Time-Space Fractional Diffusion Problems: Existence, Decay Estimates and Blow-Up of Solutions

Abstract

The aim of this paper is to study the following time-space fractional diffusion problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t^\beta u+(-\Delta )^\alpha u+(-\Delta )^\alpha \partial _t^\beta u=\lambda f(x,u) +g(x,t) &{}\text{ in } \Omega \times {\mathbb {R}}^{+},\\ u(x,t)=0\ \ &{}\text{ in } ({\mathbb {R}}^N{\setminus }\Omega )\times {\mathbb {R}}^+,\\ u(x,0)=u_0(x)\ &{}\text{ in } \Omega ,\\ \end{array}\right. } \end{aligned}$$

where $\Omega \subset {\mathbb {R}}^N$ is a bounded domain with Lipschitz boundary, $(-\Delta )^{\alpha }$ is the fractional Laplace operator with $0<\alpha <1$, $\partial _t^{\beta }$ is the Riemann-Liouville time fractional derivative with $0<\beta <1$, $\lambda$ is a positive parameter, $f:\Omega \times {\mathbb {R}}\rightarrow {\mathbb {R}}$ is a continuous function, and $g\in L^2(0,\infty ;L^2(\Omega ))$. Under natural assumptions, the global and local existence of solutions are obtained by applying the Galerkin method. Then, by virtue of a differential inequality technique, we give a decay estimate of solutions. Moreover, the blow-up property of solutions is also investigated.

1 Introduction and the Main Results

Let $\Omega$ be a bounded domain in ${\mathbb {R}}^N$ $(N\ge 1)$ with Lipschitz boundary. We consider the following time-space fractional problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t^\beta u+(-\Delta )^\alpha u+(-\Delta )^\alpha \partial _t^\beta u=\lambda f(x,u) +g(x,t) &{}\text{ in } \Omega \times {\mathbb {R}}^{+},\\ u(x,t)=0\ \ &{}\text{ in } ({\mathbb {R}}^N{\setminus }\Omega )\times {\mathbb {R}}^+,\\ u(x,0)=u_0(x)\ &{}\text{ in } \Omega , \end{array}\right. } \end{aligned}$$

(1.1)

where $0<\beta ,\alpha <1$, $N>2\alpha$, $\lambda \ne 0$ is a given parameter and $g\in L^{2}(0,\infty ;L^2(\Omega ))$ is a given function. We assume that the reaction f is a continuous function on $\Omega \times [0,\infty )$ and $f(x,\xi )=0$ for all $x\in \Omega$ and $\xi \le 0$. Moreover, for all $\xi >0$, f satisfies

$(f_1)$:

$f(x,\xi )=|\xi |^{p-2}\xi$ and $2<p\le 2_\alpha ^*:=\frac{2N}{N-2\alpha },$

or the following Lipschitz condition:

$(f_2)$:

there exists ${\mathcal {L}}>0$ such that

$$\begin{aligned} |f(x,\xi _1)-f(x,\xi _2)|\le {\mathcal {L}} |\xi _1-\xi _2| \ \ \text {for all}\ \xi _1,\xi _2\in {\mathbb {R}}\ \ \mathrm{and}\ x\in \Omega . \end{aligned}$$

Here, the order $\beta$ of Riemann-Liouville fractional operator $\partial _t^{\beta }$ is defined by

$$\begin{aligned} \partial _t^{\beta }u=\partial _t(J^{1-\beta }(u-u(0))), \end{aligned}$$

where $J^{1-\beta }$ denotes the $1-\beta$ order Riemann-Liouville fractional integral operator and it is given by

$$\begin{aligned} J^{1-\beta }(u-u(0))=\frac{1}{\Gamma (\beta )}\int _0^t (t-\tau )^{\beta -1}(u(\tau )-u(0))d\tau . \end{aligned}$$

Here $\Gamma$ is the usual Gamma function. The fractional Laplace operator $(-\Delta )^\alpha$, up to a normalization constant, is defined by

$$\begin{aligned}(-\Delta )^\alpha \varphi (x)= 2\lim _{\varepsilon \rightarrow 0^{+}}\int _{{\mathbb {R}}^{N}\setminus B_\varepsilon (x)}\frac{\varphi (x)-\varphi (y)}{|x-y|^{N+2\alpha }}\,dy, \ \ x\in {\mathbb {R}}^{N}\end{aligned}$$

for all $\varphi \in C^\infty _0({\mathbb {R}}^{N})$. Here, $B_\varepsilon (x)=\{y\in {\mathbb {R}}^N:|y-x|<\varepsilon \}.$ For more properties related to the fractional Laplacian and fractional Sobolev spaces as well as for applications of variational methods to fractional problems, we refer to [3].

The fractional operators and related differential equations have important applications in many areas such as physics [15], mechanics chemistry, population dynamic [4, 5], anomalous diffusion [29] and so on. Time fractional differential equations can be used to describe some problems with memory effects. Moreover, both time and space fractional differential equations have been exploited for anomalous diffusion or dispersion where particles spread at a rate inconsistent with Brown motion, see [9]. In the case of time fractional derivatives, particles with “memory effect” propagates slowly, which we call anomalous subdiffusion. Different from the former, spatial fractional diffusion equations are used to describe macroscopic transport and usually result in superdiffusion phenomenon. So far, the works on problems involving the fractional Laplacian and its variants are quite large, here we just list a few, see [8, 10, 20, 21, 31,32,33,34] and the references cited there.

To the best of our knowledge, until recently there has been still very little works on deal with the existence, decay estimates and blow-up of solutions for time-space fractional problems like (1.1). In [30], Vergara and Zacher considered the following time fractional diffusion problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t(k*(u-u_0))-\mathrm{div}(A(x,t)Du)=0,\ \ t>0,\ x\in \Omega , \\ u=0\ \ t>0.x\in \partial \Omega , \end{array}\right. } \end{aligned}$$

where $k*(u-u_0)=\int _0^tk(t-\tau )(u(\tau )-u_0)d\tau$ and $k\in L_{1,loc}({\mathbb {R}}_+)$. Some useful fundamental identities were obtained. Based on these identities, the existence and decay estimates of weak solutions were obtained. In particular, the decay estimates of weak solutions were given by using the sub-supersolution method. In [17], Li et al. studied the following time-space fractional Keller-Segel equation

$$\begin{aligned} ^C_0D_t^\beta \rho +(-\Delta )^{\frac{\alpha }{2}}\rho +\nabla \cdot (\rho B(\rho ))=0, \end{aligned}$$

where $^C_0D_t^\beta$ denotes the Caputo derivative, $B(\rho )=-s_{n,\gamma }\int _{{\mathbb {R}}^n} \frac{x-y}{|x-y|^{n-\gamma +2}}\rho (y)dy$ is the Riesz potential with a singular kernel. The authors obtained the existence and uniqueness of mild solutions. Moreover, the authors discussed the properties of the mild solutions, such as mass conservation and blow-up behaviors. In [1], Bekkai et al. studied the following Cauchy problem involving the Caputo derivative and the fractional Laplacian

$$\begin{aligned} {\left\{ \begin{array}{ll} ^C_0D_t^\alpha u+(-\Delta )^{\frac{\beta }{2}}u =\frac{1}{\Gamma (1-\alpha )}\int _0^t(t-s)^{-\alpha }e^{u(s)}ds,\ \ x\in {\mathbb {R}}^N,\ t>0,\\ u(x,0)=u_0(x),\ \ x\in {\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$

(1.2)

$N\ge 1,$ $0<\alpha <1$, $0<\beta \le 2$. First the existence of mild solutions of (1.2) was obatined by the Banach contraction mapping principle. Then the authors proved that the mild solution is also the weak solution. Furthermore, the authors showed the local weak solutions blow up in finite time by choosing suitable test function. See also [6, 23, 36, 39] for similar discussions of the blow-up properties of solutions. Very recently, Fu and Zhang [11] considered the following time-space fractional Kirchhoff problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \partial _t^\beta u+M(\Vert u\Vert _{H_0^\alpha (\Omega )}^2)(-\Delta )^\alpha u =\gamma |u|^{\rho }u+g(x,t) &{}\text{ in } \Omega \times {\mathbb {R}}^{+},\\ u(x,t)=0\ \ &{}\text{ in } ({\mathbb {R}}^N{\setminus }\Omega )\times {\mathbb {R}}^+,\\ u(x,0)=u_0(x)\ &{}\text{ in } \Omega , \end{array}\right. } \end{aligned}$$

where $M:[0,\infty )\rightarrow [0,\infty )$ is a continuous function. Under suitable assumptions, the authors obtained the global existence of solutions by using the Galerkin method. Furthermore, a decay estimate of solutions was established.

On the other hand, when $\alpha$, $\beta$ and s limit to 1, the Eq. (1.1) reduces to the following equation

$$\begin{aligned} \displaystyle \partial _t u-\Delta u-\Delta \partial _t u=\lambda |u|^{p-2}u +g(x,t),\ \ \partial _t u=\frac{\partial u}{\partial t}, \end{aligned}$$

(1.3)

which is called pseudo-parabolic equation. Equations like (1.3) can be used to describe many important physical processes, such as unidirectional propagation of nonlinear, long waves [2, 27], the aggregation of population [25] and semiconductors [14]. The study of Eq. (1.3) received much more attention in the past years, see [12, 18, 35].

Recently, Tuan et al. [28] studied the initial boundary value problem and Cauchy problem of Caputo time-fractional pseudo-parabolic equations

$$\begin{aligned} \displaystyle {\mathbb {D}}_t^\alpha (u-m\Delta u) +(-\Delta )^\sigma u ={\mathcal {N}}(u), \end{aligned}$$

(1.4)

where ${\mathbb {D}}_t^\alpha$ denotes the Caputo time fractional derivative. The local well-posedness of Eq. (1.4) was established. Further, the finite time blow-up of solutions was also obtained. In [24], Nguyen et al. considered a class of pseudoparabolic equations with the nonlocal condition and the Caputo derivative and obtained the existence and uniqueness of the mild solution. In [7], Chaoui and Rezgui dealt with a time fractional pseudoparabolic equation with fractional integral condition. By the Rothe time discretization scheme, the existence of weak solution was obtained. Moreover, the uniqueness of weak solution as well as some regularity results were obtained.

Inspired by the above papers, we discuss in this work the existence, uniqueness, decay estimates of weak solutions and solutions that blow up in finite time for problem (1.1) involving the time-space fractional operators. Since our problem is nonlocal, our discussion is more elaborate than the papers in the literature. Comparing with the papers in the literature, the main feature of this paper is that the problem (1.1) contains the Riemann-Liouville time fractional derivative and the fractional Laplacian. Definitely, this paper is the first time to deal with the local existence and global nonexistence of solutions for problems involving the fractional Laplacian and the Riemann-Liouville time fractional derivative.

Definition 1.1

We say that $u\in L^{\infty }(0,T;H_0^\alpha (\Omega ))$ with $\partial _t^{\beta }u \in L^2(0,T;L^2(\Omega ))$ is a weak subsolution (supsolution) of problem (1.1) if $u(x,0)\le (\ge )u_0(x)$ and

$$\begin{aligned}&\int _{\Omega }\varphi \partial _t^{\beta }u dx+\int _{{\mathbb {R}}^{2N}}\frac{(u(x,t)-u(y,t))(\varphi (x)-\varphi (y))}{|x-y|^{N+2\alpha }} dxdy\\&\qquad +\int _{{\mathbb {R}}^{2N}}\frac{(\partial _t^\beta u(x,t)-\partial _t^\beta u(y,t))(\varphi (x)-\varphi (y))}{|x-y|^{N+2\alpha }} dxdy\\&\quad \le (\ge )\lambda \int _{\Omega } f(x,u)\varphi dx +\int _{\Omega }g\varphi dx \end{aligned}$$

for any $0\le \varphi \in H_0^\alpha (\Omega )$ and a.e. $t\in (0,T)$. u is a weak solution if and only u is both a subsolution and a supsolution. Here, we call u is a global weak solution of problem (1.1), if the equality in above holds for any $0<T<\infty$; u is a local weak solution, if there exists $T_0>0$ such that the equality in Definition 1.1 holds for $0<T\le T_0$.

