=\frac{2d}{d+2\theta}$, $\hat{p}=\frac{2d}{d-2}$,
$\hat{q}=\frac{2d}{2+2\theta}$, we have

$\frac{1}{\hat{p}}+\frac{1}{\hat{q}}=\frac{1}{q}.$

Applying general Hölder inequality with $u\in L^{(p-1)\hat{q}}(\Omega),v\in
L^{\hat{p}}(\Omega)$, we gain

$\displaystyle\||u|^{p-1}v\|_{L^{q}}$
$\displaystyle\leq\||u|^{p-1}\|_{L^{\hat{q}}}\|v\|_{L^{\hat{p}}}$
$\displaystyle=\|u\|^{p-1}_{L^{(p-1)\hat{q}}}\|v\|_{L^{\hat{p}}}.$

Assume that $(p-1)\hat{q}\leq\hat{p}$, which means

$p\leq\frac{d+2\theta}{d-2}.$

By Lemma 2.3, we have

$\mathbb{H}^{1}\subset H_{0}^{1}(\Omega)\subset L^{\hat{p}}(\Omega)\subset
L^{(p-1)\hat{q}}(\Omega).$

Therefore

$\displaystyle\||u|^{p-1}v\|_{L^{q}}\leq
C\|u\|^{p-1}_{\mathbb{H}^{1}}\|v\|_{\mathbb{H}^{1}},$

where $C$ is a positive constant which does not depend on $u$ and $v$.
Moreover, we have

$L^{q}(\Omega)\subset H^{-\theta}\subset\mathbb{H}^{-\theta}.$

Hence,

$\displaystyle\||u|^{p-1}v\|_{\mathbb{H}^{-\theta}}\leq
C\|u\|^{p-1}_{\mathbb{H}^{1}}\|v\|_{\mathbb{H}^{1}}.$ (3.5)

When $u,v\in C([0,T];\mathbb{H}^{1})$, applying the inequality (3.5), we have

$\displaystyle\||u(t)|^{p}-|v(t)|^{p}\|_{\mathbb{H}^{-\theta}}\leq
C(\|u(t)\|^{p-1}_{\mathbb{H}^{1}}+\|v(t)\|^{p-1}_{\mathbb{H}^{1}})\|u(t)-v(t)\|_{\mathbb{H}^{1}}.$
(3.6)

For $\mathcal{H}u(t)=\ell*u$, we have

$\|(\chi\cdot\nabla)[\mathcal{H}u(t)-\mathcal{H}v(t)]\|\leq\|\chi\|_{\infty}\cdot\|\mathcal{H}u(t)-\mathcal{H}v(t)\|_{\mathbb{H}^{1}}.$

Because $L^{2}(\Omega)\subset\mathbb{H}^{-\theta}$, we get

$\displaystyle\|(\chi\cdot\nabla)[\mathcal{H}u(t)-\mathcal{H}v(t)]\|_{\mathbb{H}^{-\theta}}\leq
C_{\theta}\|\chi\|_{\infty}\cdot\|\mathcal{H}u(t)-\mathcal{H}v(t)\|_{\mathbb{H}^{1}},$
(3.7)

with $C_{\theta}$ is a positive constant. Combining (3.6) and (3.7), it leads
to

$\displaystyle\|f(u(t),\mathcal{H}u(t))-f(v(t),\mathcal{H}v(t)\|_{\mathbb{H}^{-\theta}}$
$\displaystyle\leq
C(\|u(t)\|^{p-1}_{\mathbb{H}^{1}}+\|v(t)\|^{p-1}_{\mathbb{H}^{1}})\|u(t)-v(t)\|_{\mathbb{H}^{1}}$
$\displaystyle\quad+C_{\theta}\|\chi\|_{\infty}\cdot\|\mathcal{H}u(t)-\mathcal{H}v(t)\|_{\mathbb{H}^{1}}.$

Thus, the function $f$ satisfies condition (F) with $\mu=1$, $\theta=\delta$
and

$\displaystyle
L_{f}(\rho)=2C\rho^{p-1},\;K_{f}(\rho)=C_{\theta}\|\chi\|_{\infty}.$

Then, we get the conclusion of Theorem 3.1 và 3.3 when $\|\chi\|_{\infty}$ is
sufficiently small.

##  4\. Application

Consider the following problem of identifying parameter

$\displaystyle\partial_{t}u-(1+D_{t}^{\\{m\\}})\Delta u$
$\displaystyle=g(x)p(t)+f_{1}(u)\text{ in }\Omega,t\in(0,T),$ (4.1)
$\displaystyle u$ $\displaystyle=0\text{ on }\partial\Omega,\;t\geq 0,$ (4.2)
$\displaystyle u(0)$ $\displaystyle=\xi\text{ in }\Omega,$ (4.3)
$\displaystyle\int_{\Omega}\kappa(x)u(t,x)dx$
$\displaystyle=\psi(t),\;t\in[0,T],$ (4.4)

where $p(t)$, $t\in[0,T]$, is an unknown parameter, $g\in L^{2}(\Omega)$ is
given. In this model, (4.4) is the complementary measurement with $\kappa\in
H_{0}^{1}(\Omega)$ such that $(g,\kappa)\neq 0$, $\psi\in W^{1,1}(0,T)$.

