Qualitative properties of solutions to a nonlinear time-space fractional diffusion equation

Abstract

In the present paper, we study the Cauchy-Dirichlet problem to a nonlocal nonlinear diffusion equation with polynomial nonlinearities $\mathcal {D}_{0|t}^{\alpha }u+(-\varDelta )^s_pu=\gamma |u|^{m-1}u+\mu |u|^{q-2}u,\,\gamma ,\mu \in \mathbb {R},\,m>0,q>1,$ involving time-fractional Caputo derivative $\mathcal {D}_{0|t}^{\alpha }$ and space-fractional p-Laplacian operator $(-\varDelta )^s_p$. We give a simple proof of the comparison principle for the considered problem using purely algebraic relations, for different sets of $\gamma ,\mu ,m$ and q. The Galerkin approximation method is used to prove the existence of a local weak solution. The blow-up phenomena, existence of global weak solutions and asymptotic behavior of global solutions are classified using the comparison principle.

1 Introduction

In this paper, we study the initial-boundary value problem for the nonlinear time-space fractional diffusion equation

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {D}_{0|t}^{\alpha }u+(-\varDelta )^s_pu=\gamma |u|^{m-1}u+\mu |u|^{q-2}u,\, \,(x,t)\in \varOmega \times (0,T),\\ {}\\ u(x,t)=0,\,x\in \mathbb {R}^N\setminus \varOmega ,\,t\in (0,T),\\ {}\\ u(x,0)=u_0(x),\, x\in \varOmega ,\end{array}\right. \end{aligned}$$

(1.1)

where $\varOmega \subset \mathbb {R}^N$ is a smoothly bounded domain; $s\in (0,1), p\ge 2, m>0, q\ge 1$, $\gamma ,\mu \in \mathbb {R}$ and $\mathcal {D}_{0|t}^{\alpha }$ is the left Caputo fractional derivative of order $\alpha \in (0,1)$ (see Definition 3).

In recent years, the study of differential equations using non-local fractional operators has attracted a lot of interest. The time-space fractional diffusion equations could be applied to a wide range of applications, including finance, semiconductor research, biology and hydrogeology, continuum mechanics, phase transition phenomena, population dynamics, image process, game theory and Lévy processes, (see [3, 5, 9, 15, 17, 22, 23]) and the references therein. When a particle flow spreads at a rate that defies Brownian motion theories, both time and spatial fractional derivatives (see [16, 24, 35]) can be employed to simulate anomalous diffusion or dispersion. Recently, motivated by some situations arising in the game theory, nonlinear generalizations of the fractional Laplacian have been introduced, (see [6, 9]).

Later on, the fractional version of the p-Laplacian was studied through energy and test function methods by Chambolle and al. in [10]. The viscosity version of this non-local operator was given by Ishii and al. in [20], Bjorland and al. in [6].

In the case $\alpha =s=1,\,\gamma =-1$ the problem (1.1) coincides with a quasilinear parabolic equation which has been studied by Li et al in [25]. By using a Gagliardo-Nirenberg type inequality, the energy method and comparison principle, the phenomena of blow-up and extinction have been classified completely in the different ranges of reaction exponents.

Moreover, when $\alpha =s=1, m>1$ and the coefficients are $\gamma >0, \mu =0$, the problem (1.1) was considered by Yin and Jin in [38]. They determined the critical extinction and blow-up exponents for the homogeneous Dirichlet boundary value problem.

Vergara and Zacher in [36] have considered nonlocal in time semilinear subdiffusion equations on a bounded domain,

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {D}_{0|t}^{\alpha } u -{\text {div}}(A(x,t)\nabla u)=f(u),\,x\in \varOmega ,\,\,t>0,\\ u(x,t)=0,\,\, x\in \partial \varOmega ,\,\,t>0, \\ u(x,0)=u_0(x),\, x\in \varOmega ,\end{array}\right. \end{aligned}$$

(1.2)

where the coefficients $A=(a_{ij})$ were assumed to satisfy

$$\begin{aligned}\left( A(x,t)\xi ,\xi \right) \ge \nu |\xi |^2,\,\,\text {for a.e.}\,\,\,(x,t)\in \varOmega \times (0,+\infty )\,\,\text {and all}\,\,\, \xi \in \mathbb {R}^N.\end{aligned}$$

They proved a well-posedness result in the setting of bounded weak solutions and studied the stability and instability of the zero function in the special case where the nonlinearity vanishes at 0. In addition, they established a blow-up result for positive convex and superlinear nonlinearities.

Later on, Alsaedi et al. [2] have studied the KPP-Fisher-type reaction-diffusion equation, which is the problem (1.1) in the case $p=2, \gamma =-1, q=3$ and $\mu =m=1$, in a bounded domain. Under some conditions on the initial data, they have showed that solutions may experience blow-up in a finite time. However, for realistic initial conditions, solutions are global in time. Moreover, the asymptotic behavior of bounded solutions was analysed.

Recently, in [34], Tuan, Au and Xu studied the initial-boundary value problem for the fractional pseudo-parabolic equation with fractional Laplacian

$$\begin{aligned} \left\{ \begin{array}{l} \mathcal {D}_{0|t}^{\alpha } (u-m \varDelta u)+(-\varDelta )^s u=\mathcal {N}(u), \, x\in \varOmega ,\,t>0,\\ u(x,t)=0,\, x\in \partial \varOmega , t>0,\\ u(x,0)=u_0(x),\, x\in \varOmega ,\end{array}\right. \end{aligned}$$

(1.3)

where $s\in (0,1), m>0$ is a constant, and $\mathcal {N}(u)$ is the source term satisfying one of the following conditions:

• (a) $\mathcal {N}(u)$ is a globally Lipschitz function;

• (b) $\mathcal {N}(u)=|u|^{p-2}u,\,p\ge 2;$

• (c) $\mathcal {N}(u)=|u|^{p-2}u\log |u|,\,\,p\ge 2.$

For the above cases, they proved the existence of a unique local mild solution and finite time blow-up solution to equation (1.3). Because of the nonlocality of the equation, the authors believe that proving the existence of a weak solution using the Galerkin method for equation (1.3) is problematic.

Motivated by the above results, in this paper we consider the time and space fractional quasilinear parabolic equation (1.1).

Using the Galerkin method, we prove the existence of a local weak solution to problem (1.1).

This, in turn, partially answers the question posed in [34] about the existence of a local weak solution to the fractional pseudo-parabolic equation. In addition, a comparison principle to problem (1.1) is obtained, and we have investigated results on the blow-up and global solution using this concept.

2 Preliminaries

2.1 The fractional Sobolev space

In this subsection, let us recall some necessary definitions and useful properties of the fractional Sobolev space.

Let $s\in (0,1)$ and $p\in [1,+\infty )$ be real numbers, and let the fractional critical exponent be defined as $\displaystyle p_c^*=\frac{Np}{N-sp}$ if $sp<N$ or $p_c^*=\infty$, otherwise.

One defines the fractional Sobolev space as follows

$$\begin{aligned} W^{s,p}(\mathbb {R}^N):=\biggl \{u\in L^p(\mathbb {R}^N), \frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\mathbb {R}^N\times \mathbb {R}^N)\biggr \}. \end{aligned}$$

This is the Banach space between $L^p(\mathbb {R}^N)$ and $W^{1,p}(\mathbb {R}^N)$, endowed with the norm

$$\begin{aligned}\Vert u\Vert _{W^{s,p}(\mathbb {R}^N)}:=\Vert u\Vert _{L^p{(\mathbb {R}^N})} +\biggl (\int _{\mathbb {R}^N}\int _{\mathbb {R}^N}\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dxdy\biggr )^\frac{1}{p}.\end{aligned}$$

Let $\varOmega$ be an open set in $\mathbb {R}^N$ and let $\mathcal {W}=(\mathbb {R}^N\times \mathbb {R}^N)\backslash ((\mathbb {R}^N\backslash \varOmega )\times (\mathbb {R}^N\backslash \varOmega ))$. It is obvious that $\varOmega \times \varOmega$ is strictly contained in $\mathcal {W}$.

