1 Introduction
In this paper we study an initial–Neumann boundary value problem for the following non-autonomous stochastic partial differential equation of parabolic type in a cylinder domain $D\times [0,T]$, driven by an infinite-dimensional fractional noise:
Here $D\subset {\mathbb{R}^{d}}$ is a bounded domain with the boundary $\partial D$ of class ${C^{2+\beta }}$ with some $\beta \in (0,1)$, ${W^{H}}$ is an ${L^{2}}(D)$-valued fractional Brownian motion with the Hurst index $H\in (\frac{1}{2},1)$, $k=\left\{{k_{i,j}}\right\}:\overline{D}\times [0,T]\to {\mathbb{R}^{d\times d}}$ is a matrix-valued field, $n(k)(x):=k(x,t)n(x)$ denotes the conormal vector-field, and the last relation in (1) refers to the conormal derivative of u relative to k, that is
(1)
\[\begin{aligned}{}du(x,t)& =\big(\operatorname{div}\big(k(x,t)\nabla u(x,t)\big)+f\big(u(x,t)\big)\big)dt+h\big(u(x,t)\big){W^{H}}(x,dt),\\ {} (x,t)& \in D\times (0,T],\\ {} u(x,0)& =\varphi (x),\hspace{1em}x\in D,\\ {} \frac{\partial u(x,t)}{\partial n(k)}& =0,\hspace{1em}(x,t)\in \partial D\times (0,T].\end{aligned}\]
\[ \frac{\partial u(x,t)}{\partial n(k)}={\sum \limits_{i,j=1}^{d}}{k_{i,j}}(x,t){n_{i}}(x)\frac{\partial }{\partial {x_{j}}}u(x,t),\]
$n(x)\in {\mathbb{R}^{d}}$ is an outer normal vector to $\partial D$.Equations similar to (1) were studied extensively in literature, so we will mention only several most relevant articles here. The articles [5] and [7] considered heat equations with additive and multiplicative fractional noise, respectively. The articles [1, 10, 18] are devoted to general non-linear evolution equations with fractional noise; however, the equations are considered in some functional spaces, and the assumptions on the coefficients imposed there do not cover general nonlinear equations of the form (1). The problem (1) was considered in [14] and then in [16], where the notions of variational and mild solutions were introduced. The article [14] established the existence of a variational solution to this equation, but the uniqueness was shown under the assumption that the function h is affine. In [16], it was shown that a variational solution to (1) is a mild solution too, however, the uniqueness, both in the variational and in the mild sense, was established also under the assumption that h is affine, moreover, a rather restrictive assumption $H>\frac{d+1}{d+2}$ on the Hurst exponent was imposed.
Our goal is to extend the uniqueness results of [16] to the case of arbitrary $H\in (\frac{1}{2},1)$ and non-affine h. Specifically, we prove the uniqueness of a mild solution, assuming that h and its derivative ${h^{\prime }}$ are Lipschitz continuous. Since the existence of a variational solution is known from [14], and [16] established that each variational solution is a mild solution, we get existence and uniqueness of a variational solution too, thus finally answering a question posed in [14]. We also show that the solution to (1) has finite moments of any order.
It is worth to mention that a similar uniqueness result holds in the case where the function h in front of ${W^{H}}$ depends on t sufficiently regularly, say, Hölder continuous with exponent greater than $1/2$. However, since our main reference for existence results are the articles [14, 16], in which h is assumed to be independent of t, we will follow this assumption.
The paper is organized as follows. In Section 2, we formulate the main hypotheses, and give the definition of a mild solution and basic facts on an ${L^{2}}(D)$-valued fractional Brownian process and stochastic integration with respect to it. Section 3 contains auxiliary results concerning the parabolic Green’s function. In Section 4, we give a priori upper bounds for mild solutions. Finally, the main result on existence and uniqueness of a mild solution is proved in Section 5.
2 Preliminaries
This section is devoted to the precise statement of the problem (1). We introduce necessary notation, give the definition of a mild solution, and formulate the assumptions for its existence and uniqueness.
2.1 Notational conventions
Throughout the article, $\left|\hspace{0.1667em}\cdot \hspace{0.1667em}\right|$ will denote absolute value of a number, Euclidean norm of a vector or operator norm of a matrix; exact meaning will always be clear from the context. We will use the symbol C for a generic constant, the precise value of which is not important and may vary between different equations and inequalities.
2.2 Norms and spaces
Let ${\left\| \hspace{0.1667em}\cdot \hspace{0.1667em}\right\| _{2}}$ and ${\left\| \hspace{0.1667em}\cdot \hspace{0.1667em}\right\| _{\infty }}$ be the norms in ${L^{2}}(D)$ and ${L^{\infty }}(D)$, respectively. Denote for $\alpha \in (0,1)$, $u:D\times [0,T]\to \mathbb{R}$ and $t\in [0,T]$
\[\begin{aligned}{}{\left\| u\right\| _{\alpha ,1,t}}& :=\underset{x\in D}{\sup }\underset{s\in [0,t]}{\sup }{\int _{0}^{s}}\frac{\left|u(x,s)-u(x,v)\right|}{{(s-v)^{\alpha +1}}}\hspace{0.1667em}dv,\\ {} {\left\| u\right\| _{\alpha ,\infty ,t}}& :=\underset{s\in [0,t]}{\sup }{\left\| u(\cdot ,s)\right\| _{\infty }}+{\left\| u\right\| _{\alpha ,1,t}},\\ {} {\left\| u\right\| _{\alpha ,2,t}}& :={\Bigg(\underset{s\in [0,t]}{\sup }{\left\| u(\cdot ,s)\right\| _{2}^{2}}+{\int _{0}^{t}}{\Bigg({\int _{0}^{s}}\frac{{\left\| u(\cdot ,s)-u(\cdot ,v)\right\| _{2}}}{{(s-v)^{\alpha +1}}}\hspace{0.1667em}dv\Bigg)^{2}}ds\Bigg)^{1/2}}.\end{aligned}\]
Denote by ${\mathcal{B}^{\alpha ,2}}(0,T;{L^{2}}(D))$ the Banach space of measurable mappings $u:D\times [0,T]\to \mathbb{R}$ such that ${\left\| u\right\| _{\alpha ,2,T}^{2}}<\infty $.Let ${H^{1}}(D)$ be the Sobolev space of functions $f:D\to \mathbb{R}$ equipped with the norm ${\left\| f\right\| _{1,2}}={({\left\| f\right\| _{2}^{2}}+{\left\| \nabla f\right\| _{2}^{2}})^{1/2}}$. Also introduce the space ${L^{2}}(0,T;{H^{1}}(D))$ of measurable mappings $u:[0,T]\to {H^{1}}(D)$ such that
2.3 Assumptions on the coefficients and on the initial value
-
(A1) The coefficients ${k_{ij}}$ satisfy the following assumptions:
-
(i) ${k_{i,j}}={k_{j,i}}$ for all $i,j=1,\dots ,d$;
-
(ii) ${k_{i,j}}\in {C^{\beta ,{\beta ^{\prime }}}}(\overline{D}\times [0,T])$ for some ${\beta ^{\prime }}\in (\frac{1}{2},1]$ and for all $i,j=1,\dots ,d$;
-
(iii) $\frac{\partial }{\partial {x_{l}}}{k_{i,j}}\in {C^{\beta ,\beta /2}}(\overline{D}\times [0,T])$ for all $i,j,l=1,\dots ,d$;
-
(v) $(x,t)\mapsto {\displaystyle\sum \limits_{i=1}^{d}}{k_{i,j}}(x,t){n_{i}}(x)\in {C^{1+\beta ,(1+\beta )/2}}(\partial D\times [0,T])$ for each j;
-
(vi) the conormal vector-field $(x,t)\mapsto n(k)(x,t)=k(x,t)n(x)$ is outward pointing, nowhere tangent to $\partial D$ for every t.