In this problem, we require further that the kernel $m$ fulfils:

  1. (M⋆)

Assumption (M) holds and $m^{\prime}\in L^{1}(0,T)$.

Then, problem (4.1) is rewritten as follows

$\displaystyle\partial_{t}u-(1+m_{0})\Delta u-m_{1}*\Delta u$
$\displaystyle=g(x)p(t)+f_{1}(u),\;m_{0}=m(0),m_{1}=m^{\prime}.$

Combining with (4.4), we obtain

$\psi^{\prime}+(1+m_{0})(\nabla u,\nabla\kappa)+m_{1}*(\nabla
u,\nabla\kappa)=(g,\kappa)p(t)+(f_{1}(u),\kappa).$

Therefore,

$p(t)=(g,\kappa)^{-1}[\psi^{\prime}+(1+m_{0})(\nabla
u,\nabla\kappa)+m_{1}*(\nabla u,\nabla\kappa)-(f_{1}(u),\kappa)].$

Set

$f_{2}(u,\mathcal{H}u):=(1+m_{0})(\nabla
u,\nabla\kappa)+(\nabla(m_{1}*u),\nabla\kappa)-(f_{1}(u),\kappa),$

where $\mathcal{H}$ is the convolution operator with the kernel $m_{1}$.

Assume that $f_{1}:\mathbb{H}^{1}\to L^{2}(\Omega)$ verifies the condition

$\|f_{1}(u)-f_{1}(v)\|\leq L_{1}\|u-v\|_{\mathbb{H}^{1}},\text{ for all
}u,v\in\mathbb{H}^{1}.$

Then, for all $u,v\in C([0,T];\mathbb{H}^{1})$, we have

$\displaystyle|f_{2}(u,\mathcal{H}u)-f_{2}(v,\mathcal{H}v)|$
$\displaystyle\leq(1+m_{0})\|\kappa\|_{\mathbb{H}^{1}}\|u-v\|_{\mathbb{H}^{1}}$
$\displaystyle\qquad+\|m_{1}*(u-v)\|_{\mathbb{H}^{1}}\|\kappa\|_{\mathbb{H}^{1}}+L_{1}\|\kappa\|\|u-v\|_{\mathbb{H}^{1}}.$

Hence, $f(u,\mathcal{H}u):=g(g,\kappa)^{-1}f_{2}(u,\mathcal{H}u)+f_{1}(u)$
verifies the conditions of Theorem 3.2 with

$\displaystyle L_{f}^{*}$
$\displaystyle=|(g,\kappa)^{-1}|\|g\|[(1+m_{0})\|\kappa\|_{\mathbb{H}^{1}}+L_{1}\|\kappa\|]+L_{1},$
$\displaystyle K_{f}^{*}$
$\displaystyle=|(g,\kappa)^{-1}|\|g\|\|\kappa\|_{\mathbb{H}^{1}}.$

Therefore, problem (4.1)-(4.4) is solvable in the sense that there exists a
unique mild solution $(u,p)\in C([0,T];\mathbb{H}^{1})\times C([0,T])$.