The proof of the following existence results relies on the contract mapping theorem and the Galerkin method.

Theorem 1.2

Assume that $0\le u_{0}\in H_{0}^{\alpha }(\Omega )$, $g\in L^2(0,\infty ;L^2(\Omega ))$ and $f(x,\xi )=0$ for all $x\in \Omega$ and $\xi \le 0$. If f satisfies $(f_1)$, then problem (1.1) admits a local nonnegative weak solution. If f satisfies $(f_2)$, then problem (1.1) has a unique global weak solution.

The following theorem shows the asymptotic behavior of global solutions to problem (1.1).

Theorem 1.3

Assume that $g\equiv 0$ and $f(x,\xi )=0$ for all $x\in \Omega$ and $\xi \le 0$. If $0\le u_0\in H_0^{\alpha }(\Omega )$ and $u_0(x)\le \eta _0\varphi _1(x)$ with $\eta _0>0$ for all $x\in \Omega$, and f satisfies $(f_2)$, then the unique solution of problem (1.1) satisfies the following decay estimates

$$\begin{aligned} 0\le u(x,t)\le \frac{c_2\varphi _1(x)}{1+t^{\beta }}\ \ \mathrm{for\ all}\ t\ge 0\ \ \mathrm{and}\ x\in \Omega , \end{aligned}$$

where $c_2>0$ and $\varphi _1>0$ is the eigenfunction corresponding to the first eigenvalue of the fractional Laplacian.

We also discuss the global nonexistence of local solutions for problem (1.1).

Theorem 1.4

Assume that $0\le u_{0}\in H_{0}^{\alpha }(\Omega )$, $g=0$ and $f(x,\xi )=0$ for all $x\in \Omega$ and $\xi \le 0$. Suppose that f satisfies $(f_1)$, and $\int _\Omega u_0(x)\varphi _1(x)dx>\left( \frac{\lambda _1}{\lambda }\right) ^{1/(p-2)}$, where $c_2>0$ and $\varphi _1>0$ is the eigenfunction corresponding to the first eigenvalue $\lambda _1$. Then the nonnegative weak solutions of problem (1.1) blow up in finite time.

In what follows, the letters c, $c_{i}$, C, $C_{i}$, $i=1, 2, \ldots ,$ denote positive constants which vary from line to line, but are independent of terms that take part in any limit process. Furthermore, for any $p\ge 1$ we denote $\Vert u\Vert _{p}=\Vert u\Vert _{L^{p}(\Omega )}.$

2 Preliminaries

In this section, we provide some basic results which will be used in the next sections. The fractional Sobolev space $H^\alpha ({\mathbb {R}}^N)$ is defined as

$$\begin{aligned} H^\alpha ({\mathbb {R}}^N)=\left\{ u\in L^2({\mathbb {R}}^N): \iint _{{\mathbb {R}}^{2N}} \frac{(u(x)-u(y))^2}{|x-y|^{N+2\alpha }}dxdy<\infty \right\} . \end{aligned}$$

endowed with the norm

$$\begin{aligned} \Vert u\Vert _{H^\alpha ({\mathbb {R}}^N)}^2=\iint _{{\mathbb {R}}^{2N}}\frac{(u(x)-u(y))^2}{|x-y|^{N+2\alpha }}dxdy+\Vert u\Vert _{L^2({\mathbb {R}}^N)}^2. \end{aligned}$$

$H_0^\alpha (\Omega )$ is defined as

$$\begin{aligned} H_0^\alpha (\Omega )=\{u\in H^\alpha ({\mathbb {R}}^N):u=0 \ a.e. \ \mathrm{in} \ {\mathbb {R}}^N \backslash {\Omega }\}. \end{aligned}$$

in the sequel, we take

$$\begin{aligned} \Vert u\Vert _{H_0^\alpha (\Omega )}^2=\iint _{{\mathbb {R}}^{2N}} \frac{(u(x)-u(y))^2}{|x-y|^{N+2\alpha }}dxdy. \end{aligned}$$

$H_0^\alpha (\Omega )$ is a Hilbert space in which a scalar product is given by

$$\begin{aligned} \langle u,v\rangle _\alpha = \iint _{{\mathbb {R}}^{2N}} \frac{(u(x)-u(y))(v(x))-v(y))}{|x-y|^{N+2\alpha }}dxdy \end{aligned}$$

for any $u,v\in H_0^\alpha (\Omega )$.

Denote by

$$\begin{aligned} 0<\lambda _1<\lambda _2\le \cdots \le \lambda _k \le \lambda _{k+1} \le \cdots <+\infty \end{aligned}$$

the distinct eigenvalues of the fractional Laplace operator and let $\omega _k$ be the eigenfunction corresponding to $\lambda _k$ of the following eigenvalue problem

$$\begin{aligned} {\left\{ \begin{array}{ll} (-\Delta )^\alpha u=\lambda u, \qquad &{} x\in \Omega ,\\ u=0, &{} x\in {\mathbb {R}}^N\backslash \Omega . \end{array}\right. } \end{aligned}$$

We obtain for $k\in {\mathbb N}$,

$$\begin{aligned} \lambda _k=\min _{u\in P_k\backslash \{0\}} \frac{\int _{{\mathbb {R}}^{2N}} \frac{(u(x)-u(y))^2}{|x-y|^{N+2\alpha }}dxdy}{\int _{\Omega }|u(x)|^2dx}, \end{aligned}$$

where $P_1=H_0^\alpha (\Omega )$ and

$$\begin{aligned} P_k=\left\{ u\in H_0^\alpha (\Omega ): (u,w_k)_{H_0^\alpha (\Omega )}=0,\forall j=1,2,...,k-1\right\} ,\ k\ge 2. \end{aligned}$$

Lemma 2.1

([3]). Let $2_\alpha ^*=\frac{2N}{N-2\alpha }$. For any $q\in [1,2_\alpha ^*]$, the embedding $H_0^\alpha (\Omega )\hookrightarrow L^q(\Omega )$ is continuous. Furthermore, the embedding is compact if $q\in [1,2_\alpha ^*)$.

The Yosida approximation of the time-fractional derivative operator is an useful tool to deal with problems with Caputo fractional derivative operators. For more details, we refer to [30, 37, 38]. Let $1\le p< \infty , 0<\beta <1$ and X be a real Banach space. Define fractional derivative operator

$$\begin{aligned} Bu=\frac{d}{dt}(g_{1-\beta } *u),\qquad D(B)=\{u\in L^p([0,T];X):g_{1-\beta }*u \in W_0^{1,p}([0,T];X)\}, \end{aligned}$$

where $g_{1-\beta }$ is given by

$$\begin{aligned} g_{1-\beta }(t)= {\left\{ \begin{array}{ll} \frac{1}{\Gamma (\beta )}t^{\beta -1}\ \ \mathrm{if}\ t>0,\\ 0\ \ \ \ \ \ \mathrm{if}\ t\le 0. \end{array}\right. } \end{aligned}$$

Its Yosida approximation $B_n$ defined by $B_n=nB(n+B)^{-1}$ ($n\in {\mathbb {N}}$) possesses the property that for any $u\in D(B)$, $B_nu\rightarrow Bu$ strongly in $L^p([0,T];X)$ as $n\rightarrow \infty$. Here, we collect some important properties of $g_{1-\beta }$ and $B_n$ which are listed in the following:

• The kernel $g_{1-\beta ,n}$ is nonnegative and nonincreasing for all $n\in N$, and $g_{1-\beta ,n}\in W^{1,1}([0,T])$.

• $g_{1-\beta ,n}\rightarrow g_{1-\beta }$ in $L^1([0,T])$ and $B_nu \rightarrow Bu$ in $L^p([0,T];X)$ as $n\rightarrow \infty$.

Lemma 2.2

([30]). Assume that H is a real Hilbert space and $T>0$ is a real number. Then for any $k\in W^{1,1}([0,T])$ and $u\in L^2(0,T;H)$, the following identity holds

$$\begin{aligned} \left( \frac{d}{dt}(k*u)(t),u(t)\right) _H&=\frac{1}{2}\frac{d}{dt}(k*|u(\cdot )|_H^2)(t)+\frac{1}{2}k(t)|u(t)|_H^2\\&\quad +\frac{1}{2}\int _{0}^{t}[-\dot{k}(s)]|u(t)-u(t-s)|_H^2ds,\quad \mathrm{a.e.}\ t\in (0,T). \end{aligned}$$

Remark 1

Obviously, if k is a nonincreasing and nonnegative function in $W^{1,1}([0,T])$, then we obtain for any $u\in L^2(0,T;H)$ that

$$\begin{aligned} \left( \frac{d}{dt}(k*u)(t),u(t)\right) _H&\ge \frac{1}{2}\frac{d}{dt}(k*|u(\cdot )|_H^2)(t),\quad \mathrm{a.e.}\ t\in (0,T). \end{aligned}$$

Lemma 2.3

([30]). Let $H\in C^1({\mathbb {R}})$ and $k\in W^{1,1}([0,T])$, for a sufficiently smooth function u, then there holds for a.e. $t\in (0,T)$

$$\begin{aligned} \dot{H}(u(t))\frac{d}{dt}(k*u)(t)&=\frac{d}{dt}(k*H(u))(t)+[-H(u(t))+\dot{H}(u(t))u(t)]k(t) \\&\quad +\int _{0}^{t}[H(u(t-s))-H(u(t))\\&\quad -\dot{H}(u(t))(u(t)-u(t-s))][-\dot{k}(s)]ds \end{aligned}$$

Definition 2.4

(see [19]). Let $q>0$ be a real number and $0<T\le \infty$. We say that a function $\omega :{\mathbb {R}}^+\rightarrow {\mathbb {R}}$ $({\mathbb {R}}^+=[0,\infty ))$ satisfies a condition (q), if

$$\begin{aligned} e^{-qt}[\omega (u)]^q\le R(t)\omega (e^{-qt}u^q)\ \ \mathrm{for\ all}\ u\in {\mathbb {R}}^+,\ t\in [0,T), \end{aligned}$$

where R(t) is a continuous and nonnegative function.

Clearly, if $\omega (u)=u^r,r>0$, then $\omega$ satisfies the condition (q) with any $q>1$, and $R(t)=e^{(r-1)qt}$.

Lemma 2.5

(see [19, Theorem 1]). Let a be a nondecreasing, nonnegative $C^1$-function on [0, T), F be a continuous, nonnegative function on [0, T), $\omega : {\mathbb {R}}^+\rightarrow {\mathbb {R}}$ be a continuous, nondecreasing function, $\omega (0)=0, \omega (u)>0$ on [0, T), and u be a continuous, nonnegative function on [0, T) with

$$\begin{aligned} u(t)\le a(t)+\int _0^t(t-\tau )^{\beta -1}F(\tau )\omega (u(\tau ))d\tau ,\ \ t\in [0, T), \end{aligned}$$

where $\beta >0$. Then the following assertions hold:

(i) Suppose that $\beta >1/2$ and $\omega$ satisfies the condition (q) with $q=2$. Then

$$\begin{aligned} u(t)\le e^t\{\Omega ^{-1}[\Omega (2a(t)^2)+g_1(t)]\}^{1/2},\ \ t\in [0, T_1], \end{aligned}$$

where

$$\begin{aligned} g_1(t)=\frac{\Gamma (2\beta -1)}{4^{\beta -1}}\int _0^t R(\tau )F^2(\tau )d\tau , \end{aligned}$$

$\Gamma$ is the gamma function, $\Omega (v)=\int _{v_0}^v\frac{dy}{\omega (y)}$, $v_0>0$, $\Omega ^{-1}$ is the inverse of $\Omega$, and $T_1\in {\mathbb {R}}^+$ is such that $\Omega (2a(t)^2)+g_1(t)\in Dom(\Omega ^{-1})$ for all $t\in [0,T_1]$.