Denote

$$\begin{aligned} W^{s,p}(\varOmega ):=\biggl \{u\in L^p(\varOmega ),\, u=0\,\, \text {in}\,\, \mathbb {R}^N\backslash \varOmega ,\, \frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\mathcal {W})\biggr \}. \end{aligned}$$

The space $W^{s,p}(\varOmega )$ is also endowed with the norm

$$\begin{aligned}\Vert u\Vert _{W^{s,p}(\varOmega )}:=\Vert u\Vert _{L^p(\varOmega )} +[u]_{W^{s,p}(\varOmega )},\end{aligned}$$

where the term

$$\begin{aligned}_{W^{s,p}(\varOmega )}:=\biggl (\int _\varOmega \int _\varOmega \frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dxdy\biggr )^\frac{1}{p}<\infty \end{aligned}$$

is the so-called Gagliardo semi-norm of u, which was introduced by Gagliardo [13] to describe the trace spaces of Sobolev maps.

We refer to [28] and [7], where one can find a description of the most useful properties of the fractional Sobolev spaces $W^{s,p}(\varOmega )$. In the literature, fractional Sobolev-type spaces are also called Aronszajn, Gagliardo or Slobodeckij spaces, by the name of the people who first introduced them, practically concurrently (see [4, 14, 31]).

For Gagliardo semi-norms, the next result is a Poincaré inequality. This is standard, but we should also always pay careful attention to the sharp constants dependence on s.

Lemma 1

([28], Theorem 6.7) Let $s\in (0,1)$ and $p\in [1,+\infty )$ be such that $sp<N$. Let $\varOmega \subseteq \mathbb {R}^N$ be an extension domain for $W^{s,p}$. Then, there exists a positive constant $C=C(N,p,s,\varOmega )$ such that, for any $u\in W^{s,p}(\varOmega )$, we have

$$\begin{aligned}\Vert u\Vert _{L^q(\varOmega )}\le C\Vert u\Vert _{W^{s,p}(\varOmega )},\end{aligned}$$

for any $q\in [p,p^*],$ where $\displaystyle p^*=p^*(N,s)=\frac{Np}{N-sp}$ is the so-called fractional critical exponent.

That means, the space $W^{s,p}(\varOmega )$ is continuously embedded in $L^q(\varOmega )$ for $q\in [p,p^*].$ If, in addition, $\varOmega$ is bounded, then, the space $W^{s,p}(\varOmega )$ is continuously embedded in $L^q(\varOmega )$ for $q\in [1,p^*].$

Lemma 2

([29], Lemma 2.1. Fractional Gagliardo–Nirenberg inequality) Let $p>1, \tau >0, N,q\ge 1, 0<s<1$ and $0 <a\le 1$ be such that

$$\begin{aligned}\frac{1}{\tau }=a\left( \frac{1}{p}-\frac{s}{N}\right) +\frac{1-a}{q}.\end{aligned}$$

We have

$$\begin{aligned}\Vert u\Vert _{L^\tau (\mathbb {R}^N)}\le C[u]^a_{W^{s,p}(\mathbb {R}^N)}\Vert u\Vert ^{(1-a)}_{L^q(\mathbb {R}^N)},\,\,\text {for}\,\,\,u\in C^1_c(\mathbb {R}^N),\end{aligned}$$

for some positive constant C independent of u.

2.2 Fractional operators

This part is devoted to the definitions and properties of fractional derivatives in time and space.

Definition 1

([21], p. 69) The left and right Riemann-Liouville fractional integrals of order $0<\alpha <1$ for an integrable function u(t) are given by

$$\begin{aligned}I^\alpha _{0|t}u(t)=\frac{1}{\varGamma \left( \alpha \right) }\int \limits _{0}^{t}{{{\left( t-s \right) }^{\alpha -1}}}u\left( s \right) ds, \,\,\,t\in (0,T]\end{aligned}$$

and

$$\begin{aligned}I^\alpha _{t|T}u(t)=\frac{1}{\varGamma \left( \alpha \right) }\int \limits _{t}^{T}{{{\left( s-t \right) }^{\alpha -1}}}u\left( s \right) ds, \,\,\,t\in [0,T).\end{aligned}$$

Definition 2

([21], p. 70) The left and right Riemann-Liouville fractional derivatives $\mathbb {D}_{0|t}^{\alpha }$ of order $\alpha \in (0,1)$, for an absolutely continuous function u(t) is defined by

$$\begin{aligned}\mathbb {D}_{0|t}^{\alpha } u(t)=\frac{d}{dt}I_{{0|t}}^{1-\alpha } u(t)=\frac{1}{\varGamma (1-\alpha )}\frac{d}{dt}\int \limits _{0}^{t}{{(t-s)^{-\alpha }}}{u}\left( s \right) ds,\,\,\, \forall t\in (0,T]\end{aligned}$$

and

$$\begin{aligned}\mathbb {D}_{t|T}^{\alpha } u(t)=-\frac{d}{dt}I_{t|T}^{1-\alpha } u(t)=-\frac{1}{\varGamma (1-\alpha )}\frac{d}{dt}\int \limits _{t}^{T}{{(s-t)^{-\alpha }}}{u}\left( s \right) ds,\,\,\, \forall t\in [0,T).\end{aligned}$$

Lemma 3

([21], Lemma 2.20) If $\alpha >0,$ then for $u\in L^1(0,T)$, the relations

$$\begin{aligned}\mathbb {D}_{0|t}^{\alpha }I_{0|t}^{\alpha } u(t)=u(t)\,\,\,\text {and}\,\,\,\mathbb {D}_{t|T}^{\alpha }I_{t|T}^{\alpha }u(t)=u(t)\end{aligned}$$

are true.

Definition 3

([21], p. 91) The $\alpha \in (0,1)$ order of left and right Caputo fractional derivatives for $u\in C^1([0,T])$ are defined, respectively, by

$$\begin{aligned}\mathcal {D}_{0|t}^{\alpha } u(t)=I_{0|t}^{1-\alpha } \frac{d}{dt}u(t)=\frac{1}{\varGamma (1-\alpha )}\int \limits _{0}^{t}{{(t-s)^{-\alpha }}}{u'}\left( s \right) ds,\,\,\, \forall t\in (0,T]\end{aligned}$$

and

$$\begin{aligned}\mathcal {D}_{t|T}^{\alpha } u(t)=-I_{t|T}^{1-\alpha }\frac{d}{dt} u(t)=-\frac{1}{\varGamma (1-\alpha )}\int \limits _{t}^{T}{{(s-t)^{-\alpha }}}{u'}\left( s \right) ds,\,\,\, \forall t\in [0,T).\end{aligned}$$

If $u\in C^1([0,T])$, then the Caputo fractional derivative can be represented by the Riemann-Liouville fractional derivative in the following form

$$\begin{aligned}\mathcal {D}_{0|t}^{\alpha } u(t)=\mathbb {D}_{0|t}^{\alpha }[u(t)-u(0)],\,\,\, \forall t\in (0,T]\end{aligned}$$

and

$$\begin{aligned}\mathcal {D}_{t|T}^{\alpha } u(t)=\mathbb {D}_{t|T}^{\alpha }[u(t)-u(T)],\,\,\, \forall t\in [0,T).\end{aligned}$$

Lemma 4

([39], Corollary 4.1) Let $T>0$ and let U be an open subset of $\mathbb {R}$. Let further $u_0\in U,$ $k\in H^1_1(0,T), H\in C^1(U)$ and $u\in L^1(0,T)$ with $u(t)\in U,$ for a. a. $t\in (0,T)$. Suppose that the functions $H(u), H'(u)u$, and $H'(u)(k_t*u)$ belong to $L^1(0,T)$ (which is the case if, e.g., $u\in L^\infty (0,T)$). Assume in addition that k is nonnegative and nonincreasing and that H is convex. Then

$$\begin{aligned}H'(u(t))\frac{d}{dt}(k*[u-u_0])(t)\ge \frac{d}{dt}(k*[H(u)-H(u_0)])(t),\,\,\,t\in (0,T).\end{aligned}$$