## References

  * [1] E. Bazhlekova, B. Jin, R. Lazarov, Z. Zhou, An analysis of the Rayleigh-Stokes problem for a generalized second-grade fluid, Numer. Math. 131 (2015), no. 1, 1-31.
  * [2] X. Bi, S. Mu, Q. Liu, Q. Liu, B. Liu, P. Zhuang, J. Gao, H. Jiang, X. Li, B. Li, Advanced implicit meshless approaches for the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, Int. J. Comput. Methods 15 (2018), no. 5, 1850032, 27 pp.
  * [3] P.K. Bhattacharyya, Distributions. Generalized functions with applications in Sobolev spaces. De Gruyter Textbook. Walter de Gruyter & Co., Berlin, 2012.
  * [4] M. Bonforte, Y. Sire, J.L.Vázquez, Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst. 35 (2015), no. 12, 5725-5767.
  * [5] C.M. Chen, F. Liu, K. Burrage, Y. Chen, Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative, IMA J. Appl. Math. 78 (2013), no. 5, 924-944.
  * [6] C.M. Chen, F. Liu, V. Anh, Numerical analysis of the Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivatives, Appl. Math. Comput. 204 (2008), no. 1, 340-351.
  * [7] Ph. Clément, J. A. Nohel, Asymptotic behavior of solutions of nonlinear Volterra equations with completely positive kernels, SIAM J. Math. Anal. 12 (1981), 514-535.
  * [8] F. Demengel, G. Demengel, Functional spaces for the theory of elliptic partial differential equations. Translated from the 2007 French original by Reinie Erné. Universitext. Springer, London; EDP Sciences, Les Ulis, 2012.
  * [9] E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math. 136 (2012), no.5, 521-573.
  * [10] C. Fetecau, M. Jamil, C. Fetecau, D. Vieru, The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid, Z. Angew. Math. Phys. 60 (2009), no. 5, 921-933.
  * [11] M. Khan, The Rayleigh-Stokes problem for an edge in a viscoelastic fluid with a fractional derivative model, Nonlinear Anal. Real World Appl. 10 (2009), no. 5, 3190-3195.
  * [12] T.D. Ke, N.N. Thang, On regularity and stability for a class of nonlocal evolution equations with nonlinear perturbations, Commun. Pure Appl. Anal. 21 (2022), no. 3, 817-835.
  * [13] D. Lan, Regularity and stability analysis for semilinear generalized Rayleigh-Stokes equations, Evol. Equ. Control Theory 11 (2022), no. 1, 259-282.
  * [14] N.H.Luc, D.Lan, D.O’Regan, N.A.Tuan, Y. Zhou, On the initial value problem for the nonlinear fractional Rayleigh-Stokes equation, J. Fixed Point Theory Appl. 23 (2021), no. 4, Paper No. 60, 28 pp.
  * [15] N.H. Luc, N.H. Tuan, Y. Zhou, Regularity of the solution for a final value problem for the Rayleigh-Stokes equation, Math. Methods Appl. Sci. 42 (2019), no. 10, 3481-3495.
  * [16] R.K. Miller, On Volterra integral equations with nonnegative integrable resolvents, J. Math. Anal. Appl. 22 (1968), 319-340.
  * [17] T. B. Ngoc, N. H. Luc, V. V. Au, N. H. Tuan and Y. Zhou, Existence and regularity of inverse problem for the nonlinear fractional Rayleigh-Stokes equations, Math. Methods Appl. Sci. 44 (2021), 2532-2558.
  * [18] J. Prüss, Evolutionary Integral Equations and Applications. Monographs in Mathematics 87, Birkhäuser, Basel, 1993.
  * [19] F. Shen, W. Tan, Y. Zhao, T. Masuoka, The Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative model, Nonlinear Anal. Real World Appl. 7 (2006), no. 5, 1072-1080.
  * [20] N.H. Tuan, Y. Zhou, T.N. Thach, N.H. Can, Initial inverse problem for the nonlinear fractional Rayleigh-Stokes equation with random discrete data, Commun. Nonlinear Sci. Numer. Simul. 78 (2019), 104873, 18 pp.
  * [21] Y. Zhou, J.N. Wang, The nonlinear Rayleigh-Stokes problem with Riemann-Liouville fractional derivative, Math. Meth. Appl. Sci. 44 (2021), 2431-2438.


# Reinforcement Learning for Traffic Signal Control: Comparison with
Commercial Systems

Alvaro Cabrejas-Egea  Raymond Zhang  Neil Walton  The Alan Turing Institute,
96 Euston Rd, London NW1 2DB, United Kingdom  MathSys CDT, University of
Warwick, Coventry CV4 7AL, United Kingdom  Ecole Normale Superieure de Paris-
Saclay, 4 Avenue des Sciences, 91190 Gif-sur-Yvette, France  University of
Manchester, Oxford Rd, Manchester M13 9PL, United Kingdom

###### Abstract

Recently, Intelligent Transportation Systems are leveraging the power of
increased sensory coverage and computing power to deliver data-intensive
solutions achieving higher levels of performance than traditional systems.
Within Traffic Signal Control (TSC), this has allowed the emergence of Machine
Learning (ML) based systems. Among this group, Reinforcement Learning (RL)
approaches have performed particularly well. Given the lack of industry
standards in ML for TSC, literature exploring RL often lacks comparison
against commercially available systems and straightforward formulations of how
the agents operate. Here we attempt to bridge that gap. We propose three
different architectures for TSC RL agents and compare them against the
currently used commercial systems MOVA, SurTrac and Cyclic controllers and
provide pseudo-code for them. The agents use variations of Deep Q-Learning and
Actor Critic, using states and rewards based on queue lengths. Their
performance is compared in across different map scenarios with variable
demand, assessing them in terms of the global delay and average queue length.
We find that the RL-based systems can significantly and consistently achieve
lower delays when compared with existing commercial systems.

###### keywords:

Modelling and Simulation; Urban Traffic Control; Reinforcement Learning

capbtabboxtable[][]

<EMAIL_ADDRESS>- Pre-print version, submitted to CIT2021.