(ii) Let $\beta \in (0,1/2]$ and $\omega$ satisfies the condition (q) with $q=z+2$, where $z=\frac{1-\beta }{\beta }$. Then

$$\begin{aligned} u(t)\le e^t\{\Omega ^{-1}\left[ \Omega (2^{q-1}a(t)^{q})+g_2(t)\right] \}^{1/q},\ \ t\in [0, T_1], \end{aligned}$$

where

$$\begin{aligned}&g_2(t)=2^{q-1}K_z^q\int _0^t F(\tau )^q R(\tau )d\tau ,\ \ K_z=\left[ \frac{\Gamma (1-\gamma p)}{p^{1-\gamma p}}\right] ^{1/p},\\&\gamma =\frac{z}{z+1},\ p=\frac{z+2}{z+1}, \end{aligned}$$

$T_1\in {\mathbb {R}}^+$ is such that $\Omega (2^{q-1}a(t)^{q})+g_2(t)\in Dom (\Omega ^{-1})$ for all $t\in [0,T_1]$.

In particular, if $w(u)=u$, then there holds

Lemma 2.6

(see [19, Theorem 2]). Let $0<T\le \infty$, a(t), F(t) be as in Lemma 2.5, and let u(t) be a continuous, nonnegative function on [0, T) with

$$\begin{aligned} u(t)\le a(t)+\int _0^t(t-\tau )^{\beta -1}F(\tau )u(\tau )d\tau , \end{aligned}$$

where $\beta >0$. Then the following assertions hold

(i) If $\beta >1/2$, then

$$\begin{aligned} u(t)\le \sqrt{2}a(t)\exp \left( \frac{2\Gamma (2\beta -1)}{4^\beta }\int _0^tF(\tau )^2d\tau +t\right) ,\ \ t\in [0,T). \end{aligned}$$

(ii) If $\beta =\frac{1}{z+1}$ for some $z\ge 1$, then

$$\begin{aligned} u(t)\le (2^{q-1})^{1/q}a(t)\exp \left( \frac{2^{q-1}}{q}K_z\int _0^tF(\tau )^qd\tau +t\right) ,\ \ t\in [0,T), \end{aligned}$$

where $K_z$ is defined by $K_z=[\frac{\Gamma (1-\gamma p)}{p^{1-\gamma p}}]^{1/p}$, $p=\frac{z+2}{z+1},$ $q=z+2.$

Lemma 2.7

Define an operator

$$\begin{aligned} I_{1-\beta }(u)=\frac{1}{\Gamma (\beta )}\int _0^t (t-\tau )^{\beta -1}u(\tau )d\tau \end{aligned}$$

for any $u\in L^2(0,T;L^2(\Omega ))$. If $\beta \ge 1/2$, then $I_{1-\beta }:L^2(0,T;L^2(\Omega ))\rightarrow L^2(0,T;L^2(\Omega ))$ is a bounded linear operator; If $0<\beta <1/2$, then $I_{1-\beta }:L^r(0,T;L^r(\Omega ))\rightarrow L^r(0,T;L^r(\Omega ))$ is a bounded linear operator, where $r>\frac{1}{\beta }$.

Proof

If $\beta \ge \frac{1}{2}$, then the Hölder inequality implies that

$$\begin{aligned} \left( \int _0^t (t-\tau )^{\beta -1}u(\tau )d\tau \right) ^2&\le \int _0^t (t-\tau )^{2(\beta -1)}d\tau \int _0^t u^2(\tau )d\tau \\&=\frac{1}{2\beta -1}t^{2\beta -1}\int _0^t u^2(\tau )d\tau . \end{aligned}$$

Thus,

$$\begin{aligned} \int _0^T\int _\Omega (I_{1-\beta }(u))^2dxdt \le C \int _0^T\int _\Omega u^2dxdt, \end{aligned}$$

which yields the desired result.

Now we consider the case $0<\beta <1/2$. By the Hölder inequality, we have

$$\begin{aligned}&\int _0^t (t-\tau )^{\beta -1}u(\tau )d\tau \\&\quad =\int _0^t (t-\tau )^{\beta -1}e^{\tau } e^{-\tau }u(\tau )d\tau \\&\quad \le \left( \int _0^t (t-\tau )^{q(\beta -1)}e^{q\tau }d\tau \right) ^{\frac{1}{q}}\left( \int _0^t u^{q^\prime }(\tau )e^{-q^\prime \tau }d\tau \right) ^{\frac{1}{q^\prime }}\\&\quad =\left( \frac{e^{qt}}{q^{1-(1- \beta )q}} \int _0^t \tau ^{-(1-\beta )q}e^{-\tau }d\tau \right) ^{1/q} \left( \int _0^t u^{q^\prime }(\tau )e^{-q^\prime \tau }d\tau \right) ^{\frac{1}{q^\prime }}\\&\quad \le \left( \frac{e^{qt}}{q^{1-(1- \beta )q}} \Gamma (1-(1-\beta )q)\right) ^{1/q}\left( \int _0^t u^{q^\prime }(\tau )d\tau \right) ^{\frac{1}{q^\prime }}, \end{aligned}$$

where $q>1$ satisfying $1-(1-\beta )q>0$, and $\frac{1}{q}+\frac{1}{q^\prime }=1$. Observe that $1-(1-\beta )q>0$. Thus, we get

$$\begin{aligned} \int _0^T\int _\Omega (I_{1-\beta }(u))^{q^\prime }dxdt \le C\int _0^T\int _\Omega |u|^{q^\prime }dxdt, \end{aligned}$$

which ends the proof. $\square$

Lemma 2.8

(see [26] Fractional integration by parts). Let $\alpha >0$, $p\ge 1$, $q\ge 1$ and $\frac{1}{p}+\frac{1}{q}\le 1+\alpha$ ($p\ne 1,\ q\ne 1$ in the case when $\frac{1}{p}+\frac{1}{q}=1+\alpha$). If $\varphi \in L^p(a,b)$ and $\psi \in L^q(a,b)$, then

$$\begin{aligned} \int _a^b\varphi (x)(I_{a+}^\alpha \psi )(x)dx =\int _a^b\psi (x)(I_{b-}^\alpha \varphi )(x)dx, \end{aligned}$$

where

$$\begin{aligned} (I_{a+}^\alpha \phi )(x)=\frac{1}{\Gamma (\alpha )}\int _a^x\frac{\psi (t)}{(x-t)^{1-\alpha }}dt \end{aligned}$$

and

$$\begin{aligned} (I_{b-}^\alpha \varphi )(x)=\frac{1}{\Gamma (\alpha )}\int _x^b \frac{\varphi (t)}{(t-x)^{1-\alpha }}dt. \end{aligned}$$

Using Lemma 2.8, we can obtain the following result.

Lemma 2.9

Let $\alpha >0$ and $1\le p$. Assume that $\varphi \in C_0^1(0,T)$ and $\psi \in L^p(0,T)$. Then

$$\begin{aligned} \int _0^T\varphi (x)\partial _x^\alpha \psi (x)dx =-\frac{1}{\Gamma (\alpha )}\int _0^T\int _x^T\frac{\varphi ^\prime (t)}{(t-x)^{1-\alpha }}dt (\psi (x)-\psi (0))dx. \end{aligned}$$

In order to show the existence of solutions to problem (1.1), we give some properties of the operator $L: H_{0}^{\alpha }(\Omega )\rightarrow (H_{0}^{\alpha }(\Omega ))'$ defined by

$$\begin{aligned} \langle L(u), v\rangle =\langle u,v\rangle _{\alpha } =\iint _{{\mathbb {R}}^{2N}}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+2\alpha }}dxdy, \end{aligned}$$

for all $u,v\in H_0^\alpha (\Omega )$.

Lemma 2.10

The operator $L: H_{0}^{\alpha }(\Omega )\rightarrow (H_{0}^{\alpha }(\Omega ))'$ is a monotone and linear bounded functional. Moreover, $\Vert L(u)\Vert _{(H_{0}^{\alpha }(\Omega ))'} \le [u]_{\alpha }$ for all $u\in H_{0}^{\alpha }(\Omega ).$

Proof

Let $u,v\in H_{0}^{\alpha }(\Omega )$, we have

$$\begin{aligned} \langle L(u)-L(v), u-v \rangle =[u-v]_\alpha ^2\ge 0. \end{aligned}$$

Thus, L is monotone. Clearly, L is a linear functional. It remains to show that $\Vert Lu\Vert _{(H_{0}^{\alpha }(\Omega ))'} \le [u]_{\alpha }.$ It follows from the Hölder inequality, we have

$$\begin{aligned} \langle L(u), v\rangle \le [u]_{\alpha } [v]_{\alpha }, \end{aligned}$$

which means that $\Vert L(u)\Vert _{(H_{0}^{\alpha }(\Omega ))'} \le [u]_{\alpha }$. The proof is now complete. $\square$

Lemma 2.11

The operator $L:H_{0}^{\alpha }(\Omega )\rightarrow (H_{0}^{\alpha }(\Omega ))'$ is hemicontinuous.

Proof

We are going to prove that the map $t\mapsto \langle L(u+tv), w\rangle$ is continuous on [0, 1] for all $u,v,w\in H_{0}^{\alpha }(\Omega )$, i.e,

$$\begin{aligned} \lim _{t\rightarrow 0} \langle L(u+tv), w\rangle =\langle L(u), w\rangle \end{aligned}$$

for all $w\in H_{0}^{\alpha }(\Omega ).$ We have,

$$\begin{aligned} \langle L(u+tv), w\rangle =\langle u+tv,w\rangle _{\alpha }. \end{aligned}$$

We define $G_t:[0,1]\rightarrow {\mathbb {R}}$ by

$$\begin{aligned} G_{t}(x,y)=\frac{((u+tv)(x)-(u+tv)(y))}{|x-y|^{N+2\alpha }}(w(x)-w(y)) \end{aligned}$$

and set

$$\begin{aligned} G(x,y)=\frac{(u(x)-u(y))}{|x-y|^{N+2\alpha }}(w(x)-w(y)). \end{aligned}$$

Obviously, $\lim _{t\rightarrow 0}G_{t} (x,y)=G(x,y)$ and there exists $h\in L^{1}({\mathbb {R}}^{2N})$ such that $|G_{t}(x,y)|\le h(x,y)$. Thus, by the Lebesgue dominated convergence theorem, we obtain the desired result. $\square$

To discuss the compactness of approximate solutions, we need the following Lions-Aubin lemma.

Proposition 2.12

([16, Theorem 4.1]). Let $T>0, \beta \in (0,1)$ and $p\in [1,\infty ).$ Let $B_{0}, B, B_{1}$ be Banach spaces. Assume that $B_{0}\hookrightarrow B$ is compact and $B\hookrightarrow B_{1}$ is continuous. Suppose that $W\subset L_{loc}^1(0,T;B_0)$ satisfies;

• (i) There exists $C_1>0$ such that $\forall u\in W$,

  $$\begin{aligned} \sup _{t\in (0,T)}\frac{1}{\Gamma (\beta )}\int _0^t(t-s)^{\beta -1} \Vert u\Vert _{B_0}^p(s)ds\le C_1. \end{aligned}$$

• (ii) There exist $r\in (\frac{p}{1+p\gamma },\infty )\bigcap [1,\infty )$ and $C_3>0$ such that $\forall u\in W$, there is an assignment of initial value $u_0$ for u such that the weak Caputo derivative satisfies

  $$\begin{aligned} \Vert D_c^\beta u\Vert _{L^r(0,T; B_1)}\le C_3. \end{aligned}$$

  Then W is relatively compact in $L^{p}(0,T,B)$.