Lemma 5

([1], Lemma 1) For $0<\alpha <1$ and any function u(t) absolutely continuous and real-valued on [0, T], one has the inequality

$$\begin{aligned}\begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha } (u^2)(t)&\le u(t)\mathcal {D}_{0|t}^{\alpha } u(t). \end{aligned}\end{aligned}$$

Property 1

([21], p. 95-96) If $0<\alpha <1$, $u\in AC^1[0,T]$ or $u\in C^1[0,T]$, then

$$\begin{aligned} I^{\alpha }_{0|t}(\mathcal {D}^{\alpha }_{0|t}u)(t)=u(t)-u(0) \end{aligned}$$

and

$$\begin{aligned} \mathcal {D}^{\alpha }_{0|t}(I^{\alpha }_{0|t}u)(t)=u(t), \end{aligned}$$

hold almost everywhere on [0, T]. In addition,

$$\begin{aligned}\mathcal {D}^{1-\alpha }_{0|t}\int _0^t\mathcal {D}^{\alpha }_{0|\tau }u(\tau )d\tau =\biggl (I^{\alpha }_{0|t}\frac{d}{dt}I^1_{0|t}I^{1-\alpha }_{0|t}\frac{d}{dt}u\biggr )(t)=u(t)-u(0).\end{aligned}$$

Property 2

([21], Lemma 2.7) Let $0<\alpha <1$ and $u\in C^1[0,T], \varphi \in L^p(0,T)$. Then the integration by parts for Caputo fractional derivatives has the form

$$\begin{aligned} \int _0^T\left[ \mathcal {D}_{0|t}^{\alpha } u\right] (t)\varphi (t)dt=\int _0^T u(t)\left[ \mathbb {D}_{t|T}^{\alpha }\varphi \right] (t)dt+\left[ I^{1-\alpha }_{t|T}\varphi \right] (t)u(t)\biggr |_0^T. \end{aligned}$$

Definition 4

([32]. Lemma 5.1) The fractional p-Laplacian operator for $s\in (0,1), p>1$ and $u\in W^{s,p}(\varOmega )$, is defined by

$$\begin{aligned} (-\varDelta )^s_pu(x)=C_{N,s,p}\,\text {P.V.}\int _{\mathbb {R}^N}\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}dy, \end{aligned}$$

where

$$\begin{aligned} C_{N,s,p}=\frac{sp2^{2s-2}}{\pi ^{\frac{N-1}{2}}}\frac{\varGamma (\frac{N+sp}{2})}{\varGamma (\frac{p+1}{2})\varGamma (1-s)} \end{aligned}$$

(2.1)

is a normalization constant and “P.V.” is an abbreviation for “in the principal value sense”. Since they will not play a role in this work, we omit the P.V. sense. However, let us stress that these constants guarantee:

$$\begin{aligned}\begin{aligned}&(-\varDelta )^s_pu(x)\xrightarrow []{s\rightarrow 1^-}-\varDelta _pu(x),\,\,\,\text {for all}\,\,\,\, p\in [2,\infty ), \\ {}&(-\varDelta )^s_pu(x)\xrightarrow []{p\rightarrow 2^+}(-\varDelta )^su(x),\,\,\,\text {for all}\,\,\,\, s\in (0,1). \end{aligned}\end{aligned}$$

Definition 5

([26], Theorem 5) We say that $u\in W^{s,p}_0(\varOmega )$ is an (s, p) - eigenfunction associated to the eigenvalue $\lambda$ if u satisfies the Dirichlet problem

$$\begin{aligned} \left\{ \begin{array}{l} (-\varDelta )^s_pu(x)=\lambda |u(x)|^{p-2}u(x),\,\,\, x\in \varOmega ,\\ u(x)=0,\,\,\, x\in \mathbb {R}^N\setminus \varOmega ,\end{array}\right. \end{aligned}$$

(2.2)

weakly, it means that

$$\begin{aligned} \int _{\varOmega }\int _{\varOmega }\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))}{|x-y|^{N+sp}}(\psi (x)-\psi (y))dxdy\\=\lambda \int _{\varOmega }|u(x)|^{p-2}u(x)\psi (x)dx, \end{aligned}$$

for every $\psi \in W^{s,p}_0(\varOmega )$. If we set as

$$\begin{aligned}\varSigma _p(\varOmega ):=\biggl \{u\in W^{s,p}_0(\varOmega ):\int _{\varOmega }|u(x)|^{p}dx=1\biggr \},\end{aligned}$$

then the nonlinear Rayleigh quotient determines the first eigenvalue

$$\begin{aligned} \lambda _1(\varOmega ):=\min _{u\in \varSigma _p(\varOmega )}\int _{\varOmega }\int _{\varOmega }\frac{|u(x)-u(y)|^p}{|x-y|^{N+sp}}dxdy.\end{aligned}$$

Lemma 6

([26], Lemma 15) Assume that for all j, if we have

$$\begin{aligned}\varOmega _1\subset \varOmega _2\subset \varOmega _3\subset ...\subset \varOmega ,\,\,\,\,\varOmega =\bigcup \varOmega _j.\end{aligned}$$

Then

$$\begin{aligned}\lim _{j\rightarrow \infty }\lambda _1(\varOmega _j)=\lambda _1(\varOmega ).\end{aligned}$$

Note that the minimization problem is not quite the same if $\mathbb {R}^N\times \mathbb {R}^N$ is replaced by $\varOmega \times \varOmega$ in the integral. This choice has the advantage that the property

$$\begin{aligned}\lambda _1(\varOmega ^*)\le \lambda _1(\varOmega ),\,\,\,\text {if}\,\,\,\,\varOmega \subset \varOmega ^*\end{aligned}$$

is evident for subdomains. By changing coordinates it implies

$$\begin{aligned}\lambda _1(\varOmega ^*)=k^{\alpha p-N}\lambda _1(k\varOmega ^*),\,\,\,k>0.\end{aligned}$$

This asserts that small domains have large first eigenvalues (see [26] references therein).

Lemma 7

([7], Lemma 2.4. Fractional Poincaré inequality)

Let $1 \le p <\infty$ and $s \in (0, 1)$, $\varOmega \subset \mathbb {R}^N$ be an open and bounded set. Then, it holds

$$\begin{aligned} \Vert u\Vert ^p_{L^p(\varOmega )}\le \lambda _1(\varOmega )[u]^p_{W^{s,p}(\varOmega )}\,\,\text {for}\,\,\,u\in C_0^\infty (\varOmega )\end{aligned}$$

and we have the lower bound

$$\begin{aligned}\lambda _1(\varOmega )\ge \frac{1}{\mathcal {I}_{N,s,p(\varOmega )}},\end{aligned}$$

where the geometric quantity $\mathcal {I}_{N,s,p(\varOmega )}$ is defined by

$$\begin{aligned}\mathcal {I}_{N,s,p(\varOmega )}=\min \biggl \{\frac{\text {diam} (\varOmega \cup B)^{N+sp}}{|B|},\,\,B\subset \mathbb {R}^N\setminus \varOmega \,\,\text {is a ball}\biggr \}.\end{aligned}$$

We define the inner product of the operator $(-\varDelta )^s_p$ for $u,v\in W^{s,p}(\varOmega )$ as

$$\begin{aligned} \langle (-\varDelta )^s_pu,v\rangle =\int _{\varOmega }\int _{\varOmega }\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(v(x)-v(y))}{|x-y|^{N+sp}}dxdy.\nonumber \\ \end{aligned}$$

(2.3)

Definition 6

A function $u=u(x,t)\in W^{s,p}(\varOmega ;L^\infty (0,T))\cap L^2(\varOmega ;L^\infty (0,T))$ is called a weak solution of (1.1) if the following identity holds

$$\begin{aligned}\begin{aligned} \int _0^T\!\int \limits _\varOmega \mathcal {D}_{0|t}^\alpha u\varphi dxdt&\!+\!\int _0^T\int \limits _{\varOmega }\int \limits _{\varOmega }\frac{|u(x)\!-\!u(y)|^{p\!-\!2}(u(x)\!-\!u(y))}{|x-y|^{N+sp}}(\varphi (x)\!-\!\varphi (y))dxdydt\\ {}&=\gamma \int _0^T\int _\varOmega |u|^{m-1}u\varphi dxdt+\mu \int _0^T\int _\varOmega |u|^{q-2}u\varphi dxdt, \end{aligned}\end{aligned}$$

almost everywhere in $t\in [0,T]$, for any $\varphi =\varphi (x,t)\in W_0^{s,p}(\varOmega ;L^\infty (0,T))$, such that $\varphi \ge 0$ in $\varOmega$, $\varphi =0$ on $\partial \varOmega$.