##  1 Introduction

Traffic Signal Control (TSC) can be used to ensure the safe and efficient
utilisation of the road network at junctions, where traffic can change
directions and merge, having to manage conflicting individual priorities with
the global needs of the network. Traffic congestion has a major financial
impact. A study by INRIX [2019] shows that traffic congestion in 2019 cost
£$6.9$ billion in the UK alone, with similar patterns being observed in other
developed countries. Cities around the globe are starting to explore the
deployment of smart Urban Traffic Controllers (UTCs) that use real time data
to adjust their stage schedule and green time duration. Traditionally, fixed
time plans have been used, with systems that optimise the green time splits in
a deterministic manner, requiring costly site-specific knowledge and typical
demand profiles to provide effective control. These methods are not easily
scalable and deteriorate over time as the traffic demand changes (Bell and
Bretherton [1986]). With the development of induction loops, real time
actuated UTCs were created in two variants: those that optimise single
isolated intersections with systems such as MOVA (Vincent and Peirce [1988]),
and those that cover multiple intersections such as SCOOT (Hunt et al.
[1982]). To remedy the scalability problem, other systems are based on local
rules, generating a self-organising area traffic controllers, such as SurTrac
(Smith and Barlow [2013]). With the breakthrough of Deep Reinforcement
Learning (DRL) on complex problems such as Atari games or Go (Mnih et al.
[2013], Silver et al. [2017]), attention has turned towards adapting these
approaches to generate industry-grade controllers for traditionally noisy and
systems such as TSC.

This paper aims to reproduce some of the results of the main and most
successful RL approaches on intersections of increasing complexity, while
comparing different architectures of DRL TSC agents, since, given the
complexity of their implementation, most available literature only deals with
a single class.

##  2 State of the Art

We discuss the state of the art in reinforcement learning in signal control.
We discuss the state of the art systems for commercial simulation and traffic
signal control which we analyse in this study.

###  2.1 Previous Work

RL is an area of ML aiming to imitate how biological entities learn, where an
agent evolves in an unknown environment, learning how to perform with no prior
information, based on its interactions said environment. The agent aims to
maximise a reward signal it receives as a feedback for its actions. RL methods
have been applied to TSC in experimental setups. While there is a variety of
approaches in the literature that craft successful RL-based TSC systems, most
do no present comparisons against commercial systems that are the concern of
this paper.

Recent works Gao et al. [2017], Wan and Hwang [2018], Mousavi et al. [2017]
use neural networks as function approximators to avoid the dimensionality and
computing limitations of table based methods in large state-action spaces,
showing DRL TSC can be more efficient than some earlier methods. The first two
use discreet cell encoding vectors to represent the system, which are passed
to a Convolutional Neural Network (CNN), whereas the second directly uses
pixels in the same manner. Gao et al. [2017] compared the results against a
fixed time and longest-queue-first systems, finding RL to perform better,
while Wan and Hwang [2018] found similar results comparing against a fixed
time system. Liang et al. [2018], used the same initial approach and compared
against two different fixed-time systems, ranking better than both and
providing some early evidence of the benefits of using Double DQN (Hasselt
[2010]), Duelling architecture (wang2015dueling) and Prioritised Experience
Replay (PER) (Schaul et al. [2016]). Genders and Razavi [2018] evaluated
different state representations, finding little difference in the performance
of the agents as a result of the change in the magnitudes observed. In
Stevanovic and Martin [2008] the authors compare SCOOT with a Genetic
Algorithm-based control method. It is shown that SCOOT’s performance can be
surpassed by more adaptive Genetic Algorithms that, in turn, tend to be less
effective at learning than RL methods.

Despite these previous works, most results are hard or impossible to reproduce
given the lack of industry standards in terms of simulators, performance
metrics, the lack of availability of commercial algorithms for comparison and
the fierce protection of their internal workings, and the lack of open-source
code of proposed RL models.

###  2.2 Commercial Traffic Signal Control Simulators and Optimisers

PTV Vissim. Our simulations are conducted on PTV Vissim, a state-of-the-art
commerical traffic simulator that is able to produce a wide variety of traffic
demands over an array of signal controls and road traffic scenarios. We
interface our RL algorithms via COM interface, allowing the interface with
Tensorflow to construct deep learning agents.

MOVA (Microprocessor Optimised Vehicle Actuation, Vincent and Peirce [1988])
is a traffic controller designed by TRL Software. It aims to reduce delay on
isolated junctions. The basic functioning of MOVA involves two induction loop
detectors estimating the flow of vehicles in each lane. The system makes a
virtual cell representation of the lanes within MOVA, and then it computes a
performance index based on the delays calculated. If the index results lower
than a certain threshold, the signal is changed to the next stage, otherwise
the stage it is extended.

Surtrac (Scalable URban TRaffic Control, Smith and Barlow [2013]) is
decentralised, with each intersection allocating green time independently and
asynchronously based on incoming flows. Each intersection is controlled by a
local scheduler and communicates projected outflows to the downstream
neighbouring junctions, modelling vehicles as a sequence of clusters. This
allows for locally balancing competing flows while creating ”green corridors”
by finding an optimal sequence such that the input jobs (ordered clusters) are
cleared while minimising the joint waiting time.

##  3 Methods

Traffic Control as a Markov Decision Process. The control problem can be
formulated as a Markov Decision Process (MDP) defined in terms of a 5-tuple: A
set of possible environment states $s\in\mathcal{S}$, a set of actions of the
agent $a\in\mathcal{A}$, a stochastic transition function $\forall
a\in\mathcal{A},\mathcal{T}^{a}_{s,s^{\prime}}\triangleq\mathbb{P}(s_{t+1}=s^{\prime}|s_{t}=s,a_{t}=a)$,
a scalar real valued reward function $R(s_{t},s_{t+1},a_{t})$ providing a
performance measure to the transition generated by progressing into the state
$s_{t+1}$ after taking action $a_{t}$ while in state $s_{t}$, and a discount
factor $\gamma$ that will provide the balance between immediate exploitation
and approaches that aim to maximise returns over time. In the case of TSC, the
MDP is modelled as partially observable, following an unknown stochastic
transition function.