Remark 2

The Caputo derivative of an absolutely continuous function u is defined as follows

$$\begin{aligned} D_c^\beta u=\frac{1}{\Gamma (\beta )} \int _0^t\frac{u^\prime (\tau )}{(t-\tau )^{1-\beta }}d\tau . \end{aligned}$$

In view of the definition of the Riemann-Liouville derivative, we know that the Riemann-Liouville derivative and the Caputo derivative have the relationship

$$\begin{aligned} D_c^\beta u=\partial _t^\beta u. \end{aligned}$$

3 Existence and Uniqueness of Weak Solutions

In this section by means of the Galerkin method, we establish the existence of local solutions to the problem (1.1). Assume that $\{\omega _k\}$ is an orthonormal basis in $L^2(\Omega )$ and

$$\begin{aligned} u_{0m}=\sum _{j=1}^mb_{mj}w_{j}\rightarrow u_0 \quad \mathrm{in} \; H_0^\alpha (\Omega ), \end{aligned}$$

then we shall find Galerkin approximation solutions $u_m=u_m(t)$ of the following form

$$\begin{aligned} u_m(t)=\sum _{j=1}^ma_{mj}(t)w_j,\quad m=1,2,\ldots , \end{aligned}$$

where $a_{mj}$ satisfies that

$$\begin{aligned} {\left\{ \begin{array}{ll} (\displaystyle \partial _t^\beta u_m,w_j)+\langle u_m,w_j\rangle _\alpha +\langle \partial _t^\beta u_m,w_j\rangle _\alpha =(\lambda f(x,u_m),w_j)+(g,w_j)\\ a_{mj}(0)=b_{mj}, \end{array}\right. } \end{aligned}$$

(3.1)

for $j=1,2,\ldots ,m.$ Here, $(\cdot ,\cdot )$ denotes the inner product of $L^2(\Omega )$. Problem (3.1) is a nonlinear fractional ordinary differential system. Next, we show that problem (3.1) has a unique local solution for every $m\in N$.

   First, by Lemmas 2.2 and 2.3, we give a prior estimate for problem (3.1).

Lemma 3.1

Suppose that $u_m= \sum _{j=1}^ma_{mj}w_j$ solves problem (3.1). If $f(x,\xi )=|\xi |^{p-2}\xi$ and $2<p\le 2_\alpha ^*$, then there exist $T^*>0$ and ${\mathcal {C}}_1>0$ such that

$$\begin{aligned}&\Vert u_m\Vert _{H_0^\alpha (\Omega )}\le {\mathcal {C}}_1\ \ \mathrm{for\ all}\ t\in [0, T^*) \ \mathrm{and}\ m\ge 1. \end{aligned}$$

(3.2)

Proof

Multiplying (3.1) by $\partial _t^{\beta }a_{mj}$ and summing j from 1 to m, we have

$$\begin{aligned}&(\partial _t^\beta u_m,\partial _t^\beta u_m)+((-\Delta )^\alpha u_m,\partial _t^\beta u_m)+((-\Delta )^\alpha \partial _t^\beta u_m,\partial _t^\beta u_m)\nonumber \\&\quad =\lambda (|u_m|^{p-2}u_m,\partial _t^{\beta }u_m)+(g,\partial _t^{\beta }u_m). \end{aligned}$$

(3.3)

By the Yosida approximation of time Riemamm-Liouville fractional derivative, Lemma 2.2 and Hölder’s inequality, one can deduce that

$$\begin{aligned} ((-\Delta )^\alpha u_m,\partial _t^\beta u_m)&=(u_m,\partial _t^\beta u_m)_{H_0^\alpha (\Omega )} \\&=(\frac{d}{dt}(g_{1-\beta ,n}*u_m),u_m)_{H_0^\alpha (\Omega )}-g_{1-\beta ,n}(u_{0\,m},u_m)_{H_0^\alpha (\Omega )}-R_{mn}(1)\\&\ge \frac{1}{2}\frac{d}{dt}(g_{1-\beta ,n}*\Vert u_m\Vert _{H_0^\alpha (\Omega )}^2)-\frac{1}{2}g_{1-\beta ,n}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2-R_{mn}(1) \end{aligned}$$

where

$$\begin{aligned} R_{mn}(1)=\left( \frac{d}{dt}(g_{1-\beta ,n}*(u_m-u_{0m}))-\frac{d}{dt}(g_{1-\beta }*(u_m-u_{0m})),u_m\right) _{H_0^\alpha (\Omega )}. \end{aligned}$$

By using the Hölder inequality and Young inequality, we deduce

$$\begin{aligned} (|u_m|^{p-2}u_m,\partial _t^{\beta }u_m)&\le \Vert |u_m|^{p-2}u_m\Vert _{L^{q^\prime }(\Omega )}\Vert \partial _t^{\beta }u_m\Vert _{L^q(\Omega )} \\&\le {\mathcal {S}}_*\Vert u_m\Vert _{L^{(p-1)q^\prime }(\Omega )}^{p-1}\Vert \partial _t^{\beta }u_m\Vert _{H_0^\alpha (\Omega )} \\&\le \mathcal S_*(C_\varepsilon \Vert u_m\Vert _{L^{(p-1)q^\prime }(\Omega )}^{2(p-1)} +\varepsilon \Vert \partial _t^{\beta }u_m\Vert _{H_0^\alpha (\Omega )}^2) \\&\le \mathcal S_*(C_\varepsilon \Vert u_m\Vert _{H_0^\alpha (\Omega )}^{2(p-1)}+\varepsilon \Vert \partial _t^{\beta }u_m\Vert _{H_0^\alpha (\Omega )}^2) \end{aligned}$$

where $q=2_\alpha ^*,\quad q^\prime =\frac{q}{q-1}=\frac{2N}{N+2\alpha }$ and ${\mathcal {S}}_*$ is the embedding constant from $H_0^\alpha (\Omega )$ to $L^{2_\alpha ^*}(\Omega )$. Here, we have used the fact that $(p-1)q^\prime \le 2_\alpha ^*$, thanks to $p\le 2_\alpha ^*.$ We also get

$$\begin{aligned} (g,\partial _t^{\beta }u_m)&\le \Vert g\Vert _{L^2(\Omega )}\Vert \partial _t^{\beta }u_m\Vert _{L^2(\Omega )} \\&\le \frac{1}{2}\Vert g\Vert _{L^2(\Omega )}^2+\frac{1}{2}\Vert \partial _t^{\beta }u_m\Vert _{L^2(\Omega )}^2. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned}&\frac{1}{2}\Vert \partial _t^{\beta }u_m\Vert _{L^2(\Omega )}^2 +\Vert \partial _t^{\beta }u_m\Vert _{H_0^s(\Omega )}^2+\frac{1}{2}\frac{d}{dt}(g_{1-\beta ,n}*\Vert u_m\Vert _{H_0^\alpha (\Omega )}^2) \nonumber \\&\quad \le {\mathcal {S}}_* \lambda (C_{\varepsilon }\Vert u_m\Vert _{H_0^\alpha (\Omega )}^{2(p-1)} +\varepsilon \Vert \partial _t^{\beta }u_m\Vert _{H_0^\alpha (\Omega )}^2) \nonumber \\&\qquad +\frac{1}{2}g_{1-\beta ,n}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2 +R_{mn}(1)+\frac{1}{2}\Vert g\Vert _{L^2(\Omega )}^2 \end{aligned}$$

(3.4)

Choose $\varepsilon$ small enough such that $S_* \lambda \varepsilon <1$. Then it follows from (3.4) that

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}(g_{1-\beta ,n}*\Vert u_m\Vert _{H_0^\alpha (\Omega )}^2) \nonumber \\&\quad \le {\mathcal {S}}_* \lambda C_{\varepsilon }\Vert u_m\Vert _{H_0^\alpha (\Omega )}^{2(p-1)} +\frac{1}{2}g_{1-\beta ,n}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2 +R_{mn}(1)+\frac{1}{2}\Vert g\Vert _{L^2(\Omega )}^2. \end{aligned}$$

(3.5)

Convolving (3.5) with $g_{\beta }$ and letting n go to $\infty$ and selecting an appropriate subsequence(if necessary), it leads to

$$\begin{aligned} \Vert u_m\Vert _{H_0^\alpha (\Omega )}^2 \le C_1 g_{\beta }*\Vert u_m\Vert _{H_0^\alpha (\Omega )}^{2(p-1)}+\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2 +\frac{t^{1-\beta }}{(1-\beta )\Gamma (1-\beta )}\Vert g\Vert _{L^{\infty }(0,\infty ;L^2(\Omega ))}^2, \end{aligned}$$

(3.6)

where $C_1=2\lambda {\mathcal {S}}_*C_{\varepsilon }$,

In Lemma 2.5, let $u(t)=\Vert u_m\Vert _{H_0^\alpha (\Omega )}^2$, $w(u)=u^{p-1}$, $F(s)=\frac{C_1}{\Gamma (\beta )}$, $a(t)=\sup _{m\ge 1}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2 +\frac{t^{1-\beta }}{(1-\beta )\Gamma (1-\beta )}\Vert g\Vert _{L^{\infty }(0,\infty ;L^2(\Omega ))}^2.$ When $\beta >\frac{1}{2}$, $R(t)=e^{2(p-2)t}$,

$$\begin{aligned} g_1(t)&=\frac{\Gamma (2\beta -1)}{4^{\beta -1}}\int _0^tR(\tau )F^2(\tau )d\tau =C_2(e^{2(p-2)t}-1), \\ \Omega (2a^2)&=\int _{v_0}^{2a(t)^2}\frac{1}{y^{p-1}}dy=-\frac{1}{(p-2)(2a(t)^2)^{p-2}}+C_3, \\ \Omega ^{-1}(v)&=\left( v_0^{-\frac{1}{p-2}}-(p-2)v\right) ^{-\frac{1}{p-2}}, \end{aligned}$$

where $C_2=\frac{4^{\beta -1}C_1^2}{2(p-2)\Gamma (2\beta -1)\Gamma ^2(\beta )}>0$, $C_3=\frac{1}{(p-2) v_0^{p-2}}>0$. Then we deduce from (3.6) that

$$\begin{aligned} u(t)\le e^t\left( \frac{1}{\frac{1}{(2a(t)^2)^{p-2}} +C_2(p-2)-C_2(p-2)e^{2(p-2) t}}\right) ^{\frac{1}{2(p-2)}}. \end{aligned}$$

Considering the definition of $\Omega ^{-1}$ and using $\Omega (2a^2(t))+g_1(t)\in Dom \Omega ^{-1}$, we get

$$\begin{aligned} -\frac{1}{(p-2)(2a(t)^2)^{p-2}}+C_3+C_2(e^{2(p-2)t}-1)<C_3, \end{aligned}$$

which implies that

$$\begin{aligned} e^{2(p-2)t}&\le \frac{1}{C_2(p-2)(2a(t)^2)^{p-2}}+1\\&\le \frac{1}{C_2(p-2)(2C_0^2)^{p-2}}+1, \end{aligned}$$

where $C_0=\sup _{m\ge 1}\Vert u_{0m}\Vert _{H_0^\alpha (\Omega )}^2.$ Then we have $t<\frac{\ln C_4}{2(p-2)}:=T_1$, where

$$\begin{aligned} C_4=\frac{1}{C_2(p-2)(2C_0^2)^{p-2}}+1>1. \end{aligned}$$

Therefore, for the case $\beta >\frac{1}{2}$, there exists $T_1>0$ such that

$$\begin{aligned} \Vert u_m\Vert _{H_0^\alpha (\Omega )}^2\le C_5, \end{aligned}$$

for all $0<t<T_1$.