-
-
(A2) The initial condition $\varphi \in {C^{2+\beta }}(\overline{D})$ satisfies the conormal boundary condition relative to k.
-
(A3) $f,h,{h^{\prime }}:\mathbb{R}\to \mathbb{R}$ are globally Lipschitz continuous functions.
Remark 1.
The global Lipshitz assumption implies that f and h are of linear growth:
It is worth to mention that all results of the article can be proved assuming linear growth and only local Lipschitz continuity of f and ${h^{\prime }}$ with some extra technical work. We decided to impose the global Lipschitz continuity assumption for the sake of simplicity and because it does not lead to a considerable loss of generality.
2.4 ${L^{2}}(D)$-valued fractional Brownian process and stochastic integration with respect to it
Let us briefly recall the definition of an ${L^{2}}(D)$-valued fractional Brownian process and the corresponding stochastic integral, introduced in [10]. Assume that $\{{\lambda _{j}},j\hspace{0.1667em}\in \hspace{0.1667em}\mathbb{N}\}$ is a sequence of positive real numbers and $\left\{{e_{j}},j\in \mathbb{N}\right\}$ is an orthonormal basis of ${L^{2}}(D)$ such that
Let $(\varOmega ,\mathcal{F},\mathsf{P})$ be a complete probability space. For a fixed $T>0$ let $\mathbb{F}={\left\{\mathcal{F}\right\}_{t\in [0,T]}}$ be a filtration satisfying the standard assumptions. Let ${B_{j}^{H}}=\{{B_{j}^{H}}(t),t\ge 0\}$, $j\in \mathbb{N}$, be a sequence of one-dimensional, independent fractional Brownian motions with the Hurst parameter $H\in (1/2,1)$, defined on $(\varOmega ,\mathcal{F},\mathbb{F},\mathsf{P})$ and starting at the origin. Following [10], define ${L^{2}}(D)$-valued fractional Brownian process ${W^{H}}=\left\{{W^{H}}(\cdot ,t),t\ge 0\right\}$ by
\[ {W^{H}}(\cdot ,t)={\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}{e_{j}}(\cdot ){B_{j}^{H}}(t),\]
where the series converges a. s. in ${L^{2}}(D)$.In this article we consider a pathwise stochastic integration with respect to ${W^{H}}$ in the fractional (generalized Lebesgue–Stieltjes) sense. Alternatively, one can look at the so-called Skorokhod (white-noise) integral. However, with the Skorokhod definition, it is difficult to solve even stochastic ordinary differential equations, see e.g. [8].
Fix $\alpha \in (1-H,1/2)$. Let $\varPhi =\left\{\varPhi (t),t\in [0,T]\right\}$ be an adapted stochastic process taking values in the space of linear bounded operators on ${L^{2}}(D)$ such that
for some constant ${C_{\alpha }}>0$.
\[ \underset{j\in \mathbb{N}}{\sup }{\int _{0}^{T}}\hspace{-0.1667em}\Bigg(\frac{{\left\| \varPhi (t){e_{j}}\right\| _{2}}}{{t^{\alpha }}}+{\int _{0}^{t}}\frac{{\left\| (\varPhi (t)-\varPhi (s)){e_{j}}\right\| _{2}}}{{(t-s)^{\alpha +1}}}ds\Bigg)dt<\infty .\]
Following [10] (see also [14, 16]), we introduce the integral with respect to an ${L^{2}}(D)$-valued fractional Brownian process by
\[ {\int _{a}^{b}}\varPhi (s)\hspace{0.1667em}d{W^{H}}(s):={\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}{\int _{a}^{b}}\varPhi (s){e_{j}}\hspace{0.1667em}d{B_{j}^{H}}(s),\]
where the integrals with respect to ${B_{j}^{H}}$, $j\in \mathbb{N}$, are understood as pathwise generalized Lebesgue–Stieltjes integrals. Such integrals are defined in terms of fractional derivatives, the detailed exposition of this approach can be found, e. g., in the book [12, Section 2.1]. We mention only that under above assumptions, the generalized Lebesgue–Stieltjes integral ${\int _{a}^{b}}\varPhi (s){e_{j}}\hspace{0.1667em}d{B_{j}^{H}}(s)$ is well defined and admits the bound
(3)
\[\begin{aligned}{}& \left|{\int _{a}^{b}}\varPhi (s){e_{j}}\hspace{0.1667em}d{B_{j}^{H}}(s)\right|\\ {} & \hspace{1em}\le {C_{\alpha }}{\left\| {B_{j}^{H}}\right\| _{\alpha ,0,b}}{\int _{a}^{b}}\Bigg(\frac{\left|\varPhi (s){e_{j}}\right|}{{(s-a)^{\alpha }}}+{\int _{a}^{s}}\frac{\left|(\varPhi (s)-\varPhi (v)){e_{j}}\right|}{{(s-v)^{\alpha +1}}}\hspace{0.1667em}dv\Bigg)ds\end{aligned}\]2.5 Mild solution
Following [16], we understand a solution to the problem (1) in a mild sense. Its definition uses the notion of the parabolic Green’s function $G(x,t,y,s)$, $x,y\in \overline{D}$, $0\le s<t\le T$, associated with the principal part of (1) (see, e. g., [2–4, 9]). For every $(y,s)\in D\times (0,T]$, $G(x,t,y,s)$ is a classical solution to the linear initial-boundary value problem
with
(4)
\[\begin{aligned}{}{\partial _{t}}G(x,t;y,s)& =\operatorname{div}\big(k(x,t){\nabla _{x}}G(x,t;y,s)\big),\hspace{1em}(x,t)\in D\times (0,T],\\ {} \frac{\partial G(x,t;y,s)}{\partial n(k)}& =0,\hspace{1em}(x,t)\in \partial D\times (0,T],\end{aligned}\]
\[ {\int _{D}}G(\cdot ,s;y,s)\varphi (y)\hspace{0.1667em}dy:=\underset{t\downarrow s}{\lim }{\int _{D}}G(\cdot ,t;y,s)\varphi (y)\hspace{0.1667em}dy=\varphi (\cdot ).\]
In the next section we consider the properties of G in detail. Definition 1 ([16]).
Fix $H\in (1/2,1)$ and $\alpha \in (1-H,1/2)$. An ${L^{2}}(D)$-valued random field $\left\{u(\cdot ,t),t\in [0,T]\right\}$ is a mild solution to the problem (1) if the following two conditions are satisfied:
-
(1) $u\in {L^{2}}(0,T;{H^{1}}(D))\cap {\mathcal{B}^{\alpha ,2}}(0,T;{L^{2}}(D))$ a. s.
-
(2) The relationholds a. s. for every $t\in [0,T]$ as an equality in ${L^{2}}(D)$.
(5)
\[\begin{aligned}{}u(\cdot ,t)& ={\int _{D}}G(\cdot ,t;y,0)\varphi (y)\hspace{0.1667em}dy+{\int _{0}^{t}}{\int _{D}}G(\cdot ,t;y,s)f\big(u(y,s)\big)\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} & \hspace{1em}+{\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}{\int _{0}^{t}}{\int _{D}}G(\cdot ,t;y,s)h\big(u(y,s)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(s)\end{aligned}\]
3 Properties of Green’s function
In this section we collect several upper bounds for Green’s function G, needed for the proof of the main result.
Denote
It is known from [2, 3] that under assumptions (A1) and (A2) G is a continuous function, twice continuously differentiable in x, once continuously differentiable in t. Moreover, G satisfies the heat kernel estimates
for $\mu =({\mu _{1}},\dots ,{\mu _{d}})$, ${\mu _{1}},\dots ,{\mu _{d}},\nu \in \mathbb{N}\cup \left\{0\right\}$, and ${\left|\mu \right|_{1}}+2\nu \le 2$ with ${\left|\mu \right|_{1}}={\sum _{j=1}^{d}}{\mu _{j}}$. In particular, for ${\left|\mu \right|_{1}}=\nu =0$, we have
The inequality (7) is sometimes called the Gaussian property of G ([15, 16]).