2.3 Notations

We recall standard notations, which will be used in the sequel. If $\varOmega$ is a bounded and open set in $\mathbb {R}^N$ $(\varOmega \subseteq \mathbb {R}^N)$, we denote

$$\begin{aligned}\varOmega _T=\varOmega \times (0,T).\end{aligned}$$

We include the following function space

$$\begin{aligned} \varPi =\{u, \mathcal {D}_{0|t}^\alpha u\in W^{s,p}(\varOmega ;L^\infty (0,T))\cap L^2(\varOmega ;L^\infty (0,T))\}, \end{aligned}$$

(2.4)

with the norm

$$\begin{aligned}\begin{aligned} \Vert u\Vert ^2_\varPi&=\Vert u\Vert ^2_{W^{s,p}(\varOmega ;L^\infty (0,T))}+\Vert u\Vert ^2_{L^2(\varOmega ;L^\infty (0,T))}\\ {}&+\Vert \mathcal {D}_{0|t}^\alpha u\Vert ^2_{W^{s,p}(\varOmega ;L^\infty (0,T))}+\Vert \mathcal {D}_{0|t}^\alpha u\Vert ^2_{L^2(\varOmega ;L^\infty (0,T))}. \end{aligned}\end{aligned}$$

3 A comparison principle

In this section we study a comparison principle for the fractional parabolic equation. We begin by presenting a weak subsolution and a weak supersolution to the problem (1.1).

Definition 7

A real-valued function

$$\begin{aligned}u=u(x,t)\in \varPi , u(x,0)\le u_0(x), u(x,t)|_{ x\in \partial \varOmega }\le 0\end{aligned}$$

is called a weak subsolution of (1.1) if the inequality

$$\begin{aligned} \begin{aligned}&\int _0^T\int _{\varOmega }\frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+sp}}(\varphi (x,t)-\varphi (y,t))dxdydt\\ {}&\le \gamma \int _0^T\int _{\varOmega }|u|^{m-1}u\varphi dxdt+\mu \int _0^T\int _{\varOmega }|u|^{q-2}u\varphi dxdt\\ {}&-\int _0^T\int _{\varOmega }[\mathcal {D}_{0|t}^\alpha u]\varphi dxdt,\end{aligned}\end{aligned}$$

(3.1)

holds for any $\varphi \in W_0^{s,p}(\varOmega ;L^\infty (0,T))$, such that $\varphi \ge 0$ in $\varOmega$, $\varphi =0$ on $\partial \varOmega$.

Similarly, a real-valued function

$$\begin{aligned}v=v(x,t)\in \varPi , v(x,0)\ge v_0(x), v(x,t)|_{ x\in \partial \varOmega }\ge 0\end{aligned}$$

is called a weak supersolution of (1.1) if it satisfies the inequality

$$\begin{aligned} \begin{aligned}&\int _0^T\int _{\varOmega }\frac{|v(x,t)-v(y,t)|^{p-2}(v(x,t)-v(y,t))}{|x-y|^{N+sp}}(\varphi (x,t)-\varphi (y,t))dxdydt\\ {}&\ge \gamma \int _0^T\int _{\varOmega }|v|^{m-1}v\varphi dxdt+\mu \int _0^T\int _{\varOmega }|v|^{q-2}v\varphi dxdt\\ {}&-\int _0^T\int _{\varOmega }[\mathcal {D}_{0|t}^\alpha v]\varphi dxdt.\end{aligned}\end{aligned}$$

(3.2)

A function is a weak solution, if it is both a weak subsolution and a weak supersolution.

Theorem 1

Let $s\in (0,1), p\ge 2$ and let $m, q, \gamma , \mu$ satisfy one of the following conditions:

$$\begin{aligned}&m\ge 1, q\ge 2, \gamma \ge 0, \mu \ge 0;\\ {}&m>0, q\ge 1, \gamma \le 0, \mu \le 0;\\ {}&m\ge 1, q\ge 1, \gamma \ge 0, \mu \le 0;\\ {}&m>0, q\ge 2, \gamma \le 0, \mu \ge 0. \end{aligned}$$

Suppose that $u, v\in \varPi$ be real-valued weak subsolution and weak supersolution of (1.1), respectively, with $u_0(x)\le v_0(x)$ for $x\in \varOmega$. Then $u\le v$ a.e. in $\varOmega _T$.

Corollary 1

Assume that $p\ge 2$ and let $m, q, \gamma , \mu$ satisfy the conditions in Theorem 1. If $u_0(x)\ge 0$ for all $x\in \varOmega ,$ then $u(x,t)\ge 0,\,x\in \varOmega ,\,t\ge 0.$

The proof of Corollary 1 follows from Theorem 1. More precisely, if $u_0(x)\ge 0$, taking 0 as a subsolution, then we have $u(x,t)\ge 0$.

Proof of Theorem 1

We choose the test function $\varphi =(u-v)_+$, where $(u-v)_+$ is the positive part of a real quantity $(u-v)_+=\max \{u-v, 0\}$. Then it follows that $\varphi (x,0)=0,$ $\varphi (x,t)|_{\partial \varOmega }=0$. By subtracting (3.2) from (3.1), we obtain for $t\in (0,T]$

$$\begin{aligned} \begin{aligned} \int _0^t\int _{\varOmega }\mathcal {D}_{0|\tau }^\alpha [u-v]\varphi dxd\tau&+\int _0^t\int _{\varOmega }[(-\varDelta )^s_pu-(-\varDelta )^s_pv]\varphi dxd\tau \\&\le \underbrace{\gamma \int _0^t\int _{\varOmega }(|u|^{m-1}u-|v|^{m-1}v)\varphi dxd\tau }_{\mathcal {A}}\\&+\underbrace{\mu \int _0^t\int _{\varOmega }(|u|^{q-2}u-|v|^{q-2}v)\varphi dxd\tau }_{\mathcal {B}}.\end{aligned}\end{aligned}$$

(3.3)

According to (2.3), we can write the last term of the left-hand side inequality (3.3) in the form

$$\begin{aligned} \begin{aligned}&\int _0^t\int _{\varOmega }[(-\varDelta )^s_pu-(-\varDelta )^s_pv]\varphi dxd\tau \\&\quad =\int _0^t\int _{\varOmega }\int _{\varOmega }\frac{|u(x)-u(y)|^{p-2}(u(x)-u(y))(\varphi (x)-\varphi (y))}{|x-y|^{N+sp}}dxdyd\tau \\&\qquad -\int _0^t\int _{\varOmega }\int _{\varOmega }\frac{|v(x)-v(y)|^{p-2}(v(x)-v(y))(\varphi (x)-\varphi (y))}{|x-y|^{N+sp}}dxdyd\tau \\&\quad =\int _0^t\int _{\varOmega }\int _{\varOmega }\frac{\mathcal {M}(u,v)(\varphi (x)-\varphi (y))}{|x-y|^{N+sp}}dxdyd\tau ,\end{aligned}\end{aligned}$$