When $\beta \in (0,\frac{1}{2})$, $R(t)=e^{(p-2)(z+2)t}$,

$$\begin{aligned}&\Omega (2^{q-1}a^q)=\int _{v_0}^{2^{q-1}a^q}\frac{1}{y^{p-1}}dy =-\frac{1}{(p-2)(2^{q-1}a^q)^{p-2}}+C_3, \\&g_2(t)=2^{q-1}K_z{q}\int _{0}^{t}F^q(\tau )R(\tau )d\tau =C_6(e^{(p-2)(z+2)t}-1), \\&\Omega ^{-1}(\Omega (2^{q-1}a^q)+g_2(t))=(\frac{1}{(2^{q-1}a^q)^{-p+2}+C_6(p-2) -C_6(p-2) e^{(p-2)(z+2)t}})^{\frac{1}{p-2}}, \end{aligned}$$

then we have

$$\begin{aligned} u(t)\le e^t\left( \frac{1}{\frac{1}{(2^{q-1}a^q)^{p-2}}+C_6(p-2)-C_6(p-2) e^{(p-2) (z+2)t}}\right) ^{\frac{1}{q(p-2)}}, \end{aligned}$$

where $z=\frac{1-\beta }{\beta }$, $q=z+2$, $K_z=(\frac{\Gamma (1-\gamma r)}{r^{1-\gamma r}})^{\frac{1}{r}}$, $\gamma =\frac{z}{z+1}$, $r=\frac{z+2}{z+1}$, $C_6=\frac{C_1^qK_z^q2^{q-1}}{\Gamma ^q(\beta )(p-2)(z+2)}>0$.

Consider the definition of $\Omega ^{-1}$, we obtain

$$\begin{aligned} C_6(e^{(p-2)(z+2)t}-1)-\frac{1}{(p-2)(2^{q-1}a^q)^{p-2}}+C_3<C_3, \end{aligned}$$

which implies that $t<\frac{\ln C_7}{(p-2)(z+2)}:=T_2$. Here, $C_7=\frac{1}{C_6(p-2)(2^{q-1}C_0^q)^{p-2}}+1>0$.

Therefore, for $\beta \in (0,\frac{1}{2})$, there exists $T_2>0$ such that $\Vert u_m\Vert _{H_0^\alpha (\Omega )}^2 \le C_8$ for all $0<t<T_2$.

In conclusion, there exist $T^*=\min \{T_1,T_2\}$ and ${\mathcal {C}}>0$ such that $\Vert u_m\Vert _{H_0^\alpha (\Omega )}\le {\mathcal {C}}$ for all $0<t<T^*$. $\square$

Lemma 3.2

Suppose that $u_m= \sum _{j=1}^ma_{mj}w_j$ solves problem (3.1). If f satisfies Lipschitz condition $(f_1)$, then for any $T>0$ there exists ${\mathcal {C}}_{2}>0$ such that

$$\begin{aligned}&\Vert u_m\Vert _{H_0^\alpha (\Omega )}\le {\mathcal {C}}_2 \ \ \mathrm{for\ all}\ t\in [0, T] \ \mathrm{and}\ m\ge 1. \end{aligned}$$

Proof

Since f satisfies the Lipschitz condition, there exists a positive constant C such that

$$\begin{aligned} |f(x,\xi )|\le C(1+|\xi |)\ \ \mathrm{for\ all}\ \xi \in {\mathbb {R}}. \end{aligned}$$

Then a similar discussion as in Lemma 3.1 gives that

$$\begin{aligned} \Vert u_m\Vert _{H_0^\alpha (\Omega )}^2&\le C g_{\beta }*\Vert u_m\Vert _{H_0^\alpha (\Omega )}^{2}+\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2\nonumber \\&\quad +\frac{C t^{1-\beta }}{\Gamma (1-\beta )} +\frac{t^{1-\beta }}{(1-\beta )\Gamma (1-\beta )}\Vert g\Vert _{L^{\infty }(0,\infty ;L^2(\Omega ))}^2. \end{aligned}$$

By Lemma 2.6, we get (i) If $\beta >1/2$, then

$$\begin{aligned} \Vert u_m\Vert _{H_0^\alpha (\Omega )}^2\le \sqrt{2}a(t)\exp \left( \left( \frac{2C_1^2\Gamma (2\beta -1)}{4^\beta \Gamma (\beta )^2}+1\right) t\right) ,\ \ t\in [0,T), \end{aligned}$$

where $a(t)=\sup _{m\ge 1}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2 +\frac{C_1 t^{1-\beta }}{\Gamma (1-\beta )} +\frac{t^{1-\beta }}{(1-\beta )\Gamma (1-\beta )}\Vert g\Vert _{L^{\infty }(0,\infty ;L^2(\Omega ))}^2.$

(ii) If $\beta =\frac{1}{z+1}$ for some $z\ge 1$, then

$$\begin{aligned} \Vert u_m\Vert _{H_0^\alpha (\Omega )}^2\le (2^{q-1})^{1/q}a(t)\exp \left( \left( \frac{2^{q-1}K_z C_1^q}{\Gamma (\beta )^q}+1\right) t\right) ,\ \ t\in [0,T), \end{aligned}$$

where $K_z=[\frac{\Gamma (1-\gamma p)}{p^{1-\gamma p}}]^{1/p}$, $\gamma =\frac{z}{z+1}$, $p=\frac{z+2}{z+1}$ and $q=z+2.$

In conclusion, for any $T>0$ there exists ${\mathcal {C}}_2>0$ such that $\Vert u_m\Vert _{H_0^\alpha (\Omega )}\le {\mathcal {C}}_2$ for all $t\in [0, T]$ and $m\ge 1$. $\square$

Based on Lemmas 3.1 and 3.2, we obtain the following estimate.

Lemma 3.3

Suppose that $u_m= \sum _{j=1}^ma_{mj}w_j$ solves (3.1), then there exists ${\mathcal {C}}_3$ such that

$$\begin{aligned} \Vert \partial _t^\beta u_m \Vert _{L^2(0,T;L^2(\Omega ))}+\Vert \partial _t^{\beta }u_m\Vert _{L^2(0,T:H_0^\alpha (\Omega ))} \le {\mathcal {C}}_3. \end{aligned}$$

Proof

Choosing $\varepsilon =\frac{1}{2{\mathcal {S}}_* \lambda }$ in (3.4), we get

$$\begin{aligned}&\frac{1}{2}\Vert \partial _t^{\beta }u_m\Vert _{L^2(\Omega )}^2 +\frac{1}{2}\Vert \partial _t^{\beta }u_m\Vert _{H_0^s(\Omega )}^2+\frac{1}{2}\frac{d}{dt}(g_{1-\beta ,n}*\Vert u_m\Vert _{H_0^\alpha (\Omega )}^2) \nonumber \\&\quad \le {\mathcal {S}}_* \lambda C_{\varepsilon }\Vert u_m\Vert _{H_0^\alpha (\Omega )}^{2(p-1)} \nonumber \\&\qquad +\frac{1}{2}g_{1-\beta ,n}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2 +R_{mn}(1)+\frac{1}{2}\Vert g\Vert _{L^2(\Omega )}^2. \end{aligned}$$

Integrating above inequality from 0 to T and letting $n\rightarrow \infty$, we obtain

$$\begin{aligned}&\Vert \partial _t^{\beta }u_m\Vert _{L^2(0,T;(L^2(\Omega )))}^2 +\Vert \partial _t^{\beta }u_m\Vert _{L^2(0,T:H_0^\alpha (\Omega ))}^2\\&\quad \le 2{\mathcal {S}}_* \lambda C_{\varepsilon } \int _{0}^{T}\Vert u_m\Vert _{H_0^\alpha (\Omega )}^{2p-2}dt+ \int _{0}^{T}g_{1-\beta ,n}\Vert u_{0\,m}\Vert _{H_0^\alpha (\Omega )}^2dt\\&\qquad +\Vert g\Vert _{L^2(0,T;L^2(\Omega ))}^2+g_{1-\beta }*\Vert u_{0,m}\Vert _{H_0^\alpha (\Omega )}^2. \end{aligned}$$

By Lemma 3.1, it yields

$$\begin{aligned} \Vert \partial _t^{\beta }u_m\Vert _{L^2(0,T;L^2(\Omega ))}^2+\Vert \partial _t^{\beta }u_m\Vert _{L^2(0,T:H_0^\alpha (\Omega ))}^2 \le C. \end{aligned}$$

For the case f satisfies the Lipschitz condition, by Lemma 3.2 and a similar discussion as above, one can obtain the desired result. $\square$

Now, we prove the local existence of solutions for system (3.1).

Theorem 3.4

Under the assumptions of Theorem 1.2, system (3.1) has a unique solution for all $t\in [0,T]$, where $0<T<T^*$ if $f=|u|^{p-2}u$ and $0<T<\infty$ if f satisfies $(f_1)$.

Proof

First, problem (3.1) is equivalent to the problem

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^\beta \psi (t)+A\psi (t)=BR(\psi (t))+BG(t)\\ \psi (0)=\xi \end{array}\right. } \end{aligned}$$

(3.7)

where $\psi (t)=(a_{mj}(t))\in {\mathbb {R}}^m$,$A=diag(\frac{\lambda _j}{1+\lambda _j})_{m\times m}, B= diag(\frac{1}{1+\lambda _j})_{m\times m},$ $\xi =(b_{mj})\in {\mathbb {R}}^m$, $\mu =(\lambda _j)\in {\mathbb {R}}^m$, $R(\psi (t))_j=\lambda (|u_m|^{p-2}u_m,wj)$, $G_j=(g,w_j)$. By Laplace transform or convoluting with $g_\beta$, we transform (3.7) into the following Volterra type system

$$\begin{aligned} \psi (t)=\xi +g_\beta *B(R(\psi (t)+G(t)))-g_\beta *A\psi (t). \end{aligned}$$

(3.8)

Therefore, we only need to prove that system (3.8) admits a unique continuous solution.

Notice

$$\begin{aligned} \Vert u_m(t)\Vert _{L^2(\Omega )}^2=\sum _{j=1}^m a_{mj}^2(t), \quad \Vert \psi (t)\Vert _{{\mathbb {R}}^m}^2=\sum _{j=1}^{m} a_{mj}^2(t). \end{aligned}$$

Then, $\Vert u_m(t)\Vert _{L^2(\Omega )}=\Vert \psi (t)\Vert _{R^m}$, $\Vert u_m(t)\Vert _{L^2(\Omega )}\le {\mathcal {S}} \Vert u_m(t)\Vert _{H_0^\alpha (\Omega )}$. Let

$$\begin{aligned} R_0={\mathcal {S}} {\mathcal {C}}_1. \end{aligned}$$

Then we obtain a prior estimate $\Vert \psi (t)\Vert _{{\mathbb {R}}^m}\le R_0$.