(6)
\[ \left|{\partial _{x}^{\mu }}{\partial _{t}^{\nu }}G(x,t;y,s)\right|\le C{(t-s)^{-({\left|\mu \right|_{1}}+2\nu )/2}}{\varPhi _{t-s}^{C}}(x-y)\]Several important properties of the parabolic Green’s function G follow from the fact that it is, for every $(x,t)\in D\times [0,T]$, a classical solution to the linear boundary value problem
for ${\left|\mu \right|_{1}}+2\nu \le 2$, and, moreover, the following convolution formula holds:
see [2, Appendix, p. 232–233] for the details.
\[\begin{aligned}{}{\partial _{s}}G(x,t;y,s)& =-\operatorname{div}\big(k(y,s){\nabla _{y}}G(x,t;y,s)\big),\hspace{1em}(y,s)\in D\times (0,T],\\ {} \frac{\partial G(x,t;y,s)}{\partial n(k)}& =0,\hspace{1em}(y,s)\in \partial D\times (0,T],\end{aligned}\]
dual to (4). In particular, along with (6) we have also
(8)
\[ \left|{\partial _{y}^{\mu }}{\partial _{s}^{\nu }}G(x,t;y,s)\right|\le C{(t-s)^{-({\left|\mu \right|_{1}}+2\nu )/2}}{\varPhi _{t-s}^{C}}(x-y)\](9)
\[ G(x,t;y,s)={\int _{D}}G(x,t;z,\sigma )G(z,\sigma ;y,s)\hspace{0.1667em}dz\hspace{1em}\text{for all}\hspace{2.5pt}\sigma \in (s,t),\]Furthermore, according to Eqs. (3.4)–(3.5) from [16], G satisfies the following inequalities for all $x,y\in D$ and $\delta \in (\frac{d}{d+2},1)$.
Proof.
Write
By (9), the equality
(13)
\[\begin{aligned}{}& G(x,t;y,v)-G(x,s;y,v)-G(x,t;y,r)+G(x,s;y,r)\\ {} & \hspace{1em}={\int _{s}^{t}}{\int _{r}^{v}}\frac{{\partial ^{2}}}{\partial \theta \hspace{0.1667em}\partial \tau }G(x,\theta ;y,\tau )\hspace{0.1667em}d\tau \hspace{0.1667em}d\theta .\end{aligned}\]
\[ \frac{{\partial ^{2}}}{\partial \theta \hspace{0.1667em}\partial \tau }G(x,\theta ;y,\tau )={\int _{D}}{\partial _{\theta }}G(x,\theta ;z,\sigma ){\partial _{\tau }}G(z,\sigma ;y,\tau )\hspace{0.1667em}dz\]
holds with $\sigma =\frac{\tau +\theta }{2}$. By applying the bounds (6) and (8) with ${\left|\mu \right|_{1}}=0$ and $\nu =1$, we see that
\[\begin{aligned}{}\left|\frac{{\partial ^{2}}}{\partial \theta \hspace{0.1667em}\partial \tau }G(x,\theta ;y,\tau )\right|& \le {\int _{D}}\left|{\partial _{\theta }}G(x,\theta ;z,\sigma ){\partial _{\tau }}G(z,\sigma ;y,\tau )\right|\hspace{0.1667em}dz\hspace{0.1667em}d\sigma \\ {} & \le C{(\theta -\sigma )^{-1}}{(\sigma -\tau )^{-1}}{\int _{D}}{\varPhi _{\theta -\sigma }^{C}}(x-z){\varPhi _{\sigma -\tau }^{C}}(z-y)\hspace{0.1667em}dz\\ {} & \le C{(\theta -\tau )^{-2}}{\varPhi _{\theta -\tau }^{C}}(x-y).\end{aligned}\]
Combining this bound with (13), we conclude the proof. □4 A priori estimates
Fix $H\in (1/2,1)$ and $\alpha \in (1-H,1/2)$. Let u be a mild solution to (1), defined by (5). Note that the random variable
\[ {\xi _{\alpha ,H,T}}:=1+{\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}{\left\| {B_{j}^{H}}\right\| _{\alpha ,0,T}}\]
is finite a. s., see [11].The goal of this section is to prove the following result.
We split the proof of Proposition 1 into two lemmas. In Lemma 2 we establish an upper bound for ${\sup _{s\in [0,t]}}{\sup _{x\in D}}\left|u(x,s)\right|$. In Lemma 3 we obtain similar estimate for ${\left\| u\right\| _{\alpha ,1,t}}$.
In the calculations below we shall often refer to the following simple formulas: for all $a>0$, $b>0$, and $0<v<t$,
where $C=\mathrm{B}(a,b)$, the beta function. The formula (14) follows directly from the definition of the beta function by the substitution $z=\frac{s-v}{t-v}$. The inequality (15) is obtained by the substitution $z=\frac{v-s}{t-v}$ as follows:
\[\begin{aligned}{}& {\int _{0}^{v}}{(t-s)^{-a-b}}{(v-s)^{b-1}}\hspace{0.1667em}ds={(t-v)^{-a}}{\int _{0}^{\frac{v}{t-v}}}\frac{{z^{b-1}}}{{(1+z)^{a+b}}}\hspace{0.1667em}dz\\ {} & \hspace{1em}\le {(t-v)^{-a}}{\int _{0}^{\infty }}\frac{{z^{b-1}}}{{(1+z)^{a+b}}}\hspace{0.1667em}dz=\mathrm{B}(a,b){(t-v)^{-a}}.\end{aligned}\]
Denote for brevity
Proof.