(3.4)

where

$$\begin{aligned} \mathcal {M}(u,v)=|u(x)-u(y)|^{p-2}(u(x)-u(y))-|v(x)-v(y)|^{p-2}(v(x)-v(y)).\nonumber \\ \end{aligned}$$

(3.5)

Hence, we can show that

$$\begin{aligned}\begin{aligned} \mathcal {M}(u,v)&(\varphi (x)-\varphi (y))\\ {}&=\left[ |u(x)-u(y)|^{p-2}(u(x)-u(y))-|v(x)-v(y)|^{p-2}(v(x)-v(y))\right] \\ {}&\times \left[ (u(x)-u(y))-(v(x)-v(y))\right] _+\end{aligned} \end{aligned}$$

is nonnegative for any $p\ge 2$, thanks to the inequality (see [27], P. 99)

$$\begin{aligned} \biggl (\frac{4}{p^2}\biggr )\biggl ||a|^{\frac{p-2}{2}}a-|b|^{\frac{p-2}{2}}b\biggr |^2\le \langle |a|^{p-2}a-|b|^{p-2}b,a-b\rangle ,\,\,\text {for}\,\,\,a,b\in \mathbb {R}^N,\nonumber \\ \end{aligned}$$

(3.6)

with $a:=u(x)-u(y),\, b:=v(x)-v(y)$ in (3.5).

Now, we will evaluate the right-side of (3.3).

Taking account the following inequality

$$\begin{aligned} ||u|^{m-1}u-|v|^{m-1}v|\le C(m)|u-v|||u|^{m-1}+|v|^{m-1}|\le L(m)|u-v|,\end{aligned}$$

(3.7)

where $L(m)= C(m)\max (\Vert u\Vert ^{m-1}_{L^\infty (\varOmega )},\Vert v\Vert ^{m-1}_{L^\infty (\varOmega )})$, we can verify that

$$\begin{aligned} \begin{aligned} \mathcal {A}&\le \gamma C(m)\int _0^t\int _{\varOmega }||u|^{m-1}+|v|^{m-1}||u-v|\varphi dxd\tau \\ {}&\le \gamma C(m)\max (\Vert u\Vert ^{m-1}_{C(\varOmega )},\Vert v\Vert ^{m-1}_{C(\varOmega )})\int _0^t\int _{\varOmega }|u-v|\varphi dxd\tau \\ {}&\le \gamma L(m)\int _0^t\int _{\varOmega }|u-v|\varphi dxd\tau ,\end{aligned}\end{aligned}$$

(3.8)

where we have used the well known inequality [28, Theorem 8.2] for any $u\in L^p(\varOmega )$ such that

$$\begin{aligned} \Vert u\Vert _{C(\varOmega )}\le \Vert u\Vert _{C^{0,\beta }(\varOmega )}\le \Vert u\Vert _{W^{s,p}(\varOmega )},\,\,\,\beta =(sp-N)/p, \end{aligned}$$

(3.9)

which gives the boundness of $\max (\Vert u\Vert ^{m-1}_{C(\varOmega )},\Vert v\Vert ^{m-1}_{C(\varOmega )})$.

In addition, from the Lipchitsz condition it follows that

$$\begin{aligned} ||u|^{q-2}u-|v|^{q-2}v|\le L(q)|u-v|,\,\,\,q\ge 1,\end{aligned}$$

(3.10)

where $L(q)= C(q)\max (\Vert u\Vert ^{q-2}_{L^\infty (\varOmega )},\Vert v\Vert ^{q-2}_{L^\infty (\varOmega )})$. Hence, from (3.9), it follows that

$$\begin{aligned} \begin{aligned} \mathcal {B}&\le \mu C(q)\int _0^t\int _{\varOmega }||u|^{q-2}+|v|^{q-2}||u-v|\varphi dxd\tau \\ {}&\le \mu C(q)\max (\Vert u\Vert ^{q-2}_{C(\varOmega )},\Vert v\Vert ^{q-2}_{C(\varOmega )}) \int _0^t\int _{\varOmega }|u-v|\varphi dxd\tau \\ {}&\le \mu L(q)\int _0^t\int _{\varOmega }|u-v|\varphi dxd\tau .\end{aligned}\end{aligned}$$

(3.11)

Combining (3.4), (3.8) and (3.11), we can rewrite the inequality (3.3) as

$$\begin{aligned} \begin{aligned}&\int _0^t\int _{\varOmega }(\mathcal {D}_{0|\tau }^\alpha [u-v])(u-v)_+ dxd\tau \\ {}&\le \left( \gamma L(m)+\mu L(q)\right) \int _0^t\int _{\varOmega }|u-v|(u-v)_+ dxd\tau .\end{aligned}\end{aligned}$$

(3.12)

Using Lemma 5, the inequality (3.12) can be rewritten in the following form

$$\begin{aligned} \begin{aligned} \frac{1}{2}\int _0^t\int _{\varOmega }\mathcal {D}_{0|\tau }^\alpha (u-v)_+^2dxd\tau&\le \left( \gamma L(m)+\mu L(q)\right) \int _0^t\int _{\varOmega }(u-v)_+^2dxd\tau .\qquad \end{aligned}\end{aligned}$$

(3.13)

At this stage, we have to consider three cases depending on $\gamma , \mu$:

$\bullet$ The case $\gamma \ge 0,\mu \ge 0.$ Applying the left Caputo fractional differentiation operator $\mathcal {D}_{0|t}^{1-\alpha }$ to both sides of (3.13) and using Property 1, we obtain

$$\begin{aligned} \frac{1}{2}\int _{\varOmega }(u-v)_+^2dx\le \left( \gamma L(m)+\mu L(q)\right) \int _{\varOmega }\int _0^t(t-\tau )^{\alpha -1}(u-v)_+^2d\tau dx.\end{aligned}$$

(3.14)

Then, from the weakly singular Gronwall’s inequality (see [19], Lemma 7.1.1 and [18], Lemma 6, p. 33)

$$\begin{aligned}\int _{\varOmega }(u-v)_+^2dx=0 \iff (u-v)_+=0,\,\,\,x\in \varOmega .\end{aligned}$$

Finally, it follows that $u\le v$ almost everywhere for $(x,t)\in \varOmega _T$.

$\bullet$ The case $\gamma \le 0,\mu \le 0.$ According to the inequality (3.13), the right-hand side integral is positive and the coefficients $\gamma ,\mu$ are non-positive, we deduce that

$$\begin{aligned}\begin{aligned} \frac{1}{2}\int _0^t\int _{\varOmega }\mathcal {D}_{0|\tau }^\alpha (u-v)_+^2 dxd\tau&\le \left( \gamma L(m)+\mu L(q)\right) \int _0^t\int _{\varOmega }(u-v)_+^2dxd\tau \\ {}&\le 0.\end{aligned}\end{aligned}$$

Therefore, repeating the similar procedure as above we obtain

$$\begin{aligned}\begin{aligned} \int _{\varOmega }(u-v)_+^2 dx=0.\end{aligned}\end{aligned}$$

Consequently, we have $u\le v$ almost everywhere for $(x,t)\in \varOmega _T$.

$\bullet$ The case $\gamma \ge 0,\mu \le 0$ or $\gamma \le 0,\mu \ge 0$. Using the inequality (3.13), it follows that

$$\begin{aligned}\begin{aligned} \frac{1}{2}\int _{\varOmega }(u-v)_+^2dx\le \gamma L(m)\int _{\varOmega }\int _0^t(t-\tau )^{\alpha -1}(u-v)_+^2d\tau dx\end{aligned}\end{aligned}$$

or

$$\begin{aligned}\begin{aligned} \frac{1}{2}\int _{\varOmega }(u-v)_+^2 dx\le \mu L(q) \int _{\varOmega }\int _0^t(t-\tau )^{\alpha -1}(u-v)_+^2d\tau dx,\end{aligned}\end{aligned}$$

respectively. By the weakly singular Gronwall’s inequality, we arrive at $u\le v$ almost everywhere for $(x,t)\in \varOmega _T$. $\square$

4 Local well-posedness

4.1 Existence of a local weak solution

In this subsection, we will prove that problem (1.1) has the local weak solution by Galerkin method.