Define the operator as

$$\begin{aligned} \Phi \psi (t)=\xi +g_\beta *B(R(\psi (t)+G(t)))-g_\beta *A\psi (t) \end{aligned}$$

for all $\psi \in E_T$, where

$$\begin{aligned} E_T=\{\psi \in C(0,T;{\mathbb {R}}^m):\Vert \psi \Vert _{C(0,T;{\mathbb {R}}^m)}\le 2R_0\}. \end{aligned}$$

Set $d(\psi _1,\psi _2)=\max _{t\in [0,T]}\Vert \psi _1(t)-\psi _2(t)\Vert _{{\mathbb {R}}^m}$. Since $C(0,T;{\mathbb {R}}^m)$ is a Banach space, it follows that $(E_T,d)$ is a complete metric space. Then

$$\begin{aligned} \Vert \Phi \psi \Vert _{{\mathbb {R}}^m} \le \Vert \xi \Vert _{{\mathbb {R}}^m}+\Vert Bg_\beta *(R(\psi (t)+G(t)))\Vert _{{\mathbb {R}}^m} +\Vert Ag_\beta *\psi (t)\Vert _{{\mathbb {R}}^m}. \end{aligned}$$

Observe that

$$\begin{aligned}&\Vert g_\beta *B(R(\psi (t)+G(t)))\Vert _{{\mathbb {R}}^m}^2\\&\quad \le \sum _{j=1}^m \left( \frac{\lambda }{1+\lambda _j}g_\beta *\int _\Omega |u_m|^{p-2}u_mw_jdx +\frac{1}{1+\lambda _j}g_\beta *\int _\Omega g\omega _jdx\right) ^2\\&\quad \le \sum _{j=1}^m \left( \frac{\lambda \Vert w_j\Vert _p}{1+\lambda _j}g_\beta *\Vert u_m\Vert ^{p-1}_p +\frac{\Vert w_j\Vert _2}{1+\lambda _j}g_\beta *\Vert g\Vert _2\right) ^2\\&\quad \le \frac{C^2}{(1-\beta )^2\Gamma ^2(1-\beta )}(R_0^{(p-1)}+R_0)^2t^{2(1-\beta )} \end{aligned}$$

and

$$\begin{aligned} \Vert Ag_\beta *\psi (t)\Vert _{{\mathbb {R}}^m}^2 \le (g_\beta *\Vert A\psi (t)\Vert _{{\mathbb {R}}^m})^2 \le \frac{\lambda _m^2 t^{2(1-\beta )}}{(1+\lambda _1)^2(1-\beta )^2\Gamma ^2(1-\beta )}\sum _{j=1}^ma_{mj}^2(t)\\ \le \frac{\lambda _m^2 t^{2(1-\beta )}}{(1+\lambda _1)^2(1-\beta )^2\Gamma ^2(1-\beta )} \Vert \psi (t)\Vert _{{\mathbb {R}}^m}^2. \end{aligned}$$

Thus, we have

$$\begin{aligned} \Vert \Phi \psi \Vert _{{\mathbb {R}}^m}&\le \Vert \xi \Vert _{{\mathbb {R}}^m}+ \frac{C}{(1-\beta )\Gamma (1-\beta )}(R_0^{(p-1)}+R_0)T^{1-\beta } \\&\quad +\frac{\lambda _m T^{1-\beta }}{(1+\lambda _1)(1-\beta )\Gamma (1-\beta )} 2R_0. \end{aligned}$$

Now we assume that T is small enough such that

$$\begin{aligned} \frac{C}{(1-\beta )\Gamma (1-\beta )}(R_0^{(p-1)}+R_0)T^{1-\beta } +\frac{\lambda _m T^{1-\beta }}{(1+\lambda _1)(1-\beta )\Gamma (1-\beta )} 2R_0<R_0. \end{aligned}$$

Then $\Vert \Psi \psi \Vert _{{\mathbb {R}}^m}\le 2R_0$, which means that $\Phi$ maps from $E_{T}$ to $E_T$.

Next we show that $\Phi$ is contractive in $E_T$.

If $2<p\le 2_\alpha ^*:=\frac{2N}{N-2\alpha }$, then by the Hölder inequality we have

$$\begin{aligned}&\Vert BR(\psi ^{(1)})-BR(\psi ^{(2)})\Vert _{{\mathbb {R}}^m}^2 \nonumber \\&\quad \le \lambda ^2 \sum _{j=1}^m\left( \frac{1}{1+\lambda _j}\right) ^2 \left( \int _{\Omega }||u_m^{(1)}|^{p-2}u_m^{(1)}-|u_m^{(2)}|^{p-2}u_m^{(2)}||\omega _j|dx\right) ^2 \nonumber \\&\quad \le \lambda ^2C \sum _{j=1}^m \left( \frac{1}{1+\lambda _j}\right) ^2 \left( \int _{\Omega }(|u_m^{(1)}|^{p-2}+|u_m^{(2)}|^{p-2})|u_m^{(1)}-u_m^{(2)}||\omega _j|dx\right) ^2\nonumber \\&\quad \le \lambda ^2C \sum _{j=1}^m \left( \frac{1}{1+\lambda _j}\right) ^2 \Vert |u_m^{(1)}|^{p-2}+|u_m^{(2)}|^{p-2}\Vert _{L^{\frac{N}{2\alpha }}(\Omega )}^2 \Vert u_m^{(1)}-u_m^{(2)}\Vert _{L^{\frac{2N}{N-2\alpha }}(\Omega )}^2\Vert \omega _j\Vert _2^2\nonumber \\&\quad \le \frac{\lambda ^2 C}{(1+\lambda _1)^2}(\Vert u_m^{(1)}\Vert _{H_0^\alpha (\Omega )}^{p-2} +\Vert u_m^{(2)}\Vert _{H_0^\alpha (\Omega )}^{p-2})^2\Vert u_m^{(1)}-u_m^{(2)}\Vert _{H_0^\alpha (\Omega )}^2 \nonumber \\&\quad \le \frac{\lambda ^2 C\lambda _m^{p-1}}{(1+\lambda _1)^2}(\Vert u_m^{(1)}\Vert _{L^2(\Omega )}^{p-2} +\Vert u_m^{(2)}\Vert _{L^2(\Omega )}^{p-1})^2\Vert u_m^{(1)}-u_m^{(2)}\Vert _{L^2(\Omega )}^2 \nonumber \\&\quad \le \frac{\lambda ^2 C\lambda _m^{p-1}}{(1+\lambda _1)^2}(\Vert \psi ^{(1)}\Vert _{{\mathbb {R}}^m}^{p-2}+\Vert \psi ^{(2)}\Vert _{{\mathbb {R}}^m}^{p-2})^2\Vert \psi ^{(1)}-\psi ^{(2)}\Vert _{{\mathbb {R}}^m}^2 \nonumber \\&\quad \le \frac{\lambda ^2 C\lambda _m^{p-1}}{(1+\lambda _1)^2}R_0^{2(p-2)}\Vert \psi ^{(1)}-\psi ^{(2)}\Vert _{{\mathbb {R}}^m}^2, \end{aligned}$$

(3.9)

thanks to the following basic inequality:

$$\begin{aligned} ||u_m^{(1)}|^{p-2}u_m^{(1)}-|u_m^{(2)}|^{p-2}u_m^{(2)}|\le C(|u_m^{(1)}|^{p-2}+|u_m^{(2)}|^{p-2})|u_m^{(1)}-u_m^{(2)}|. \end{aligned}$$

Then

$$\begin{aligned}&\Vert g_\beta *BR(\psi ^{(1)})-g_\beta *BR(\psi ^{(2)})\Vert _{{\mathbb {R}}^m}\\&\quad \le g_\beta *\Vert BR(\psi ^{(1)})-BR(\psi ^{(2)})\Vert _{{\mathbb {R}}^m}\\&\quad \le \frac{\lambda C\lambda _m^{\frac{p-1}{2}}}{1+\lambda _1}\frac{R_0^{(p-2)}T^{1-\beta }}{\Gamma (1-\beta )} d(\psi ^{(1)},\psi ^{(2)}). \end{aligned}$$

Observe that

$$\begin{aligned} \Vert A\psi ^{(1)}-A\psi ^{(2)}\Vert _{{\mathbb {R}}^m}^2&=\sum _{j=1}^m((\psi ^{(1)}-\psi ^{(2)})\frac{\lambda _j}{1+\lambda _j})^2\\&\le \left( \frac{\lambda _m}{1+\lambda _m}\right) ^2 \sum _{j=1}^m(\psi ^{(1)}-\psi ^{(2)})^2\\&= \left( \frac{\lambda _m}{1+\lambda _m}\right) ^2 \Vert \psi ^{(1)}-\psi ^{(2)}\Vert _{{\mathbb {R}}^m}^2. \end{aligned}$$

Thus,

$$\begin{aligned}&\Vert g_\beta *A\psi ^{(1)}-g_\beta *A\psi ^{(2)}\Vert _{{\mathbb {R}}^m}\nonumber \\&\quad \le \frac{\lambda _m}{1+\lambda _m} \frac{T^{1-\beta }}{\Gamma (1-\beta )}d(\psi ^{(1)},\psi ^{(2)}). \end{aligned}$$

(3.10)

Gathering (3.9) and (3.10), we arrive at

$$\begin{aligned} d(\Phi (\psi ^{(1)}),\Phi (\psi ^{(1)})) \le {\mathcal {D}} T^{1-\beta }d(\psi ^{(1)},\psi ^{(2)}),\quad \forall \ \psi ^{(1)},\psi ^{(2)}\in E_T, \end{aligned}$$

where

$$\begin{aligned} {\mathcal {D}}:=\frac{\lambda C\lambda _m^{\frac{p-1}{2}}}{1+\lambda _1}\frac{R_0^{(p-2)}}{\Gamma (1-\beta )}+ \frac{\lambda _m}{1+\lambda _m} \frac{1}{\Gamma (1-\beta )}. \end{aligned}$$

Consequently, we prove that $\Phi$ is contractive on $E_T$ provided T is small enough such that ${\mathcal {D}} T^{1-\beta }<1$. Thus, by the Banach contraction mapping theorem, we know that the map $\Phi$ has a unique fixed point on some small interval $[0,T_0]$. Therefore, we prove that system (3.7) has a unique solution on $[0,T_0]$.

On the other hand, if f satisfies the Lipschitz condition, the existence of unique solution of system (3.7) on some small interval $[0,T_0]$ can be proved similarly as above.

Finally, we show that the local solution can be extended to (0, T]. Let $T_0$ and $u_m(T_0)$ be the initial data. Then repeating the same process as above, we can get a unique continuous solution on $[T_0,2T_0]$. Divide [0, T] into $[(k-1)T_0.kT_0]$ with $k=1,2,\ldots K$ and $T/K\le T_0$. Then we can obtain a unique continuous solution in [0, T]. In conclusion, we show that system (3.1) has a unique solution in $C(0,T;{\mathbb {R}}^m)$.

$\square$

Proof of Theorem 1.2

Gathering Lemma 3.1 and Lemma 3.2, we get

$$\begin{aligned} \{u_m\}\ \ \mathrm{is\ bounded \ in} \ L^{\infty }(0,T;H_0^\alpha (\Omega )\cap L^{p}(\Omega )) \end{aligned}$$

and

$$\begin{aligned} \{\partial _t^\beta u_m\}\ \ \mathrm{is\ bounded \ in} \ L^2(0,T;H_0^\alpha (\Omega )). \end{aligned}$$

By Proposition 2.12, we deduce that there exist a subsequence (still denoted by $\{u_m\}$) and $u\in L^{\infty }(0,T;H_0^\alpha (\Omega )\cap L^{p}(\Omega ))$ such that

$$\begin{aligned}&u_m \rightharpoonup u \qquad \mathrm{weakly\ star\ in} \ L^{\infty }(0,T;H_0^\alpha (\Omega )\cap L^{p}(\Omega )),\nonumber \\&u_m \rightarrow u \qquad \ \mathrm{strongly\ in}\ L^2(0,T;L^2(\Omega )),\nonumber \\&u_m \rightarrow u \qquad \mathrm{a.e.\ in}\ (0,T)\times \Omega . \end{aligned}$$

(3.11)

By the Vitali convergence theorem, one can show that

$$\begin{aligned} \lim _{m\rightarrow \infty }\int _0^T\int _\Omega f(x,u_m)vdx =\int _0^T\int _\Omega f(x,u)vdx \end{aligned}$$

for all $v\in C^1(0,T;H_0^\alpha (\Omega ))$.