By (5),
where
Then, applying (7), we get
\[\begin{aligned}{}{I_{\varphi }}& ={\int _{D}}\left|G(x,t;y,0)\varphi (y)\right|\hspace{0.1667em}dy,\\ {} {I_{f}}& ={\int _{0}^{t}}{\int _{D}}\left|G(x,t;y,s)f\big(u(y,s)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}ds,\\ {} {I_{h}}& ={\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}\left|{\int _{0}^{t}}{\int _{D}}G(x,t;y,s)h\big(u(y,s)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(s)\right|.\end{aligned}\]
By (A2), φ is bounded. Therefore, the Gaussian property (7) implies that
It follows from the linear growth property (2) that
(19)
\[ \left|f\big(u(y,s)\big)\right|\le C\big(1+\left|u(y,s)\right|\big)\le C(1+{\left\| u\right\| _{s}}).\]Using the bound (3) for the integrals with respect to ${B_{j}^{H}}$, we may write
By the Lipschitz continuity of h,
In order to estimate ${I_{h3}}$, we use (10) together with the bound
We have
Combining (17), (18), (20)–(23), (25), we obtain
\[\begin{aligned}{}{I_{h}}& \le C{\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}{\left\| {B_{j}^{H}}\right\| _{\alpha ,0,t}}{\int _{0}^{t}}\Bigg(\frac{{\int _{D}}\left|G(x,t;y,s)h(u(y,s)){e_{j}}(y)\right|\hspace{0.1667em}dy}{{s^{\alpha }}}\\ {} & \hspace{1em}+{\int _{0}^{s}}\frac{{\int _{D}}\left|G(x,t;y,s)h(u(y,s))-G(x,t;y,v)h(u(y,v))\right|\left|{e_{j}}(y)\right|dy}{{(s-v)^{\alpha +1}}}\hspace{0.1667em}dv\Bigg)ds.\end{aligned}\]
The assumption (A4) implies that ${\sup _{j}}\left|{e_{j}}(y)\right|\le C$ for all $y\in D$. Therefore,
where
\[\begin{aligned}{}{I_{h1}}& ={\int _{0}^{t}}{s^{-\alpha }}{\int _{D}}\left|G(x,t;y,s)h\big(u(y,s)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}ds,\\ {} {I_{h2}}& ={\int _{0}^{t}}{\int _{0}^{s}}{(s-v)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,s)\right|\left|h\big(u(y,s)\big)-h\big(u(y,v)\big)\right|dy\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {I_{h3}}& ={\int _{0}^{t}}{\int _{0}^{s}}{(s-v)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,s)-G(x,t;y,v)\right|\left|h\big(u(y,v)\big)\right|dy\hspace{0.1667em}dv\hspace{0.1667em}ds.\end{aligned}\]
The term ${I_{h1}}$ can be estimated similarly to ${I_{f}}$, using the linear growth of h and the Gaussian property of G:
Since ${\left\| u\right\| _{s}}$ is non-decreasing and ${s^{-\alpha }}$ is non-increasing, we can use the rearrangement inequality [6, Theorem 378] to obtain
(22)
\[ {I_{h1}}\le C{\int _{0}^{t}}(1+{\left\| u\right\| _{s}}){(t-s)^{-\alpha }}\hspace{0.1667em}ds.\]
\[ {I_{h2}}\le {\int _{0}^{t}}{\int _{D}}\left|G(x,t;y,s)\right|{\int _{0}^{s}}\frac{\left|u(y,s)-u(y,v)\right|}{{(s-v)^{\alpha +1}}}\hspace{0.1667em}dv\hspace{0.1667em}dy\hspace{0.1667em}ds.\]
The inner integral can be bounded by ${\left\| u\right\| _{\alpha ,1,s}}$. Therefore, we get
(23)
\[ {I_{h2}}\le {\int _{0}^{t}}{\left\| u\right\| _{\alpha ,1,s}}{\int _{D}}\left|G(x,t;y,s)\right|\hspace{0.1667em}dy\hspace{0.1667em}ds\le C{\int _{0}^{t}}{\left\| u\right\| _{\alpha ,1,s}}\hspace{0.1667em}ds.\](24)
\[ \left|h\big(u(y,v)\big)\right|\le C\big(1+\left|u(y,v)\right|\big)\le C(1+{\left\| u\right\| _{v}}).\]
\[\begin{aligned}{}{I_{h3}}& \le C{\int _{0}^{t}}{(t-s)^{-\delta }}{\int _{0}^{s}}(1+{\left\| u\right\| _{v}}){(s-v)^{\delta -\alpha -1}}{\int _{D}}{\varPhi _{t-{t^{\ast }}}^{C}}(x-y)\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}{(t-s)^{-\delta }}{\int _{0}^{s}}(1+{\left\| u\right\| _{v}}){(s-v)^{\delta -\alpha -1}}\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){\int _{v}^{t}}{(t-s)^{-\delta }}{(s-v)^{\delta -\alpha -1}}\hspace{0.1667em}ds\hspace{0.1667em}dv,\end{aligned}\]
where we choose $\delta \in (\frac{d}{d+2},1)$ so that $\delta >\alpha $. Computing the inner integral by (14), we get
(25)
\[ {I_{h3}}\le C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){(t-v)^{-\alpha }}\hspace{0.1667em}dv.\]
\[ \left|u(x,t)\right|\le C{\xi _{\alpha ,H,T}}\Bigg(1+{\int _{0}^{t}}\big({\left\| u\right\| _{s}}{(t-s)^{-\alpha }}+{\left\| u\right\| _{\alpha ,1,s}}\big)ds\Bigg),\]
Since ${\left\| u\right\| _{s}}$ and ${\left\| u\right\| _{\alpha ,1,s}}$ are non-decreasing, the right-hand side here is non-decreasing as well. Indeed, using the substitution $s=zt$, the integral in the right-hand side can be rewritten in the form
\[ {t^{1-\alpha }}{\int _{0}^{1}}{\left\| u\right\| _{zt}}{(1-z)^{-\alpha }}dz+t{\int _{0}^{1}}{\left\| u\right\| _{\alpha ,1,zt}}dz.\]
Therefore, taking suprema, we arrive at (16). □Proof.
By (5),
where
\[\begin{aligned}{}{J_{\varphi }}& ={\int _{D}}\left|G(x,t;y,0)-G(x,s;y,0)\right|\left|\varphi (y)\right|\hspace{0.1667em}dy,\\ {} {J_{f}}& =\left|{\int _{0}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,t;y,v)f\big(u(y,v)\big)\hspace{0.1667em}dy\hspace{0.1667em}dv-{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,s;y,v)f\big(u(y,v)\big)\hspace{0.1667em}dy\hspace{0.1667em}dv\right|,\\ {} {J_{h}}& ={\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}\Bigg|{\int _{0}^{t}}{\int _{D}}G(x,t;y,v)h\big(u(y,v)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(v)\\ {} & \hspace{1em}-{\int _{0}^{s}}{\int _{D}}G(x,s;y,v)h\big(u(y,v)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(v)\Bigg|.\end{aligned}\]
Then
\[ {\int _{0}^{t}}\frac{\left|u(x,t)-u(x,s)\right|}{{(t-s)^{\alpha +1}}}\hspace{0.1667em}ds\le {K_{\varphi }}+{K_{f}}+{K_{h}},\]
where ${K_{\varphi }}={\int _{0}^{t}}\frac{{J_{\varphi }}}{{(t-s)^{\alpha +1}}}\hspace{0.1667em}ds$, ${K_{f}}={\int _{0}^{t}}\frac{{J_{f}}}{{(t-s)^{\alpha +1}}}\hspace{0.1667em}ds$, ${K_{h}}={\int _{0}^{t}}\frac{{J_{h}}}{{(t-s)^{\alpha +1}}}\hspace{0.1667em}ds$.Let $\delta \in (\frac{d}{d+2}\vee \alpha ,1)$ be fixed throughout the proof. Using the boundedness of φ and (11), we can write
Therefore,
Let us consider ${K_{f}}$. We have that
In order to estimate ${K^{\prime\prime }_{f}}$, we apply (19) and (11), change the order of integration and then use (14):
\[\begin{aligned}{}{J_{f}}& \le {\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)f\big(u(y,v)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}dv\\ {} & \hspace{1em}+{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\left|f\big(u(y,v)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}dv.\end{aligned}\]
Consequently,
where
\[\begin{aligned}{}{K^{\prime }_{f}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)f\big(u(y,v)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {K^{\prime\prime }_{f}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\left|f\big(u(y,v)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds.\end{aligned}\]
By (19) and the Gaussian property of G,
(27)
\[\begin{aligned}{}{K^{\prime }_{f}}& \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}(1+{\left\| u\right\| _{v}})\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){\int _{0}^{v}}{(t-s)^{-\alpha -1}}\hspace{0.1667em}ds\hspace{0.1667em}dv\\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){(t-v)^{-\alpha }}\hspace{0.1667em}dv\le C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){(t-v)^{-2\alpha }}\hspace{0.1667em}dv.\end{aligned}\](28)