Theorem 2

Let $u_{0} \in W^{s,p}_0(\varOmega ), u_0\ge 0,\, sp<N$ and let either $1<m<q-1<p-1$ or $1<q-1<m <p-1.$ Then there exists $T>0$ such that the problem (1.1) has a local real-valued weak solution $u\in \varPi$, where $\varPi$ is defined in (2.4).

Proof

$\bullet$ The case $1<m<q-1<p-1$. The space $W^{s,p}_0(\varOmega )$ is separable. Then there exists a countable linear set $\{\omega _j\}_{j\in N}$ that is everywhere dense in $W^{s,p}_0(\varOmega )$.

Let us consider the Galerkin approximations

$$\begin{aligned} u_n(x,t)=\sum _{j=1}^n v_{nj}(t)\omega _j(x),\end{aligned}$$

(4.1)

where the unknown $v_{nj}\in C^1([0,T_n])$ functions satisfy the following system of ordinary fractional differential equations:

$$\begin{aligned} \begin{aligned}&\int _\varOmega \mathcal {D}_{0|t}^{\alpha }u_{n}\omega _k dx+P(u_{n}, \omega _k) \\&\quad =\gamma \int _\varOmega |u_n|^{m-1}u_n\omega _kdx+\mu \int _\varOmega |u_n|^{q-2}u_n\omega _kdx,\,\,k=1,2,...,n, \end{aligned}\end{aligned}$$

(4.2)

supplemented by the initial condition

$$\begin{aligned} \begin{aligned} u_n(x,0)=\sum _{j=1}^n v_{nj}(0)\omega _j\xrightarrow []{n\rightarrow \infty }u_0\,\,\text {in}\,\,W_0^{s,p}(\varOmega ),\end{aligned} \end{aligned}$$

(4.3)

where

$$\begin{aligned}\begin{aligned} P(u_{n},\omega _k)&\!=\!\int _{\varOmega }\int _{\varOmega }\biggl |u_{n}(x,t)\!-\!u_{n}(y,t)\biggr |^{p\!-\!2}\!(u_{n}(x,t)\!-\!u_{n}(y,t))\frac{\omega _{k}(x)\!-\!\omega _{k}(y)}{|x-y|^{N+sp}}dxdy.\end{aligned}\end{aligned}$$

First of all, we need to prove that the system of Galerkin equations (4.2) has a solution $v_{nj}\in C^1([0, T_n]),\,j=\overline{1,n}$ for some $T_n > 0$, which depends on $n\in N$. Therefore, we note that the system of equations (4.2) can be represented in the following form

$$\begin{aligned} \begin{aligned} \sum _{j=1}^n a_{jk}\mathcal {D}_{0|t}^{\alpha }v_{nj}(t)+F_{1k}(v_n)=F_{2k}(v_n)+F_{3k}(v_n),\,\,\,\end{aligned} \end{aligned}$$

(4.4)

where $a_{jk}$ is an invertible matrix for each $n\in N$ and the functions $F_{ik}(v_n),\,i=1,2,3$ are defined by

$$\begin{aligned}\begin{aligned} F_{1k}(v_n)&\!=\!\int _{\varOmega }\int _{\varOmega }\sum _{j=1}^n\biggl |v_{nj}(t)\omega _{j}(x)\!-\!v_{nj}(t)\omega _{j}(y)\biggr |^{p-2}(v_{nj}(t)\omega _{j}(x)-v_{nj}(t)\omega _{j}(y))\\ {}&\times (\omega _{k}(x)-\omega _{k}(y)) \frac{1}{|x-y|^{N+sp}}dxdy\,\,\,\end{aligned} \end{aligned}$$

and

$$\begin{aligned}\begin{aligned} F_{2k}(v_n)=\int _\varOmega \sum _{j=1}^n|v_{nj}(t)\omega _j(x)|^{m-1}v_{nj}(t)\omega _j(x)\omega _k(x)dx,\,\,\,\end{aligned} \\\begin{aligned} F_{3k}(v_n)=\int _\varOmega \sum _{j=1}^n|v_{nj}(t)\omega _j(x)|^{q-2}v_{nj}(t)\omega _j(x)\omega _k(x)dx.\,\,\,\end{aligned} \end{aligned}$$

Next, we will prove the functions $F_{ik}(v_n),\,i=1,2,3$ are locally Lipschitz function. Indeed, we have

$$\begin{aligned}\begin{aligned}&|F_{1k}(v^1_n)-F_{1k}(v^2_n)|\\ {}&=\int _{\varOmega }\int _{\varOmega }\sum _{j=1}^n\biggl (\biggl |v^1_{nj}(t)\omega _{j}(x)-v^1_{nj}(t)\omega _{j}(y)\biggr |^{p-2}(v^1_{nj}(t)\omega _{j}(x)-v^1_{nj}(t)\omega _{j}(y))\\ {}&\!-\!\biggl |v^2_{nj}(t)\omega _{j}(x)\!-\!v^2_{nj}(t)\omega _{j}(y)\biggr |^{p\!-\!2}\!(v^2_{nj}(t)\omega _{j}(x)\!-\!v^2_{nj}(t)\omega _{j}(y))\biggr )\!\frac{\omega _{k}(x)\!-\!\omega _{k}(y)}{|x\!-\!y|^{N+sp}}\!dxdy.\end{aligned} \end{aligned}$$

Using the following inequality, for $p\ge 1,$

$$\begin{aligned}\begin{aligned} ||\varphi |^{p-2}\varphi -|\psi |^{p-2}\psi |&\le C(p)||\varphi |^{p-2}+|\psi |^{p-2}||\varphi -\psi |\\ {}&\le C(p)\max \{|\varphi |^{p-2};|\psi |^{p-2}\}|\varphi -\psi | \end{aligned} \end{aligned}$$

and the generalized Hölder inequality with parameters

$$\begin{aligned} \frac{1}{r_1}+\frac{1}{r_2}+\frac{1}{r_3}=1,\,\,\,r_1=\frac{p}{p-2},\,\,\,r_2=p,\,\,\,r_3=p,\end{aligned}$$

(4.5)

in the last equality, we obtain

$$\begin{aligned}\begin{aligned}&|F_{1k}(v^1_n)-F_{1k}(v^2_n)|\\ {}&\le \int _{\varOmega }\int _{\varOmega }\frac{\max \biggl \{|u^1_{n}(x,t)-u^1_{n}(y,t)|^{p-2};|u^2_{n}(x,t)-u^2_{n}(y,t)|^{p-2}\biggr \}}{|x-y|^{\frac{(N+sp)(p-2)}{p}}} \\ {}&\times \sum _{j=1}^n\frac{|(v^1_{nj}(t)\omega _{j}(x)-v^1_{nj}(t)\omega _{j}(y))-(v^2_{nj}(t)\omega _{j}(x)-v^2_{nj}(t)\omega _{j}(y))|}{|x-y|^{\frac{N+sp}{p}}} \\ {}&\times \frac{\omega _{k}(x)-\omega _{k}(y)}{|x-y|^{{\frac{N+sp}{p}}}}dxdy \\ {}&\le C(p)\varPhi ^{p-2}[\omega _k]_{W^{s,p}(\varOmega )}[u^1_n-u_n^2]_{W^{s,p}(\varOmega )}, \end{aligned} \end{aligned}$$

where $\varPhi ^{p-2}=\max \{[u_n^1]_{W^{s,p}(\varOmega )}; [u_n^2]_{W^{s,p}(\varOmega )}\}$ and $[\,\cdot \,]_{W^{s,p}(\varOmega )}$ is the Gagliardo semi-norm. Consequently,

$$\begin{aligned}\begin{aligned}&[u^1_n-u_n^2]_{W^{s,p}(\varOmega )} \\ {}&=\int _{\varOmega }\int _{\varOmega }\sum _{j=1}^n\frac{|v^1_{nj}(t)\omega _{j}(x)-v^1_{nj}(t)\omega _{j}(y))-(v^2_{nj}(t)\omega _{j}(x)-v^2_{nj}(t)\omega _{j}(y)|^p}{|x-y|^{N+sp}}dxdy \\ {}&=\int _{\varOmega }\int _{\varOmega }\sum _{j=1}^n\frac{|v^1_{nj}(t)-v^2_{nj}(t)|^p|\omega _{j}(x)-\omega _{j}(y)|^p}{|x-y|^{N+sp}}dxdy \\ {}&\le [w_j]_{W^{s,p}(\varOmega )}|v^1_{n}-v^2_{n}|^p. \end{aligned} \end{aligned}$$