Since $\{\partial _t^\beta u_m\}$ is bounded in $L^2(0,T;L^2(\Omega ))$, up to a subsequence we may assume that

$$\begin{aligned} \partial _t^\beta u_m\rightharpoonup \chi \ \mathrm{in}\ L^2(0,T;L^2(\Omega )). \end{aligned}$$

Next we show that $\chi =\partial _t^\beta u$. By Lemma 2.8, we obtain

$$\begin{aligned} {\int _0^T\int _\Omega \partial _t^\beta u_m\varphi dxdt} =-\int _0^T\int _\Omega \int _t^T \frac{\varphi ^\prime (s)}{(s-\tau )^{1-\beta }}ds(u_m-u_m(0))dtdx. \end{aligned}$$

It follows from (3.11) that

$$\begin{aligned} \int _0^T\int _\Omega \chi \varphi dxt&=-\int _0^T\int _\Omega \int _t^T \frac{\varphi ^\prime (s)}{(s-\tau )^{1-\beta }}ds(u-u_0)dtdx\\&=\int _0^T\int _\Omega \partial _t^\beta u\varphi dx, \end{aligned}$$

which means that $\chi =\partial _t^\beta u$. Further, by (3.1) we conclude that

$$\begin{aligned}&\int _0^T\int _\Omega \partial _t^{\beta }uvdxdt +\int _0^T\langle u,v\rangle _\alpha dt +\int _0^T\langle \partial _t^{\beta }u,v\rangle _\alpha dt\nonumber \\&\quad =\lambda \int _0^T\int _\Omega f(x,u)vdxdt+\int _0^T\int _\Omega gvdxdt \end{aligned}$$

(3.12)

for any $v\in L^2(0,T;H_0^\alpha (\Omega ))$.

Next we show that

$$\begin{aligned}&\int _\Omega \partial _t^{\beta }u\varphi dx +\langle u,\varphi \rangle _\alpha +\langle \partial _t^{\beta }u,\varphi \rangle _\alpha \nonumber \\&\quad =\lambda \int _\Omega f(x,u)\varphi dx+\int _\Omega g\varphi dx \end{aligned}$$

(3.13)

for any $\varphi \in H_0^\alpha (\Omega )$.

For any $t\in (0,T)$, let $\chi _{(0,t)}$ denote the characteristic function in (0, t). Let $\varphi \in H_0^\alpha (\Omega )$. Taking $v=\varphi \chi _{(0,t)}$ in (3.12), we get

$$\begin{aligned}&\int _0^t\int _\Omega \partial _t^{\beta }u\varphi dxdt +\int _0^t\langle u,\varphi \rangle _\alpha dt +\int _0^t\langle \partial _t^{\beta }u,\varphi \rangle _\alpha dt\nonumber \\&\quad =\lambda \int _0^t\int _\Omega f(x,u)\varphi dxdt+\int _0^t\int _\Omega g\varphi dxdt. \end{aligned}$$

(3.14)

Since

$$\begin{aligned}&\int _\Omega \partial _t^{\beta }u\varphi dx +\langle u,\varphi \rangle _\alpha +\langle \partial _t^{\beta }u,\varphi \rangle _\alpha -\lambda \int _\Omega f(x,u)\varphi dx-\int _\Omega g\varphi dx\in L^1(0,T), \end{aligned}$$

we differentiate (3.14) with respect to t and get that

$$\begin{aligned}&\int _\Omega \partial _t^{\beta }u\varphi dx +\langle u,\varphi \rangle _\alpha +\langle \partial _t^{\beta }u,\varphi \rangle _\alpha \nonumber \\&\quad =\lambda \int _\Omega f(x,u)\varphi dx+\int _\Omega g\varphi dx \end{aligned}$$

for any $\varphi \in H_0^\alpha (\Omega )$ and a.e. $t\in [0,T]$. Thus, (3.13) holds.

It follows that u is a weak solution of problem (1.1). $\square$

In the next lemma we shall show that under some assumptions the solution of problem (1.1) is nonnegative.

Lemma 3.5

If $f(x,\xi )\le 0$ for any $\xi \le 0$, $g\ge 0$ and $u_{0}\ge 0$ a.e. in $\Omega$. Then the solutions of problem (1.1) are nonnegative.

Proof

Let u be a weak solution to problem (1.1). Clearly,

$$\begin{aligned} u^{-}=\max \{-u, 0\}\in L^{\infty }(0,T; H_{0}^{\alpha }(\Omega )). \end{aligned}$$

Taking $v=-u^{-}$ in Definition 1.1, we obtain

$$\begin{aligned}&\int _{\Omega }\partial _t^\beta u(t)u^{-}(t)\,dx +\langle u(t), -u^{-}(t)\rangle _{\alpha } +\langle \partial _t^\beta u(t),-u^-(t)\rangle _{\alpha }\nonumber \\&\quad =-\int _{\Omega }f(x,u(t))u^{-}(t)\,dxdt-\int _\Omega gu^{-}(t)dx. \end{aligned}$$

(3.15)

Observe that for a.e. $x,y\in \Omega$,

$$\begin{aligned}&(u(x)-u(y))(-u^{-}(x)+u^{-}(y))\\&\quad =(u^{-}(x)-u^{-}(y))^{2}+u^{-}(x)u^{+}(y)+u^{+}(x)u^{-}(y)\\&\quad \ge |u^{-}(x)-u^{-}(y)|^{2},\quad \quad \quad \quad \quad \quad \quad \quad \quad . \end{aligned}$$

Then

$$\begin{aligned} \langle u(t), -u^{-}(t)\rangle _{\alpha }\ge \Vert u^-\Vert _{H_0^\alpha }^2. \end{aligned}$$

Moreover, $g(-u^-)\le 0$ and $f(x,u)u^{-}=0$ a.e. $x\in \Omega$.

To apply Lemma 2.2, we can use the same regularization discussion as above. For convenience, we omit this process. In view of Lemma 2.2, we have

$$\begin{aligned} -\left( \frac{d}{dt}(g_{1-\beta }*u(t)),u^{-}(t)\right) \ge \frac{1}{2}\frac{d}{dt}(g_{1-\beta }*\Vert u^{-}(t)\Vert _2^{2}) \end{aligned}$$

and

$$\begin{aligned} -\left\langle \frac{d}{dt}(g_{1-\beta }*u(t)),u^{-}(t)\right\rangle _\alpha \ge \frac{1}{2}\frac{d}{dt}(g_{1-\beta }*\Vert u^{-}(t)\Vert _{H_0^\alpha }^{2}). \end{aligned}$$

Combining these facts with (3.15), it yields

$$\begin{aligned} \frac{1}{2}\frac{d}{dt}(g_{1-\beta }*\Vert u^{-}(t)\Vert _2^{2})+ \frac{1}{2}\frac{d}{dt}(g_{1-\beta }*\Vert u^{-}(t)\Vert _{H_0^\alpha }^{2})+ \Vert u^{-}(t)\Vert ^{2}_{\alpha }\le 0. \end{aligned}$$

This implies that

$$\begin{aligned}&\frac{1}{2}(g_{1-\beta }*\Vert u^{-}(t)\Vert _2^{2})+ \frac{1}{2}(g_{1-\beta }*\Vert u^{-}(t)\Vert _{H_0^\alpha }^{2})+ \int _0^t\Vert u^{-}(t)\Vert ^{2}_{\alpha }dt\\&\quad \le \frac{1}{2}(g_{1-\beta }*\Vert u^{-}_0\Vert _2^{2})+ \frac{1}{2}(g_{1-\beta }*\Vert u^{-}_0\Vert _{H_0^\alpha }^{2}). \end{aligned}$$

Since $u_0\ge 0$ a.e. in $\Omega$, it leads to

$$\begin{aligned} g_{1-\beta }*(\Vert u^{-}(t)\Vert _2^{2}+\Vert u^{-}(t)\Vert _{H_0^\alpha }^{2}) \le 0. \end{aligned}$$

Then convoluting above inequality with $g_\beta$, it leads to

$$\begin{aligned} \Vert u^{-}(t)\Vert _2^{2}\le 0. \end{aligned}$$

Thus, we get $u^{-} (t)=0$ a.e. in $\Omega$ and for any $t>0$. Hence, $u(x,t)\ge 0$ a.e. in $\Omega$ and for any $t>0$. $\square$

At the end of this section, we study the uniqueness of solutions of problem (1.1).

Theorem 3.6

(Comparison theorem). Assume that f satisfies $(f_2)$. Let ${\underline{u}}$ be a subsolution of problem (1.1) and let ${\overline{u}}$ be a supsolution of problem (1.1). Then ${\underline{u}}\le {\overline{u}}$.

Proof

We deduce from Definition 1.1 that

$$\begin{aligned}&\int _{\Omega }\partial _t^\beta ({\underline{u}}(t)-{\overline{u}}(t))v \,dx +\langle {\underline{u}}(t)-{\overline{u}}(t), v\rangle _{\alpha } +\langle \partial _t^\beta ({\underline{u}}(t)-{\overline{u}}(t)),v\rangle _{\alpha }\nonumber \\&\quad \le \lambda \int _{\Omega }(f(x,{\underline{u}}(t))-f(x,{\overline{u}}(t)))v\,dx, \end{aligned}$$

for any $0\le v\in H_0^\alpha (\Omega )$. Taking $v=({\underline{u}}-{\overline{u}})^+=\max \{{\underline{u}}-{\overline{u}},0\}$ and applying Lemma 2.2 and $(f_2)$, it leads to

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}(g_{1-\beta }*\Vert ({\underline{u}}(t)-{\overline{u}}(t))^+\Vert _2^2) +[({\underline{u}}(t)-{\overline{u}}(t))^+]^2_{\alpha }\\&\qquad +\frac{1}{2}\frac{d}{dt}(g_{1-\beta }*(\Vert ({\underline{u}}(t)-{\overline{u}}(t))^+\Vert _\alpha ^2))\nonumber \\&\quad \le \lambda \mathcal L\int _{\Omega }|({\underline{u}}(t)-{\overline{u}}(t))^+|^2\,dx. \end{aligned}$$

Set $Y(t)=\Vert ({\underline{u}}(t)-{\overline{u}}(t))^+\Vert _2^2+[({\underline{u}}(t)-{\overline{u}}(t))^+]_\alpha ^2$. Then

$$\begin{aligned} \frac{d}{dt}(g_{1-\beta }*Y(t))\le 2\lambda {\mathcal {L}} Y(t). \end{aligned}$$

Convoluting above inequality with $g_\beta$, it follows that

$$\begin{aligned} Y(t)\le 2\lambda {\mathcal {L}} g_\beta *Y(t). \end{aligned}$$

By Lemma 2.6 and $Y(0)=0$, one can get that $Y(t)=0$. Thus, we get ${\underline{u}}\le {\overline{u}}$ a.e. in $\Omega \times (0,T)$, which ends the proof. $\square$

Lemma 3.7

Assume that f satisfies $(f_2)$. Then the solution of problem (1.1) is unique.

Proof

Assume that $u_1$ and $u_2$ are two solutions of problem (1.1). Then we deduce from 1.1 that

$$\begin{aligned}&\int _{\Omega }\partial _t^\beta (u_1(t)-u_2(t))v \,dx +\langle u_1(t)-u_2(t), v\rangle _{\alpha } +{\langle \partial _t^\beta (u_1(t)-u_2(t)),v\rangle _{\alpha }}\nonumber \\&\quad =\lambda \int _{\Omega }(f(x,u_1(t))-f(x,u_2(t)))v\,dx, \end{aligned}$$

for any $v\in H_0^\alpha (\Omega )$. Taking $v=u_1-u_2$, we get

$$\begin{aligned}&\int _{\Omega }(u_1(t)-u_2(t))\partial _t^\beta (u_1(t)-u_2(t))\,dx +[u_1(t)-u_2(t)]^2_{\alpha }\\&\qquad +\langle \partial _t^\beta (u_1(t)-u_2(t)),u_1(t)-u_2(t)\rangle _{s}\\&\quad =\lambda \int _{\Omega }(f(x,u_1(t))-f(x,u_2(t)))u_1(t)-u_2(t))\,dx. \end{aligned}$$

Using a similar discussion as above and applying Lemma 2.2 and $(f_2)$, it leads to

$$\begin{aligned}&\frac{1}{2}\frac{d}{dt}(g_{1-\beta }*\Vert (u_1(t)-u_2(t)\Vert _2^2) +[u_1(t)-u_2(t)]^2_{\alpha } +\frac{1}{2}\frac{d}{dt}(g_{1-\beta }*([u_1(t)-u_2(t)]_\alpha ^2))\nonumber \\&\quad \le \lambda {\mathcal {L}}\int _{\Omega }|u_1(t)-u_2(t)|^2\,dx. \end{aligned}$$

Set $Z(t)=\Vert (u_1(t)-u_2(t)\Vert _2^2+[u_1(t)-u_2(t)]_\alpha ^2$. Then

$$\begin{aligned} \frac{d}{dt}(g_{1-\beta }*Z(t))\le 2\lambda {\mathcal {L}} Z(t). \end{aligned}$$

Convoluting above inequality with $g_\beta$, it follows that

$$\begin{aligned} Z(t)\le \lambda {\mathcal {L}} g_\beta *Z(t), \end{aligned}$$

which together with Lemma 2.2 and $Z(0)=0$ yields that $Z(t)=0$. Thus, we get $u_1=u_2$ a.e. in $\Omega \times (0,T)$, which ends the proof. $\square$

4 Decay Estimates of Solutions

In this section, we give a decay estimate for problem (1.1) in which f satisfies $(f_2)$.