\[\begin{aligned}{}{K^{\prime\prime }_{f}}& \le C{\int _{0}^{t}}{(t-s)^{\delta -\alpha -1}}{\int _{0}^{s}}{(s-v)^{-\delta }}(1+{\left\| u\right\| _{v}}){\int _{D}}{\varPhi _{{v^{\ast }}-v}^{C}}(x-y)\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}{(t-s)^{\delta -\alpha -1}}{\int _{0}^{s}}{(s-v)^{-\delta }}(1+{\left\| u\right\| _{v}})\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){\int _{v}^{t}}{(t-s)^{\delta -\alpha -1}}{(s-v)^{-\delta }}\hspace{0.1667em}ds\hspace{0.1667em}dv\\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){(t-v)^{-\alpha }}\hspace{0.1667em}dv\le C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){(t-v)^{-2\alpha }}\hspace{0.1667em}dv.\end{aligned}\]Next, consider ${K_{h}}$. We have
where
\[\begin{aligned}{}{J_{h}}& \le {\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}\Bigg|{\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,t;y,v)h\big(u(y,v)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(v)\Bigg|\\ {} & \hspace{1em}+{\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}\Bigg|{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\big(G(x,t;y,v)-G(x,s;y,v)\big)h\big(u(y,v)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(v)\Bigg|\\ {} & =:{J^{\prime }_{h}}+{J^{\prime\prime }_{h}}.\end{aligned}\]
The integrals with respect to fractional Brownian motions can be bounded similarly to ${I_{h}}$, applying (3) and (A4). We obtain
(29)
\[\begin{aligned}{}{J^{\prime }_{h}}& \le C{\xi _{\alpha ,H,t}}{\int _{s}^{t}}\Bigg(\frac{{\int _{D}}\left|G(x,t;y,v)h(u(y,v))\right|dy}{{(v-s)^{\alpha }}}\\ {} & \hspace{1em}+{\int _{s}^{v}}\frac{{\int _{D}}\left|G(x,t;y,v)h(u(y,v))-G(x,t;y,r)h(u(y,r))\right|dy}{{(v-r)^{\alpha +1}}}\hspace{0.1667em}dr\Bigg)dv\\ {} & \le C{\xi _{\alpha ,H,T}}({J_{h1}}+{J_{h2}}+{J_{h3}}),\end{aligned}\]
\[\begin{aligned}{}{J_{h1}}& ={\int _{s}^{t}}{(v-s)^{-\alpha }}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)h\big(u(y,v)\big)\right|dy\hspace{0.1667em}dv,\\ {} {J_{h2}}& ={\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{s}^{v}}{(v-r)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,v)\right|\left|h\big(u(y,v)\big)-h\big(u(y,r)\big)\right|dy\hspace{0.1667em}dr\hspace{0.1667em}dv,\\ {} {J_{h3}}& ={\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{s}^{v}}{(v-r)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,v)-G(x,t;y,r)\right|\left|h\big(u(y,r)\big)\right|dy\hspace{0.1667em}dr\hspace{0.1667em}dv.\end{aligned}\]
Similarly,
\[\begin{aligned}{}{J^{\prime\prime }_{h}}& \le C{\xi _{\alpha ,H,s}}{\int _{0}^{s}}\Bigg({v^{-\alpha }}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\left|h\big(u(y,v)\big)\right|dy\\ {} & \hspace{1em}+{\int _{0}^{v}}{(v-r)^{-\alpha -1}}{\int _{D}}\big|\big(G(x,t;y,v)-G(x,s;y,v)\big)h\big(u(y,v)\big)\\ {} & \hspace{1em}-\big(G(x,t;y,r)-G(x,s;y,r)\big)h\big(u(y,r)\big)\big|dy\hspace{0.1667em}dr\Bigg)dv\\ {} & \le C{\xi _{\alpha ,H,T}}({J_{h4}}+{J_{h4}}+{J_{h6}}),\end{aligned}\]
where
\[\begin{aligned}{}{J_{h4}}& ={\int _{0}^{s}}{v^{-\alpha }}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\left|h\big(u(y,v)\big)\right|dy\hspace{0.1667em}dv,\\ {} {J_{h5}}& ={\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{0}^{v}}{(v-r)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\\ {} & \hspace{1em}\times \left|h\big(u(y,v)\big)-h\big(u(y,r)\big)\right|dy\hspace{0.1667em}dr\hspace{0.1667em}dv,\\ {} {J_{h6}}& ={\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{0}^{v}}{(v-r)^{-\alpha -1}}{\int _{D}}\big|G(x,t;y,v)-G(x,s;y,v)\\ {} & \hspace{1em}-G(x,t;y,r)+G(x,s;y,r)\big|\left|h\big(u(y,r)\big)\right|dy\hspace{0.1667em}dr\hspace{0.1667em}dv.\end{aligned}\]
Now it remains to estimate ${K_{hi}}={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{J_{hi}}\hspace{0.1667em}ds$, $i=1,2,\dots ,6$.
In order to bound ${K_{h1}}$ we apply successively (24), (7) and (15) (with $a=2\alpha $, $b=1-\alpha $):
By the Lipschitz continuity of h,
(30)
\[\begin{aligned}{}{K_{h1}}& \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}{(v-s)^{-\alpha }}(1+{\left\| u\right\| _{v}}){\int _{D}}\left|G(x,t;y,v)\right|dy\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}{(v-s)^{-\alpha }}(1+{\left\| u\right\| _{v}})\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){\int _{0}^{v}}{(t-s)^{-\alpha -1}}{(v-s)^{-\alpha }}\hspace{0.1667em}ds\hspace{0.1667em}dv\\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){(t-v)^{-2\alpha }}\hspace{0.1667em}dv.\end{aligned}\]
\[ {K_{h2}}\le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)\right|{\int _{s}^{v}}\frac{\left|u(y,v)-u(y,r)\right|}{{(v-r)^{\alpha +1}}}\hspace{0.1667em}dr\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds.\]
According to the definition, the inner integral can be bounded by ${\left\| u\right\| _{\alpha ,1,v}}$. Then we use the Gaussian property of G to obtain
\[\begin{aligned}{}{K_{h2}}& \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}{\left\| u\right\| _{\alpha ,1,v}}{\int _{D}}\left|G(x,t;y,v)\right|\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}{\left\| u\right\| _{\alpha ,1,v}}\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}{\left\| u\right\| _{\alpha ,1,v}}{\int _{0}^{v}}{(t-s)^{-\alpha -1}}\hspace{0.1667em}ds\hspace{0.1667em}dv\\ {} & \le C{\int _{0}^{t}}{\left\| u\right\| _{\alpha ,1,v}}{(t-v)^{-\alpha }}\hspace{0.1667em}dv.\end{aligned}\]
In order to estimate ${K_{h3}}$, we use (24) and (10), and then (14):
(31)
\[\begin{aligned}{}{K_{h3}}& \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}{(t-v)^{-\delta }}{\int _{s}^{v}}{(v-r)^{\delta -\alpha -1}}(1+{\left\| u\right\| _{r}})\\ {} & \hspace{1em}\times {\int _{D}}{\varPhi _{t-{t^{\ast }}}^{C}}(x-y)dy\hspace{0.1667em}dr\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}{(t-v)^{-\delta }}{\int _{s}^{v}}{(v-r)^{\delta -\alpha -1}}(1+{\left\| u\right\| _{r}})\hspace{0.1667em}dr\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}(1+{\left\| u\right\| _{r}}){\int _{r}^{t}}{(t-v)^{-\delta }}{(v-r)^{\delta -\alpha -1}}\hspace{0.1667em}dv\hspace{0.1667em}dr\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}(1+{\left\| u\right\| _{r}}){(t-r)^{-\alpha }}\hspace{0.1667em}dr\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}(1+{\left\| u\right\| _{r}}){(t-r)^{-\alpha }}{\int _{0}^{r}}{(t-s)^{-\alpha -1}}\hspace{0.1667em}ds\hspace{0.1667em}dr\\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{r}}){(t-r)^{-2\alpha }}\hspace{0.1667em}dr.\end{aligned}\]The term ${K_{h4}}$ can be bounded similarly with the help of (24), (11) and (14):
From the Lipschitz continuity of h we get