Reinforcement Learning. The goal of the agents is to maximise their future
discounted return, $G_{t}=\sum_{t=0}^{+\infty}\gamma^{t}r(s_{t},a_{t})$ with
$\gamma$ $\in[0,1]$ by learning a policy $\pi$, parametrised by the weights
$\theta$ of the neural network performing the approximation of the reward
function and mapping states to actions:
$\pi:\mathcal{S}\rightarrow\mathcal{A}$. The reward function maps an action
given a state to a reward scalar value:
$r:\mathcal{S}\times\mathcal{A}\rightarrow\mathbb{R}$. The action-value
function or Q-value is $Q_{\pi}(s,a)=\mathbb{E}_{\pi}[G_{t}|s_{t}=s,a_{t}=a]$
representing the total episodic return by following $\pi$ after being in state
$s$ and taking action $a$.

Value-based Reinforcement Learning Methods - DQN. Tabular value-based methods,
such as Q-Learning, attempt to learn an optimal policy
$Q_{\pi}^{*}=\max_{\pi}\mathbb{E}[r_{t}|s_{t}=s,a_{t}=a]$ by iteratively
performing Bellman updates on the Q-values of the individual state-action
pairs:

$Q_{\pi}(s_{t},a_{t})\leftarrow
Q_{\pi}(s_{t},a_{t})+\alpha\big{(}y_{t}-Q_{\pi}(s_{t+1},a_{t+1})\big{)},\,\,\,\,\textrm{with}\,\,\,\,y_{t}=r_{t}+\gamma\max_{a^{\prime}}Q_{\pi}(s_{t+1},a^{\prime},\theta^{\prime}),$
(1)

where $\alpha$ is the learning rate and $y_{t}$ is the Temporal Difference
(TD) target for the value function.

Deep Q-Network (DQN) agents are an evolution of Q-Learning. The purpose of the
agent is to find an approximation of $Q_{\pi}*$ by tuning the weights $\theta$
of a neural network. The agent keeps a second neural network, the target
network, parametrised by the weights vector $\theta^{\prime}$ which is used to
generate the TD targets.

The experience replay memory increases training stability, obtaining samples
that cover a wider range of situations and that can be used several times for
gradient descent. Three additional modules have been applied to the agent to
improve performance, Double Q Learning, PER, and Dueling Architecture.The DQN
variants implemented are described on the algorithms displayed in Figs. 1(a)
and 1(b), and use the hyperparameters described in Fig. 2(b).

(a)

(b)

Fig. 1: (a): DDQN Pseudocode. (b): DDDQN Pseudocode.

Policy Gradient Reinforcement Learning Methods - A2C.  Policy Gradient in RL
is based on the idea that obtaining a direct policy $\pi(s)$ mapping states to
actions can be easier than estimating the value function or the state-action
values. It has an added benefit in that it can learn stochastic policies,
generating a probability distribution over the potential actions. The goal is
to find the policy that maximises the reward. To do so one has to perform
gradient ascent on the performance measure
$J=\sum_{a\in\mathcal{A}}\mathbb{E}[Q(s_{0},a)\pi(a|s)]$.

The Advantage Actor Critic (A2C) method tries to reduce the variance in the
policy method by combining the direct mapping from actions with the value-
based approximation method. The goal is to learn an actor

$\pi_{\theta}=\mathbb{P}_{\theta}[a_{t}=a|s_{t}=s],\,\,\,\,\textrm{and a
critic}\,\,\,\,V_{\pi}^{\theta}(s)=\mathbb{E}_{\theta}[G_{t}|s_{t}=s],$ (2)

both of which are parametrised by the neural network weights vector $\theta$.
The pseudocode for the A2C agent can be found in Fig.2(a) and its
hyperparameters are displayed in Fig.2(c).

(a)

(b)

(c)

Fig. 2: (a): A2C Pseudocode. (b): DDQN/DDDQN Hyperparameters. (c): A2C
Hyperparameters.

###  3.1 State, Action and Rewards of the agents

Agent Configuration.  All the experiments here presented use the same
descriptions for state and reward calculation, differing in the number of
actions available to them. The state of an intersection of $l$ lanes will be
presented to the agents as a state vector $s\in\mathbb{R}^{l+1}$, in which
each component represents the length of the queue of vehicles measured
upstream from the traffic light in metres. The last component will be current
stage being implemented.