At this stage using

$$\begin{aligned}|a-b|^p\le |a-b||a-b|^{p-1}\le 2^{p-2}|a+b|^{p-1}|a-b|,\,\,\,p>2, a,b\in \mathbb {R},\end{aligned}$$

in the last term of the previous inequality, and recalling $v^1_{nj}, v^2_{nj}\in C^1([0, T_n])$ we arrive at

$$\begin{aligned}\begin{aligned}&|F_{1k}(v^1_n)-F_{1k}(v^2_n)|\le C(p)\varPhi ^{p-2}[\omega _k]_{W^{s,p}(\varOmega )}\max \{|v^1_n|^{p-1};|v_n^2|^{p-1}\}|v^1_n-v_n^2|. \end{aligned} \end{aligned}$$

Accordingly, using the inequalities (3.7) and (3.9) to $F_{2k}(v_n)$, for $k,j=\overline{1,n}$, we deduce that

$$\begin{aligned}\begin{aligned}&|F_{2k}(v^1_n)-F_{2k}(v^2_n)|\\ {}&\le \int _\varOmega \sum _{j=1}^n||v^1_{nj}(t)\omega _j(x)|^{m\!-\!1}v^1_{nj}(t)\omega _j(x)\!-\!|v^2_{nj}(t)\omega _j(x)|^{m\!-\!1}v^2_{nj}(t)\omega _j(x)||\omega _k(x)|dx \\ {}&\le \max \{\Vert v^1_nw_j\Vert _{C(\varOmega )}^{m-1};\Vert v_n^2w_j\Vert _{C(\varOmega )}^{m-1}\}|v^1_n-v^2_n|\int _\varOmega |\omega _j(x)||\omega _k(x)|dx \\ {}&\le \max \{\Vert v^1_nw_j\Vert _{C(\varOmega )}^{m-1};\Vert v_n^2w_j\Vert _{C(\varOmega )}^{m-1}\}\Vert w_j\Vert _{L^2(\varOmega )}\Vert w_k\Vert _{L^2(\varOmega )}|v^1_n-v^2_n|.\,\,\,\end{aligned} \end{aligned}$$

Similarly, from (3.9) and (3.10) we obtain an estimate for $F_{3k}(v_n)$, for $k,j=\overline{1,n}$, in the following form

$$\begin{aligned}\begin{aligned}&|F_{3k}(v^1_n)-F_{3k}(v^2_n)|\\ {}&\le \int _\varOmega \sum _{j=1}^n||v^1_{nj}(t)\omega _j(x)|^{q-2}v^1_{nj}(t)\omega _j(x)\!-\!|v^2_{nj}(t)\omega _j(x)|^{q-2}v^2_{nj}(t)\omega _j(x)||\omega _k(x)|dx \\ {}&\le \max \{\Vert v^1_nw_j\Vert _{C(\varOmega )}^{q-2};\Vert v_n^2w_j\Vert _{C(\varOmega )}^{q-2}\}|v^1_n-v^2_n|\int _\varOmega |\omega _j(x)||\omega _k(x)|dx \\ {}&\le \max \{\Vert v^1_nw_j\Vert _{C(\varOmega )}^{q-2};\Vert v_n^2w_j\Vert _{C(\varOmega )}^{q-2}\}\Vert w_j\Vert _{L^2(\varOmega )}\Vert w_k\Vert _{L^2(\varOmega )}|v^1_n-v^2_n|.\,\,\,\end{aligned} \end{aligned}$$

From Lemma 1, the space $W^{s,p}(\varOmega )$ is continuously embedded in $L^2(\varOmega )$. Indeed, the right-hand side of $F_{ik}(v_n),\,\,i=1,2,3,\,k=\overline{1,n}$ is continuous with respect to $t\in [0, T_n]$ and locally Lipschitz function with respect to $v_n(t)$.

Therefore, due to [21, Theorem 3.25] the Cauchy problem for the system of equations (4.4) has a unique solution $v_{nj}\in C^1([0, T_n]),\,j=\overline{1,n}$ for some $T_n > 0$, which depends on $n\in N$.

Multiplying the expression (4.2) by $v_{nk}(t)$ and performing the summation over $k=1,...,n,$ it follows that

$$\begin{aligned}\begin{aligned}&\int _\varOmega u_{n}\mathcal {D}_{0|t}^{\alpha }u_{n}dx+[u_n]^p_{W^{s,p}(\varOmega )}=\gamma \int _\varOmega |u_n|^{m+1}dx+\mu \int _\varOmega |u_n|^{q}dx.\end{aligned}\end{aligned}$$

Applying the fractional Poincaré inequality from Lemma 7 and the inequality in Lemma 5 to the previous identity, we get

$$\begin{aligned} \begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx&+\frac{1}{\lambda _1(\varOmega )}\int _\varOmega |u_n|^{p}dx \\ {}&\le \gamma \int _\varOmega |u_n|^{m+1}dx+\mu \int _\varOmega |u_n|^{q}dx.\end{aligned}\end{aligned}$$

(4.6)

At this stage we have to consider different cases of coefficients $\gamma$ and $\mu$.

$\bullet$ The case $\gamma ,\,\mu >0$. Thanks to the inequality (see [30], P. 417),

$$\begin{aligned} z^{b+c-1}\le \varepsilon z^c+C(a,b)\varepsilon ^{-\frac{a-b}{b-1}}z^{a+c-1},\,\,a>b,c>1,\,\,\,\text {and}\,\,\, z\ge 0,\,\,\varepsilon >0,\qquad \end{aligned}$$

(4.7)

for $a=q-1,\,b=m,$ and $c=2$ in (4.6) we obtain

$$\begin{aligned} \gamma \int _\varOmega |u_{n}|^{m+1}dx\le \gamma \varepsilon \int _\varOmega |u_{n}|^2dx+\gamma C(q,m)\varepsilon ^{-\frac{q-1-m}{m-1}}\int _\varOmega |u_{n}|^qdx.\end{aligned}$$

(4.8)

Consequently, it follows that

$$\begin{aligned} \begin{aligned} \mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx&\le \gamma \varepsilon \int _\varOmega |u_{n}|^2dx-\frac{1}{\lambda _1(\varOmega )}\int _\varOmega |u_n|^{p}dx\\ {}&+\biggl (\gamma C(q,m)\varepsilon ^{-\frac{q-1-m}{m-1}}+\mu \biggr )\int _\varOmega |u_n|^{q}dx.\end{aligned}\end{aligned}$$

(4.9)

Due to the inequality (4.7) for $a=p-1,\,b=q-1$ and $c=2$, it holds

$$\begin{aligned}\int _\varOmega |u_{n}|^qdx\le {\tilde{\varepsilon }} \int _\varOmega |u_{n}|^2dx+C(p,q){\tilde{\varepsilon }}^{-\frac{p-q}{q-1}}\int _\varOmega |u_{n}|^pdx. \end{aligned}$$