\[\begin{aligned}{}{K_{h4}}& \le C{\int _{0}^{t}}{(t-s)^{\delta -\alpha -1}}{\int _{0}^{s}}(1+{\left\| u\right\| _{v}}){v^{-\alpha }}{(s-v)^{-\delta }}{\int _{D}}{\varPhi _{{v^{\ast }}-v}^{C}}(x-y)dy\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}{(t-s)^{\delta -\alpha -1}}{\int _{0}^{s}}(1+{\left\| u\right\| _{v}}){v^{-\alpha }}{(s-v)^{-\delta }}\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){v^{-\alpha }}{\int _{v}^{t}}{(t-s)^{\delta -\alpha -1}}{(s-v)^{-\delta }}\hspace{0.1667em}ds\hspace{0.1667em}dv\\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){v^{-\alpha }}{(t-v)^{-\alpha }}\hspace{0.1667em}dv.\end{aligned}\]
Since $(1+{\left\| u\right\| _{v}}){(t-v)^{-\alpha }}$ is non-decreasing and ${v^{-\alpha }}$ is non-increasing, using the rearrangement inequality, we obtain
(32)
\[ {K_{h4}}\le C{\int _{0}^{t}}(1+{\left\| u\right\| _{v}}){(t-v)^{-2\alpha }}\hspace{0.1667em}dv.\]
\[\begin{aligned}{}{K_{h5}}& \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\\ {} & \hspace{1em}\times {\int _{0}^{v}}\frac{\left|u(y,v)-u(y,r)\right|}{{(v-r)^{\alpha +1}}}\hspace{0.1667em}dr\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds.\end{aligned}\]
From (11), (14) it follows that
\[\begin{aligned}{}{K_{h5}}& \le C{\int _{0}^{t}}{(t-s)^{\delta -\alpha -1}}{\int _{0}^{s}}{\left\| u\right\| _{\alpha ,1,v}}{(s-v)^{-\delta }}{\int _{D}}{\varPhi _{{v^{\ast }}-v}^{C}}(x-y)\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}{(t-s)^{\delta -\alpha -1}}{\int _{0}^{s}}{\left\| u\right\| _{\alpha ,1,v}}{(s-v)^{-\delta }}\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & =C{\int _{0}^{t}}{\left\| u\right\| _{\alpha ,1,v}}{\int _{v}^{t}}{(t-s)^{\delta -\alpha -1}}{(s-v)^{-\delta }}\hspace{0.1667em}ds\hspace{0.1667em}dv\\ {} & \le C{\int _{0}^{t}}{\left\| u\right\| _{\alpha ,1,v}}{(t-v)^{-\alpha }}\hspace{0.1667em}dv.\end{aligned}\]
Finally, we estimate ${K_{h6}}$, using (12), (15), (18) and (24):
(33)
\[\begin{aligned}{}{K_{h6}}& \le C{\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}{\int _{0}^{v}}(1+{\left\| u\right\| _{r}}){(v-r)^{-\alpha -1}}\\ {} & \hspace{1em}\times {\int _{r}^{v}}{\int _{s}^{t}}{(\theta -\tau )^{-2}}{\int _{D}}{\varPhi _{\theta -\tau }^{C}}(x-y)dy\hspace{0.1667em}d\theta \hspace{0.1667em}d\tau \hspace{0.1667em}dy\hspace{0.1667em}dr\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{\theta }}){\int _{0}^{\theta }}{(t-s)^{-\alpha -1}}\\ {} & \hspace{1em}\times {\int _{0}^{s}}{\int _{0}^{v}}{(\theta -\tau )^{-2}}{\int _{0}^{\tau }}{(v-r)^{-\alpha -1}}dr\hspace{0.1667em}d\tau \hspace{0.1667em}dv\hspace{0.1667em}ds\hspace{0.1667em}d\theta \\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{\theta }}){\int _{0}^{\theta }}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}{\int _{0}^{v}}{(\theta -\tau )^{-2}}{(v-\tau )^{-\alpha }}d\tau \hspace{0.1667em}dv\hspace{0.1667em}ds\hspace{0.1667em}d\theta \\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{\theta }}){\int _{0}^{\theta }}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}{(\theta -v)^{-\alpha -1}}dv\hspace{0.1667em}ds\hspace{0.1667em}d\theta \\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{\theta }}){\int _{0}^{\theta }}{(t-s)^{-\alpha -1}}{(\theta -s)^{-\alpha }}ds\hspace{0.1667em}d\theta \\ {} & \le C{\int _{0}^{t}}(1+{\left\| u\right\| _{\theta }}){(t-\theta )^{-2\alpha }}d\theta .\end{aligned}\]Combining the obtained bounds for ${K_{\varphi }}$, ${K^{\prime }_{f}}$, ${K^{\prime\prime }_{f}}$ and ${K_{hi}}$, $i=1,\dots ,6$, we obtain
\[\begin{aligned}{}& {\int _{0}^{t}}\frac{\left|u(x,t)-u(x,v)\right|}{{(t-v)^{\alpha +1}}}\hspace{0.1667em}dv\le C{\xi _{\alpha ,H,T}}\\ {} & \hspace{1em}\times \Bigg(1+{\int _{0}^{t}}\big({\left\| u\right\| _{v}}{(t-v)^{-2\alpha }}+{\left\| u\right\| _{\alpha ,1,v}}{(t-v)^{-\alpha }}\big)dv\Bigg).\end{aligned}\]
Since the integral in the right-hand side can be rewritten in the form
\[ {t^{1-2\alpha }}{\int _{0}^{1}}{\left\| u\right\| _{zt}}{(1-z)^{-2\alpha }}dz+{t^{1-\alpha }}{\int _{0}^{1}}{\left\| u\right\| _{\alpha ,1,zt}}{(1-z)^{-\alpha }}dz,\]
it is a non-decreasing function. Therefore, taking suprema, we arrive at (26). □Proof of Proposition 1.
Lemmata 2 and 3 allow us to use a kind of two-dimensional Grönwall argument, proposed in [17, Lemma 4.1]. Namely, for some $\lambda >0$, which will be chosen later, define
Similarly, from (26) we get
Let K be the largest of the constants in (34) and (35); without loss of generality we can assume that $K\ge 1$. Setting $\lambda ={(4K\xi )^{1/(1-\alpha )}}$ and plugging it into (35), we obtain
\[ {f_{1}}(\lambda )=\underset{t\in [0,T]}{\sup }{e^{-\lambda t}}{\left\| u\right\| _{t}},\hspace{1em}{f_{2}}(\lambda )=\underset{t\in [0,T]}{\sup }{e^{-\lambda t}}{\left\| u\right\| _{\alpha ,1,t}}\]
and denote for shortness $\xi ={\xi _{\alpha ,H,T}}$. From (16) we get
(34)
\[\begin{aligned}{}{f_{1}}(\lambda )& \le C\xi \Bigg(1+\underset{t\in [0,T]}{\sup }{e^{-\lambda t}}{\int _{0}^{t}}\big({e^{\lambda s}}{f_{1}}(\lambda ){(t-s)^{-\alpha }}+{e^{\lambda s}}{f_{2}}(\lambda )\big)ds\Bigg)\\ {} & \le C\xi \Bigg(1+{f_{1}}(\lambda )\underset{t\in [0,T]}{\sup }{\int _{0}^{t}}{e^{-\lambda (t-s)}}{(t-s)^{-\alpha }}ds\\ {} & \hspace{1em}+{f_{2}}(\lambda )\underset{t\in [0,T]}{\sup }{\int _{0}^{t}}{e^{-\lambda (t-s)}}ds\Bigg)\\ {} & \le C\xi \Bigg(1+{f_{1}}(\lambda ){\int _{0}^{\infty }}{e^{-\lambda u}}{u^{-\alpha }}du+{f_{2}}(\lambda ){\int _{0}^{\infty }}{e^{-\lambda u}}du\Bigg)\\ {} & \le C\xi \big(1+{\lambda ^{\alpha -1}}{f_{1}}(\lambda )+{\lambda ^{-1}}{f_{2}}(\lambda )\big).\end{aligned}\](35)
\[ {f_{2}}(\lambda )\le C\xi \big(1+{f_{1}}(\lambda ){\lambda ^{2\alpha -1}}+{f_{2}}(\lambda ){\lambda ^{\alpha -1}}\big).\]
\[\begin{aligned}{}{f_{2}}(\lambda )& \le K\xi \big(1+{\lambda ^{2\alpha -1}}{f_{1}}(\lambda )+{\lambda ^{-1}}{f_{2}}(\lambda )\big)\\ {} & =\frac{1}{4}{\lambda ^{1-\alpha }}\big(1+{\lambda ^{2\alpha -1}}{f_{1}}(\lambda )+{f_{2}}(\lambda )\big)\\ {} & =\frac{1}{4}\big({\lambda ^{1-\alpha }}+{\lambda ^{\alpha }}{f_{1}}(\lambda )+{f_{2}}(\lambda )\big),\end{aligned}\]
whence ${f_{2}}(\lambda )\le \frac{1}{3}({\lambda ^{1-\alpha }}+{\lambda ^{\alpha }}{f_{1}}(\lambda ))$. Plugging this into (34) and noting that $\lambda \ge 1$, we get
\[\begin{aligned}{}{f_{1}}(\lambda )& \le \frac{1}{4}{\lambda ^{1-\alpha }}\bigg(1+{\lambda ^{\alpha -1}}{f_{1}}(\lambda )+\frac{1}{3}{\lambda ^{-\alpha }}+\frac{1}{3}{\lambda ^{\alpha -1}}{f_{1}}(\lambda )\bigg)\\ {} & \le {\lambda ^{1-\alpha }}+\frac{1}{3}{f_{1}}(\lambda ),\end{aligned}\]
whence ${f_{1}}(\lambda )\le \frac{3}{2}{\lambda ^{1-\alpha }}=6K\xi $. It follows that
Similarly,
The statement then follows from adding these estimates. □By Fernique’s theorem, $\mathsf{E}\exp \{a{\xi ^{2}}\}<\infty $ for some $a>0$. Since $\frac{1}{1-\alpha }<2$, we have $C{\xi ^{1/(1-\alpha )}}<a{\xi ^{2}}+C$, so Proposition 1 implies existence of moments of the solution.