While marginal improvements in performance can be obtained by using different
variables for reward (Cabrejas Egea et al. , Cabrejas-Egea and Connaughton
[2020]), as per the discussion of Heydecker [2004], queues can be a reasonable
choice for states and rewards, being able to transmit useful information to
the agent relative to the mean rate of delay of the system. Based on this, the
reward after an action will be calculated as the negative sum of the length of
the queues of all lanes immediately upstream from the intersection:
$r_{t}=-\sum_{l}q_{l,t}.$

The agents were trained using a fixed vehicle demand of $400$ vehicles per
hour on each of the incoming lanes. Both DQN variants were trained for $400$
episodes, using an $\epsilon$ geometrically annealed from $1$ to $0.001$. The
A2C agents were trained for $100$ episodes until they converged. Best
performing agents in each class were selected for benchmarking and evaluated
in scenarios lasting one hour.

Agent Benchmarking.  In order to compare the agents performance, a testing
framework was defined. For each model, a demand profile will be created,
following the shape found in a _typical day_ using the methodology introduced
in Cabrejas Egea et al. [2018] and expanded in Cabrejas Egea and Connaughton
[2020]. The profile will be split on 10 segments of length 6 minutes. Each of
these will correspond with a level of demand. The demand levels are obtained
by setting the maximum demand the intersection will suffer, setting that
magnitude to coincide with the peaks of the distribution that could be found
on said _typical day_ , and will be specified in each experiment’s section.
Random seeds are updated after every simulation episode, training or testing.
The quantitative metrics on which the system will be evaluated are the Global
Cumulative Delay (deviations from free-flow time) and the Average Queue Length
generated during the evaluation.

##  4 Experimental Results

(a)

(b)

Fig. 3: (a): Cross Straight Map. (b): Cross Triple Map.

Experiment 1: Cross Straight. The first test is conducted on the simplest
junction, shown in Fig 3(a). The junction is composed on 4 lanes distributed
in 4 arms. The controller has two stages, a north-south stage and an east-west
stage, and turning is not allowed. The aim was to perform an initial
performance comparison of DRL algorithms against MOVA, SUTRAC, and a cyclic
controller. Here the goal for the agent was to exert fine adaptive timing
control while extrapolating, rather than using complicated transitions between
stages. MOVA was configured using loop detectors set in accordance to its
manual, the implementation of Surtrac follows the work of Xie et al. [2012]
.During evaluation, an average of 2120 vehicles are inserted in the model,
with 2 peaks of demand of 3000 veh/h for 6 minutes each.

Figure 6 and Table 1 show the Global Cumulative Delay and average queue length
for the network. As expected, the cyclic solution is outperformed by all
adaptive controllers. The different controllers are on a par with a slight
advantage for the DuelingDDQN which saves the community an average of 3000
seconds compared to MOVA on this hour of simulation, which represents on
average 1-2 seconds per vehicle. RL agents also seem slightly more robust
against changes in demand, producing lower slopes in the delay graphs in
sections of extreme demand.

(a)

(b)

Fig. 4: Global Cumulative Delay in Single Cross for DQN variants, A2C, Surctrac, MOVA and the reference Cyclic controller. Table 1: Cumulative Delay and Cumulative Stop Delay in seconds across Controllers on Single Cross Straight. Controller | Cumulative Delay [s] | Average Sum of Queues [m]
---|---|---
Cyclic | 143660.50 | 132.37
MOVA | 27187.53 | 60.59
SURTRAC | 29008.36 | 72.41
A2C | 26382.14 | 56.07
DDQN | 28303.94 | 50.11
DDDQN | 21286.86 | 49.42

The cyclic controller resulted in saturated lanes during both peaks and queues
in excess of $150$ metres during a great part of the simulation. MOVA suffered
two moments in which at least a sensor was saturated coinciding with the peaks
in demand, however the queues were close to lengths of around $50$ metres
during the most part of the simulator. Surtrac followed a similar pattern,
having a single lane saturated coinciding with the second peak in demand. RL
agents as suffered no saturation in any of their lanes during the length of
the evaluation. They all managed a more balanced distribution of queues in
their respective lanes, displaying a higher ability to balance loads during
peak times. Because of the simplicity of this 2 actions intersection, there is
not a lot of delay difference between adaptive UTCs. As it will be appreciated
shortly, these results will change when we consider more complex junctions.

Given the difference in performance between the adaptive and cyclic
controllers, which is expected to become greater on more complex
intersections, and the increasing difficulty in setting them in large
intersections, the cyclic controller will be omitted for the next examples.
Given that the A2C agent has been clearly outperformed in this experiment by
those based on the DQN architecture, the following experiments will focus on
the performance of this last architecture compared with commercial systems.

Experiment 2: Cross Triple - 4 actions. This junction, as shown in Fig. 3(b),
displays a much higher complexity than the intersection presented in the
previous section. It is composed of 4 incoming links of 3 lanes each. In each
incoming link, the left lane serves a dedicated nearside turning lane, the
central allows for forward travel and the right lane allows for both offside
turning and going straight. To mitigate this, the first experiment was run
with agents that would take $4$ queue inputs, plus the state of the traffic
signal as state input. The action set was consequently limited to $4$
different actions, being allowed only those that set to green the $3$ traffic
lights serving the lanes of the same incoming link. This allows for turning
vehicles, but prevents more sophisticated stages from happening. During the
hour of evaluation, the demand profile from the last experiment was used with
a scaling factor of $1.5$, an average of $3180$ vehicles were introduced to
the model, with $2$ peaks of demand of $4500$ vehicles/hour for 6 minutes
each.