Let $\varphi _1$ be the corresponding eigenfunction to the first eigenvalue $\lambda _1$ of the fractional Laplacian. Clearly, $\varphi _1>0$ and $\varphi _1\in L^\infty (\Omega )$. Let $0\le u_0(x)\le \eta _0\varphi _1(x)$, where $\eta _0>0$. Assume that there exists $h_0>0$ such that $f(x,\xi )\le h_0\xi$ for all $x\in \Omega$ and $\xi \ge 0$.

Set $v(x,t)=\varphi _1(x)\eta (t)$, where $\eta$ satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t^\beta \eta (t)+\frac{\lambda _1-h_0\lambda }{1+\lambda _1}\eta (t)=0 \ \ \text {for all}\ t\ge 0,\\ \eta (0)=\eta _0>0, \end{array}\right. } \end{aligned}$$

where $0<\lambda <\frac{\lambda _1}{h_0}$.

By [30, Theorem 7.1], there exist $c_1,c_2>0$ such that

$$\begin{aligned} \frac{c_1}{1+t^\beta }\le \eta (t)\le \frac{c_2}{1+t^{\beta }}\ \ \mathrm{for\ all}\ t\ge 0. \end{aligned}$$

(4.1)

A simple calculation gives that

$$\begin{aligned}&\partial ^\beta _tv(x,t)+(-\Delta )^\alpha v(x,t)+(-\Delta )^\alpha \partial ^\beta _tv(x,t)\\&\quad = \varphi _1(x)\partial ^\beta _t\eta (t)+\lambda _1\varphi _1(x)\eta (t)+\lambda _1\varphi _1(x)\partial ^\beta _t\eta (t)\\&\quad =\varphi _1(x)(\partial ^\beta _t\eta (t)+\lambda _1\eta (t) +\lambda _1\partial ^\beta _t\eta (t))\\&\quad =\varphi _1(x)\lambda h_0 \eta (t) =\lambda h_0v(x,t)\ge \lambda f(x,v). \end{aligned}$$

Since $v(x,0)=\varphi _1(x)\eta _0$ and $\varphi _1(x)\eta _0\ge u_0(x)$, we know that v(x, t) is a supersolution of problem (1.1). Then by Theorem 3.6, we obtain that the unique solution u of problem (1.1) satisfies

$$\begin{aligned} u(x,t)\le v(x,t) \ \ \mathrm{for\ a.e.}\ (x,t)\in \Omega \times (0,\infty ). \end{aligned}$$

Further, (4.1) implies that

$$\begin{aligned} 0\le u(x,t)\le \frac{c_2\varphi _1(x)}{1+t^{\beta }}\ \ \mathrm{for\ all}\ t\ge 0\ \ \mathrm{and}\ x\in \Omega . \end{aligned}$$

5 Blow-Up of Solutions

In this section, we consider the blow-up property of solutions of problem (1.1). Some techniques are inspired from [23] and [39]. Let $u_0\in H_0^\alpha (\Omega )$ satisfy $u_0\ge 0$. If $2<p$ and $\int _\Omega u_0\varphi _1 dx>(\frac{\lambda _1}{\lambda })^{1/(p-2)}$, then the solution of problem (1.1) blows up in finite time.

Proof of Theorem 1.4

Taking $\varphi =\varphi _1(x)\varphi _2(t)$ with $\varphi _2\in C^1(0,T)$ in Definition 1.1 and integrating over (0, T), we get

$$\begin{aligned}&\int _0^T\int _{\Omega }\partial _t^\beta u(t)\varphi _1\varphi _2\,dxdt +\int _0^T\langle u(t), \varphi _1\varphi _2\rangle _{\alpha }dt +\int _0^T\langle \partial _t^\beta u(t),\varphi _1\varphi _2\rangle _{\alpha }dt\nonumber \\&\quad =\int _0^T\lambda \int _{\Omega }|u(t)|^{p-1}\varphi _1\varphi _2\,dxdt, \end{aligned}$$

which gives that

$$\begin{aligned}&\int _0^T\int _{\Omega }\partial _t^\beta u(t)\varphi _1\varphi _2\,dxdt +\lambda _1\int _0^T\int _\Omega u(t)\varphi _1\varphi _2dxdt +\lambda _1\int _0^T \int _\Omega \partial _t^\beta u(t)\varphi _1\varphi _2dxdt\nonumber \\&\quad =\int _0^T\lambda \int _{\Omega }|u(t)|^{p-1}\varphi _1\varphi _2\,dxdt. \end{aligned}$$

Set $H(t)=\int _\Omega u(x,t)\varphi _1(x)dx$. Then

$$\begin{aligned}&(1+\lambda _1)\int _0^T\partial _t^\beta H(t)\varphi _2(t)dt +\lambda _1\int _0^T H(t)\varphi _2(t)dt\nonumber \\&\quad =\lambda \int _0^T\int _{\Omega }|u(t)|^{p-1}\varphi _1\varphi _2\,dxdt. \end{aligned}$$

It follows from Jensen’s inequality that

$$\begin{aligned}&(1+\lambda _1)\int _0^T\partial _t^\beta H(t)\varphi _2(t)dt +\lambda _1\int _0^T H(t)\varphi _2(t)dt\nonumber \\&\quad \ge \int _0^T\lambda (\int _{\Omega }u(t)\varphi _1dx)^{p-1}\varphi _2(t)dt\\&\quad =\lambda \int _0^TH(t)^{p-1}\varphi _2(t)dt, \end{aligned}$$

thanks to $p>2$. By Lemma 2.9, we have

$$\begin{aligned}&-(1+\lambda _1)\int _0^T (H(t)-H(0))I_{T-}^{\beta }\varphi _2^\prime (t)dt +\lambda _1\int _0^T H(t)\varphi _2(t)dt\nonumber \\&\quad \ge \int _0^T\lambda (\int _{\Omega }u(t)\varphi _1dx)^{p-1}\varphi _2(t)dt\nonumber \\&\quad =\lambda \int _0^TH(t)^{p-1}\varphi _2(t)dt, \end{aligned}$$

(5.1)

where

$$\begin{aligned} I_{T-}^{\beta }\varphi _2^\prime (t)=\frac{1}{\Gamma (\beta )}\int _t^T(\tau -t)^{1-\beta }\varphi _2^\prime (\tau )d\tau . \end{aligned}$$

On one hand, choose $\varphi _2=I_{T-}^\beta {\bar{\varphi }}(t)$ with ${\bar{\varphi }}\in C_0^1(0,T)$ and ${\bar{\varphi }}\ge 0$. Then we deduce from [13, Lemma 2.21 ] that

$$\begin{aligned} \int _0^T (\lambda H^{p-1}-\lambda _1H)I_{T-}^\beta {\bar{\varphi }}(t)dt&\le (1+\lambda _1)\int _0^T(H-H(0)){\bar{\varphi }}dt. \end{aligned}$$

(5.2)

Applying Lemma 2.8 to the left of (5.2), it leads to

$$\begin{aligned} \int _0^T I_{0+}^\beta (\lambda H^{p-1}-\lambda _1H){\bar{\varphi }}(t)dt&\le {(1+\lambda _1)}\int _0^T(H-H(0)){\bar{\varphi }}dt. \end{aligned}$$

The arbitrary of ${\bar{\varphi }}$ gives that

$$\begin{aligned} I_{0+}^\beta (\lambda H^{p-1}-\lambda _1H)+(1+\lambda _1)H(0)\le (1+\lambda _1)H(t). \end{aligned}$$

Since $H(0)>(\frac{\lambda _1}{\lambda })^{1/(p-2)}$, we have $H(t)>(\frac{\lambda _1}{\lambda })^{1/(p-2)}$ as t small enough. Then we have

$$\begin{aligned} H(t)\ge H(0)>(\frac{\lambda _1}{\lambda })^{1/(p-2)}\ \ \text {for all}\ t\in [0,T]. \end{aligned}$$

On the other hand, choose $\varphi _2=(1-\frac{t}{T})_+^k$, $t\in [0,T]$, $k>\max \{1,(p-1)\beta /(p-2)\}$, in (5.1). By a direct calculation (see [13]), one can show that

$$\begin{aligned} -I_{T-}^\beta \varphi _2(t)=\frac{\Gamma (k+1)}{\Gamma (k+1-\beta )}T^{-\beta }(1-\frac{t}{T})^{k-\beta }. \end{aligned}$$

Then

$$\begin{aligned} -\int _0^T(H(t)-H(0))I_{T-}^\beta \varphi _2(t)dt&=\frac{\Gamma (k+1)}{\Gamma (k+1-\beta )}T^{-\beta } \int _0^TH(t)(1-\frac{t}{T})^{k-\beta }dt\\&\quad -\frac{\Gamma (k+1)}{\Gamma (k+1-\beta )}\frac{H(0)T^{1-\beta }}{(k+1-\beta )}. \end{aligned}$$

By the Hölder inequality, one has

$$\begin{aligned}&\int _0^TH(t)(1-\frac{t}{T})^{k-\beta }dt\\&\quad \le \left( \int _0^TH^{p-1}(t)(1-\frac{t}{T})^{k}dt\right) ^{1/(p-1)} \left( \int _0^T(1-\frac{t}{T})^{(k-\beta -\frac{k}{p-1}) \frac{p-1}{p-2}}dt\right) ^{(p-2)/(p-1)}. \end{aligned}$$

Further, using the Young inequality, for any $\varepsilon >0$ we get

$$\begin{aligned}&(1+\lambda _1)\frac{\Gamma (k+1)}{\Gamma (k+1-\beta )}T^{-\beta } \int _0^TH(t)(1-\frac{t}{T})^{k-\beta }dt\\&\quad \le \varepsilon \int _0^TH^{p-1}(t)(1-\frac{t}{T})^{k}dt +C(\varepsilon )T^{1-(p-1)\beta /(p-2)}. \end{aligned}$$

We know that there exists a constant $C>0$ such that

$$\begin{aligned}&\int _0^T (\lambda H^{p-1}-\lambda _1H)\varphi _2dt+ (1+\lambda _1)\frac{\Gamma (k+1)}{\Gamma (k+1-\beta )}\frac{H(0)T^{1-\beta }}{k+1-\beta }\\&\quad \le \varepsilon \int _0^T H^{p-1}\varphi _2dt +C(\varepsilon )T^{1-(p-1)\beta /(p-2)}. \end{aligned}$$

Choosing $\varepsilon$ small enough such that $H(0)>(\frac{\lambda _1}{\lambda -\varepsilon })^{-1/(p-2)}$, we get $H(0)\le CT^{\beta -(p-1)\beta /(p-2)}$ for some $C>0$. If problem (1.1) has a global weak solution, we obtain $H(0)=0$ by letting $T\rightarrow \infty$, which contradicts $H(0)>(\frac{\lambda _1}{\lambda })^{-1/(p-2)}$. Therefore we show that the global nonexistence of problem (1.1). $\square$