5 Existence and uniqueness of mild solution
The following theorem is the main result of the article.
Under the standing assumptions, the existence of a mild solution was established in [16, Th. 2.3(a)]. Hence, it remains to prove the uniqueness.
Let u and $\tilde{u}$ be two mild solutions to the problem (1). In order to prove that u and $\tilde{u}$ coincide we shall establish that the norm
is equal to zero. The proof of this fact is carried out similarly to that of Proposition 1, using the bounds
instead of (19) and (24). Therefore we omit some details. As above, we first obtain the upper bounds for each of two terms in the right-hand side of the norm (36).
(36)
\[ {\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,T}}={\left\| u-\tilde{u}\right\| _{T}}+{\left\| u-\tilde{u}\right\| _{\alpha ,1,T}}\](37)
\[\begin{aligned}{}& \left|f\big(u(y,s)\big)-f\big(\tilde{u}(y,s)\big)\right|+\left|h\big(u(y,s)\big)-h\big(\tilde{u}(y,s)\big)\right|\\ {} & \hspace{1em}\le C\left|u(y,s)-\tilde{u}(y,s)\right|\le C{\left\| u-\tilde{u}\right\| _{s}}\end{aligned}\]Let $\xi ={\xi _{\alpha ,H,T}}$. Denote also
Corollary 1 implies that η is finite a. s.
Proof.
By (5),
where
\[\begin{aligned}{}{P_{f}}& ={\int _{0}^{t}}{\int _{D}}\left|G(x,t;y,s)\right|\left|f\big(u(y,s)\big)-f\big(\tilde{u}(y,s)\big)\right|dy\hspace{0.1667em}ds,\\ {} {P_{h}}& ={\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}\left|{\int _{0}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,t;y,s)\big(h\big(u(y,s)\big)-h\big(\tilde{u}(y,s)\big)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(s)\right|\hspace{-0.1667em}.\end{aligned}\]
Using the bound (37) and the Gaussian property of G we immediately get
Further, similarly to (21),
where
The bounds
and
are obtained analogously to the bounds (22) and (25).
(40)
\[\begin{aligned}{}{P_{h1}}& ={\int _{0}^{t}}{s^{-\alpha }}{\int _{D}}\left|G(x,t;y,s)\right|\left|h\big(u(y,s)\big)-h\big(\tilde{u}(y,s)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}ds,\\ {} {P_{h2}}& ={\int _{0}^{t}}{\int _{0}^{s}}{(s-v)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,s)\right|\\ {} & \hspace{1em}\times \left|h\big(u(y,s)\big)-h\big(\tilde{u}(y,s)\big)-h\big(u(y,v)\big)+h\big(\tilde{u}(y,v)\big)\right|dy\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {P_{h3}}& ={\int _{0}^{t}}{\int _{0}^{s}}{(s-v)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,s)-G(x,t;y,v)\right|\\ {} & \hspace{1em}\times \left|h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)\right|dy\hspace{0.1667em}dv\hspace{0.1667em}ds.\end{aligned}\](41)
\[\begin{aligned}{}{P_{h1}}& \le C{\int _{0}^{t}}{(t-s)^{-\alpha }}{\left\| u-\tilde{u}\right\| _{s}}ds\end{aligned}\](42)
\[\begin{aligned}{}{P_{h3}}& \le C{\int _{0}^{t}}{(t-s)^{-\alpha }}{\left\| u-\tilde{u}\right\| _{s}}ds\end{aligned}\]According to [13, Lemma 7.1], the assumption (A3) implies that for any ${x_{1}}$, ${x_{2}}$, ${x_{3}}$, ${x_{4}}$,
Inserting the bound (43) into (40), we get
\[\begin{aligned}{}\left|h({x_{1}})-h({x_{2}})-h({x_{3}})+h({x_{4}})\right|& \le C\left|{x_{1}}-{x_{2}}-{x_{3}}+{x_{4}}\right|\\ {} & \hspace{1em}+C\left|{x_{1}}-{x_{3}}\right|(\left|{x_{1}}-{x_{2}}\right|+\left|{x_{3}}-{x_{4}}\right|).\end{aligned}\]
Therefore, we can write
(43)
\[\begin{aligned}{}& {\int _{0}^{s}}\frac{\left|h(u(y,s))-h(\tilde{u}(y,s))-h(u(y,v))+h(\tilde{u}(y,v))\right|}{{(s-v)^{\alpha +1}}}\hspace{0.1667em}dv\\ {} & \le C{\int _{0}^{s}}\frac{\left|u(y,s)-\tilde{u}(y,s)-u(y,v)+\tilde{u}(y,v)\right|}{{(s-v)^{\alpha +1}}}\hspace{0.1667em}dv\\ {} & \hspace{1em}+C\left|u(y,s)-\tilde{u}(y,s)\right|{\int _{0}^{s}}\frac{\left|u(y,s)-u(y,v)\right|+\left|\tilde{u}(y,s)-\tilde{u}(y,v)\right|}{{(s-v)^{\alpha +1}}}\hspace{0.1667em}dv\\ {} & \le C{\left\| u-\tilde{u}\right\| _{\alpha ,1,s}}+C{\left\| u-\tilde{u}\right\| _{s}}({\left\| u\right\| _{\alpha ,1,s}}+{\left\| \tilde{u}\right\| _{\alpha ,1,s}})\\ {} & \le C\eta {\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,s}}.\end{aligned}\]Proof.