As it can be seen in Table 2 the UTC using MOVA performs poorly compared to
the DQN-based agents. During this hour of simulation RL agents halve the
cumulative delay, saving over $27$ hours of travel time for all vehicles
involved, an average of over $32$ seconds of per vehicle. The length of the
queues in those intersections controlled by RL agents during the test scenario
were lower than the ones controlled by MOVA. Further in Figure 6(b) we see
that DDQN significantly outperforms MOVA over a wide range of loads and
traffic scenarios. It can be seen that the agent using Dueling Q-Learning has
a better performance than that Dueling Double Q-Learning.

(a)

(b)

Fig. 5: (a) Global Cumulative Delay in Cross Triple - 4 actions. (b) Global
Cumulative Delay in Cross Triple - 8 actions.

(a)

(b)

Fig. 6: (a) Queue length by demand level in Cross Straight. (b) Queue length by demand level in Cross Triple. Table 2: Cumulative Delay in seconds and Average Sum of Max Queue length in meters on Cross Triple - 4 actions. Controller | Cumulative Delay [s] | Average Max Queue Length [m]
---|---|---
MOVA | 260257.65 | 179.27
DDQN | 135220.91 | 153.58
DDDQN | 155563.22 | 128.20

Experiment 3: Cross Triple - 8 actions.  In order to allow the use of a higher
variety of stages in the controllers the map was reworked. All lanes were
partitioned into their own independent links, allowing extra space for lane
changes. While these modifications allowed using information from all lanes in
an akin manner to what modern sensors would achieve, due the lane changing
limitations, direct comparisons with Experiment 2 must be handled with care.
Both models share name and rough geometry, but the lanes layout is changed and
so are the routing possibilities.

The results presented below, use DQN agents taking $12$ queue length inputs
plus the state of the signal. There are $8$ different stages. Here there are
$4$ additional stages available for non-conflicting cross traffic. No specific
stage order is enforced, and the agents are free to change between any
combination of stages.

Table 3: Cumulative Delay and Cumulative Stop Delay in seconds across Controllers on Cross Triple - 8 actions. Controller | Cumulative Delay [s] | Average Sum of Queues [m]
---|---|---
MOVA | 165456.44 | 339.41
DDQN | 72642.59 | 123.52
DDDQN | 71245.61 | 119.86

The RL agents display a similar gap in performance with MOVA as in the
previous experiment, with both classes benefiting from the increased actions
pool. RL agents manage to generate about a third of the delay produced by
MOVA. While this appears to be a great success, these results have to be put
into context. MOVA has a lot of internal parameters meant to be fine tuned by
a traffic engineer with site-specific knowledge. Our settings did produce a
successful control loop, operating in line with what was expected of the
configuration process. None of the RL agents has been fine tuned to the level
that would be expected during commercial operation. The layers and neuron
distribution wasn’t optimised, nor were the activation functions, meaning that
the RL agents can still be improved upon.

##  5 Discussion

Several neural network architectures for RL controllers were tested. The
agents did not require extensive or complex configurations to adequately
control traffic junctions, outperforming the commercial controllers. RL agents
showed great stability and robustness to control situations within their
training envelope as well as outside of it. Additionally, agents trained on
relatively low uniform demand showed they can perform better than commercial
systems during evaluation tests that included variable demand 5 times higher
than anything experienced during training.

In Experiment 1 MOVA and the RL agent following a DuelingDDQN architecture
obtained very similar results, with a slight advantage for the RL agent.
Experiment 2 implies less granularity in the data and makes the control task
more challenging. The results followed the same pattern with a DuelingDDQN
agent obtaining the best performance over an array of loadings, despite the
lower resolution in the input data. Experiment 3 saw the introduction of a
much more complex intersection. Once again RL agents obtained significantly
better results than MOVA, with the DuelingDDQN agent obtaining the lowest
global and stop delay. The gap between the performance of MOVA and RL agents
is increased here with respect to the last experiment. Most likely reasons are
higher granularity in the data and extra actions being available to the agent,
allowing it to display more complex sequences of actions.

We find that Reinforcement Learning applied to UTC can significantly
outperform current adaptive traffic controllers in realistic commercial
simulation software. These experiments provide credible evidence that
Reinforcement Learning based UTC will be part of the next generation of
traffic signal controllers.

Acknowledgements. This work was part funded by EPSRC Grant EP/L015374, by The
Alan Turing Institute and the Toyota Mobility Foundation. The authors thank
Dr. W. Chernicoff for the discussions that made this project possible.