As in the proof of Lemma 3, we can write
\[\begin{aligned}{}& {\int _{0}^{t}}\frac{\left|u(x,t)-\tilde{u}(x,t)-u(x,s)+\tilde{u}(x,s)\right|}{{(t-s)^{\alpha +1}}}\hspace{0.1667em}ds\\ {} & \le {\int _{0}^{t}}{(t-s)^{-\alpha -1}}\Bigg(\Bigg|{\int _{0}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,t;y,v)\big(f\big(u(y,v)\big)-f\big(\tilde{u}(y,v)\big)\big)\hspace{0.1667em}dy\hspace{0.1667em}dv\\ {} & \hspace{1em}-{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,s;y,v)\big(f\big(u(y,v)\big)-f\big(\tilde{u}(y,v)\big)\big)\hspace{0.1667em}dy\hspace{0.1667em}dv\Bigg|\\ {} & \hspace{1em}+{\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}\Bigg|{\int _{0}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,t;y,v)\big(h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(v)\\ {} & \hspace{1em}-{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,s;y,v)\big(h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(v)\Bigg|\Bigg)ds\\ {} & \le {Q^{\prime }_{f}}+{Q^{\prime\prime }_{f}}+{Q^{\prime }_{h}}+{Q^{\prime\prime }_{h}},\end{aligned}\]
where
\[\begin{aligned}{}{Q^{\prime }_{f}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)\right|\left|f\big(u(y,v)\big)-f\big(\tilde{u}(y,v)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {Q^{\prime\prime }_{f}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\\ {} & \hspace{1em}\times \left|f\big(u(y,v)\big)-f\big(\tilde{u}(y,v)\big)\right|\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {Q^{\prime }_{h}}& ={\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}{\int _{0}^{t}}{(t-s)^{-\alpha -1}}\\ {} & \hspace{1em}\times \left|{\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}G(x,t;y,v)\big(h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(v)\right|ds,\\ {} {Q^{\prime\prime }_{h}}& ={\sum \limits_{j=1}^{\infty }}{\lambda _{j}^{1/2}}{\int _{0}^{t}}{(t-s)^{-\alpha -1}}\Bigg|{\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{D}}\big(G(x,t;y,v)-G(x,s;y,v)\big)\\ {} & \hspace{1em}\times \big(h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)\big){e_{j}}(y)\hspace{0.1667em}dy\hspace{0.1667em}d{B_{j}^{H}}(v)\Bigg|\hspace{0.1667em}ds.\end{aligned}\]
Similarly to (27), (28) and (29), we get
\[\begin{aligned}{}{Q^{\prime }_{f}}& \le C{\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{v}}{(t-v)^{-2\alpha }}\hspace{0.1667em}dv,\\ {} {Q^{\prime\prime }_{f}}& \le C{\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{v}}{(t-v)^{-2\alpha }}\hspace{0.1667em}dv,\end{aligned}\]
and
where
\[\begin{aligned}{}{Q_{h1}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}{(v-s)^{-\alpha }}\\ {} & \hspace{1em}\times {\int _{D}}\left|G(x,t;y,v)\right|\left|h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)\right|dy\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {Q_{h2}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{s}^{v}}{(v-r)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,v)\right|\\ {} & \hspace{1em}\times \left|h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)-h\big(u(y,r)\big)+h\big(\tilde{u}(y,r)\big)\right|dy\hspace{0.1667em}dr\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {Q_{h3}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{s}^{v}}{(v-r)^{-\alpha -1}}\\ {} & \hspace{1em}\times {\int _{D}}\left|G(x,t;y,v)-G(x,t;y,r)\right|\left|h\big(u(y,r)\big)-h\big(\tilde{u}(y,r)\big)\right|dy\hspace{0.1667em}dr\hspace{0.1667em}dv\hspace{0.1667em}ds.\end{aligned}\]
Further, we estimate
\[\begin{aligned}{}{Q_{h1}}& \le C{\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{v}}{(t-v)^{-2\alpha }}\hspace{0.1667em}dv,\\ {} {Q_{h3}}& \le C{\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{v}}{(t-v)^{-2\alpha }}\hspace{0.1667em}dv,\end{aligned}\]
similarly to (30) and (31). Then applying (43), the Gaussian property of G, and integrating with respect to s, we obtain
\[\begin{aligned}{}{Q_{h2}}& \le C\eta {\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{s}^{t}}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,v}}{\int _{D}}\left|G(x,t;y,v)\right|dy\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} & \le C\eta {\int _{0}^{t}}{(t-v)^{-\alpha }}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,v}}\hspace{0.1667em}dv\le C\eta {\int _{0}^{t}}{(t-v)^{-2\alpha }}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,v}}\hspace{0.1667em}dv.\end{aligned}\]
Finally,
where
\[\begin{aligned}{}{Q_{h4}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}{v^{-\alpha }}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\\ {} & \hspace{1em}\times \left|h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)\right|dy\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {Q_{h5}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{0}^{v}}{(v-r)^{-\alpha -1}}{\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\\ {} & \hspace{1em}\times \left|h\big(u(y,v)\big)-h\big(\tilde{u}(y,v)\big)-h\big(u(y,r)\big)+h\big(\tilde{u}(y,r)\big)\right|dy\hspace{0.1667em}dr\hspace{0.1667em}dv\hspace{0.1667em}ds,\\ {} {Q_{h6}}& ={\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{0}^{v}}{(v-r)^{-\alpha -1}}{\int _{D}}\big|G(x,t;y,v)-G(x,s;y,v)\\ {} & \hspace{1em}-G(x,t;y,r)+G(x,s;y,r)\big|\left|h\big(u(y,r)\big)-h\big(\tilde{u}(y,r)\big)\right|dy\hspace{0.1667em}dr\hspace{0.1667em}dv\hspace{0.1667em}ds.\end{aligned}\]
Arguing as in the proof of Lemma 3, we can prove the bounds
\[\begin{aligned}{}{Q_{h4}}& \le C{\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{v}}{(t-v)^{-2\alpha }}\hspace{0.1667em}dv,\\ {} {Q_{h6}}& \le C{\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{v}}{(t-v)^{-2\alpha }}\hspace{0.1667em}dv,\end{aligned}\]
which are similar to (32) and (33). Finally, using (43), (11), and (14), we get
\[\begin{aligned}{}{Q_{h5}}& \le C\eta {\int _{0}^{t}}{(t-s)^{-\alpha -1}}{\int _{0}^{s}}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,v}}\\ {} & \hspace{1em}\times {\int _{D}}\left|G(x,t;y,v)-G(x,s;y,v)\right|\hspace{0.1667em}dy\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C\eta {\int _{0}^{t}}{(t-s)^{\delta -\alpha -1}}{\int _{0}^{s}}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,v}}{(s-v)^{-\delta }}\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \le C\eta {\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,v}}{(t-v)^{-\alpha }}\hspace{0.1667em}dv\\ {} & \le C\eta {\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,v}}{(t-v)^{-2\alpha }}\hspace{0.1667em}dv.\end{aligned}\]
Combining the bounds for ${Q^{\prime }_{f}}$, ${Q^{\prime\prime }_{f}}$ and ${Q_{hi}}$, $i=1,\dots ,6$, we arrive at
\[ {\int _{0}^{t}}\hspace{-0.1667em}\frac{\left|u(x,t)-\tilde{u}(x,t)-u(x,s)+\tilde{u}(x,s)\right|}{{(t-s)^{\alpha +1}}}ds\le C\xi \eta {\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,s}}{(t-s)^{-2\alpha }}ds.\]
Taking suprema we get (45). □Proof of Theorem 1.
Adding the bounds from Lemmata 4–5, we obtain that for all $t\in [0,T]$
\[\begin{aligned}{}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,t}}& \le C\xi \eta {\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,s}}{(t-s)^{-2\alpha }}ds\\ {} & \le C\xi \eta \hspace{0.1667em}{t^{2\alpha }}{\int _{0}^{t}}{\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,s}}{(t-s)^{-2\alpha }}{s^{-2\alpha }}ds.\end{aligned}\]
Then ${\left\| u-\tilde{u}\right\| _{\alpha ,\infty ,T}}=0$ by the generalized Grönwall lemma [13, Lemma 7.6]. □