## References

  * Bell and Bretherton [1986] Bell, M.C., Bretherton, R.D., 1986\.  Ageing of fixed-time traffic signal plans, in: International conference on road traffic control.
  * Cabrejas-Egea and Connaughton [2020] Cabrejas-Egea, A., Connaughton, C., 2020\.  Assessment of Reward Functions in Reinforcement Learning for Multi-Modal Urban Traffic Control under Real-World limitations.  arXiv preprint arXiv:2010.08819 arXiv:arXiv:2010.08819.
  * Cabrejas Egea and Connaughton [2020] Cabrejas Egea, A., Connaughton, C., 2020\.  Wavelet Augmented Regression Profiling (WARP): improved long-term estimation of travel time series with recurrent congestion, in: IEEE Conference on Intelligent Transportation Systems, Proceedings, ITSC.  doi:10.1109/itsc45102.2020.9294318.
  * Cabrejas Egea et al. [2018] Cabrejas Egea, A., De Ford, P., Connaughton, C., 2018.  Estimating Baseline Travel Times for the UK Strategic Road Network, in: IEEE Conference on Intelligent Transportation Systems, Proceedings, ITSC, IEEE. pp. 531–536.  doi:10.1109/ITSC.2018.8569924.
  * [5] Cabrejas Egea, A., Howell, S., Knutins, M., Connaughton, C., .  Assessment of Reward Functions for Reinforcement Learning Traffic Signal Control under Real-World Limitations, in: IEEE International Conference on Systems, Man and Cybernetics, Proceedings, SMC.  doi:10.1109/smc42975.2020.9283498.
  * Gao et al. [2017] Gao, J., Shen, Y., Liu, J., Ito, M., Shiratori, N., 2017\.  Adaptive traffic signal control: Deep reinforcement learning algorithm with experience replay and target network.  arXiv preprint arXiv:1705.02755 .
  * Genders and Razavi [2018] Genders, W., Razavi, S., 2018\.  Evaluating reinforcement learning state representations for adaptive traffic signal control.  Procedia computer science 130, 26–33.
  * Hasselt [2010] Hasselt, H.V., 2010.  Double Q-learning, in: Advances in neural information processing systems, pp. 2613–2621.
  * Heydecker [2004] Heydecker, B.G., 2004.  Objectives, stimulus and feedback in signal control of road traffic.  Journal of Intelligent Transportation Systems 8, 63–76.
  * Hunt et al. [1982] Hunt, P.B., Robertson, D.I., Bretherton, R.D., Royle, M.C., 1982\.  The SCOOT on-line traffic signal optimisation technique.  Traffic Engineering & Control 23\.
  * INRIX [2019] INRIX, 2019.  Scorecard.  URL: http://inrix.com/scorecard/.
  * Liang et al. [2018] Liang, X., Du, X., Wang, G., Han, Z., 2018.  Deep Reinforcement Learning for Traffic Light Control in Vehicular Networks XX, 1–11.  URL: http://arxiv.org/abs/1803.11115, arXiv:1803.11115.
  * Mnih et al. [2013] Mnih, V., Kavukcuoglu, K., Silver, D., Graves, A., Antonoglou, I., Wierstra, D., Riedmiller, M., 2013.  Playing atari with deep reinforcement learning.  arXiv preprint arXiv:1312.5602 .
  * Mousavi et al. [2017] Mousavi, S.S., Schukat, M., Howley, E., 2017.  Traffic light control using deep policy-gradient and value-function-based reinforcement learning.  IET Intelligent Transport Systems 11, 417–423.  doi:10.1049/iet-its.2017.0153.
  * Schaul et al. [2016] Schaul, T., Quan, J., Antonoglou, I., Silver, D., 2016\.  Prioritized experience replay.  4th International Conference on Learning Representations, ICLR 2016 - Conference Track Proceedings arXiv:1511.05952.
  * Silver et al. [2017] Silver, D., Schrittwieser, J., Simonyan, K., Antonoglou, I., Huang, A., Guez, A., Hubert, T., Baker, L., Lai, M., Bolton, A., Others, 2017.  Mastering the game of go without human knowledge.  nature 550, 354–359.
  * Smith and Barlow [2013] Smith, S.F., Barlow, G.J., 2013\.  SURTRAC : Scalable Urban Traffic Control.  Transport Researach Board .
  * Stevanovic and Martin [2008] Stevanovic, A., Martin, P.T., 2008\.  Split-cycle offset optimization technique and coordinated actuated traffic control evaluated through microsimulation.  Transportation Research Record 2080, 48–56.
  * Vincent and Peirce [1988] Vincent, R.A., Peirce, J.R., 1988\.  ’MOVA’: Traffic Responsive, Self-optimising Signal Control for Isolated Intersections.  Traffic Management Division, Traffic Group, Transport and Road Research.
  * Wan and Hwang [2018] Wan, C.H., Hwang, M.C., 2018\.  Value-based deep reinforcement learning for adaptive isolated intersection signal control.  IET Intelligent Transport Systems 12, 1005–1010.
  * Xie et al. [2012] Xie, X.F., Smith, S.F., Lu, L., Barlow, G.J., 2012\.  Schedule-driven intersection control.  Transportation Research Part C: Emerging Technologies 24, 168–189.  doi:10.1016/j.trc.2012.03.004.