Therefore, using the last inequality in (4.9) we get

$$\begin{aligned}\begin{aligned} \mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx&\le \biggl ({\tilde{\varepsilon }}\gamma C(q,m)\varepsilon ^{-\frac{q-1-m}{m-1}}+{\tilde{\varepsilon }}\mu +\gamma \varepsilon \biggr ) \int _\varOmega |u_{n}|^2dx\\ {}&+\biggl [\biggl (\gamma C(q,m)\varepsilon ^{-\frac{q-1-m}{m-1}}+\mu \biggr )C(p,q){\tilde{\varepsilon }}^{-\frac{p-q}{q-1}}-\frac{1}{\lambda _1(\varOmega )}\biggr ]\int _\varOmega |u_n|^{p}dx.\end{aligned}\end{aligned}$$

Finally, choosing the constants $\varepsilon , {\tilde{\varepsilon }}>0$ such that

$$\begin{aligned}\biggl (\gamma C(q,m)\varepsilon ^{-\frac{q-1-m}{m-1}}+\mu \biggr )C(p,q){\tilde{\varepsilon }}^{-\frac{p-q}{q-1}}-\frac{1}{\lambda _1(\varOmega )}\le 0,\end{aligned}$$

then we get the following result

$$\begin{aligned} \begin{aligned} \mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx\le C(\varepsilon ,{\tilde{\varepsilon }}) \int _\varOmega |u_{n}|^2dx,\end{aligned}\end{aligned}$$

(4.10)

where

$$\begin{aligned} C(\varepsilon ,{\tilde{\varepsilon }})={\tilde{\varepsilon }}\gamma C(q,m)\varepsilon ^{-\frac{q-1-m}{m-1}}+{\tilde{\varepsilon }}\mu +\gamma \varepsilon .\end{aligned}$$

Define $\displaystyle \varPhi (t):=\int _\varOmega |u_{n}|^2dx$, then applying the left Riemann-Liouville fractional integral operator $I_{0|t}^{\alpha }$ to both sides of (4.10) and using Property 1, we get

$$\begin{aligned}\begin{aligned} \varPhi (t)\le \varPhi (0)+C(\varepsilon ,{\tilde{\varepsilon }})\int _0^t(t-s)^{\alpha -1}\varPhi (s)ds .\end{aligned}\end{aligned}$$

Furthermore, according to Gronwall-type inequality for fractional integral equations (see [12], Lemma 4.3) we obtain

$$\begin{aligned}\varPhi (t)\le \varPhi (0)E_{\alpha ,1}(C(\varepsilon ,{\tilde{\varepsilon }}) t^\alpha )\,\,\,\text {for all}\,\,\,t\in [0,T],\end{aligned}$$

where $E_{\alpha ,1 }(z)$ is the Mittag-Leffler function, defined by

$$\begin{aligned}E_{\alpha ,1 }(z)=\sum _{k=0}^{\infty }\frac{z^k}{\varGamma (\alpha k+1)},\,z\ge 0.\end{aligned}$$

Finally, in view of Corollary 1 for real-valued u, we conclude that there exists finite $T_0>0$,

$$\begin{aligned} \Vert u_{n}(\cdot ,t)\Vert ^2_{L^2(\varOmega )}\le \Vert u_{n}(\cdot ,0)\Vert ^2_{L^2(\varOmega )}E_{\alpha ,1}(C(\gamma ,\varepsilon ) t^\alpha )=A(T),\end{aligned}$$

(4.11)

for all $t\in [0,T]$, $T<T_0$, where A(T) is a constant independent of n.

$\bullet$ The case $\gamma >0$ and $\mu \le 0$. Then from (4.6) we obtain

$$\begin{aligned}\begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx&+\frac{1}{\lambda _1(\varOmega )}\int _\varOmega |u_n|^{p}dx\le \gamma \int _\varOmega |u_n|^{m+1}dx.\end{aligned}\end{aligned}$$

Setting $a=p-1,\,b=m,$ and $c=2$ in (4.8) we can rewrite the last estimate as

$$\begin{aligned}\begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx&\le \gamma \varepsilon \int _\varOmega |u_{n}|^2dx\!+\!\biggl (\gamma C(p,m)\varepsilon ^{-\frac{p-1-m}{m-1}}\!-\!\frac{1}{\lambda _1(\varOmega )}\biggr )\int _\varOmega |u_{n}|^pdx.\end{aligned}\end{aligned}$$

By choosing the constants $\varepsilon , {\tilde{\varepsilon }}>0$ which satisfy

$$\begin{aligned}\gamma C(p,m)\varepsilon ^{-\frac{p-1-m}{m-1}}-\frac{1}{\lambda _1(\varOmega )}\le 0,\end{aligned}$$

then we get

$$\begin{aligned}\begin{aligned} \mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx\le \gamma \varepsilon \int _\varOmega |u_{n}|^2dx.\end{aligned}\end{aligned}$$

The conclusion can be derived as in the previous case.

$\bullet$ The case $\gamma \le 0$ and $\mu >0$. Accordingly from (4.6) we have

$$\begin{aligned}\begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx&+\frac{1}{\lambda _1(\varOmega )}\int _\varOmega |u_n|^{p}dx\le \mu \int _\varOmega |u_n|^{q}dx.\end{aligned}\end{aligned}$$

Next, choosing $a=p-1,\,b=q-1,$ and $c=2$ in (4.8) it follows

$$\begin{aligned}\begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx&\le \mu \varepsilon \int _\varOmega |u_{n}|^2dx+\biggl (\mu C(p,q)\varepsilon ^{-\frac{p-q}{q-2}}-\frac{1}{\lambda _1(\varOmega )}\biggr )\int _\varOmega |u_{n}|^pdx.\end{aligned}\end{aligned}$$

Now, taking $\varepsilon , {\tilde{\varepsilon }}>0$, which satisfy

$$\begin{aligned}\mu C(p,q)\varepsilon ^{-\frac{p-q}{q-2}}-\frac{1}{\lambda _1(\varOmega )}\le 0,\end{aligned}$$

we obtain the estimate

$$\begin{aligned}\begin{aligned} \mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx\le \mu \varepsilon \int _\varOmega |u_{n}|^2dx.\end{aligned}\end{aligned}$$

Similarly, the conclusion can be derived as in the previous case.

$\bullet$ The case $\gamma ,\mu \le 0$. Take into consideration the inequality (4.6) it yields

$$\begin{aligned}\begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx&+\frac{1}{\lambda _1(\varOmega )}\int _\varOmega |u_n|^{p}dx\le 0.\end{aligned}\end{aligned}$$

Using the fact that $\lambda _1$ is nonnegative we obtain

$$\begin{aligned}\begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }\int _\varOmega |u_{n}|^2dx\le 0.\end{aligned}\end{aligned}$$

Hence, applying the left Riemann-Liouville integral $I_{0|t}^{\alpha }$ to the last inequality and using Property 1, we deduce that

$$\begin{aligned}\begin{aligned}\int _\varOmega |u_{n}(x,t)|^2dx\le \int _\varOmega |u_{n}(x,0)|^2dx.\end{aligned}\end{aligned}$$

Finally, it follows that

$$\begin{aligned}\Vert u_n(\cdot ,t)\Vert _{L^2(\varOmega )}\le \Vert u_{n}(x,0)\Vert _{L^2(\varOmega )},\,\,\,\text {for all}\,\,\,t\ge 0.\end{aligned}$$

Next, multiplying the expression (4.2) by $\mathcal {D}_{0|t}^{\alpha }v_{nk}(t)$ and summing over $k=\overline{1,n}$, we obtain

$$\begin{aligned} \begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}\Vert ^2_{L^2(\varOmega )}&+ P(u_{n},\mathcal {D}_{0|t}^{\alpha }u_{n}(t)) \\ {}&=\gamma \int _\varOmega |u_n|^{m-1}u_n\mathcal {D}_{0|t}^{\alpha }u_{n}dx+\mu \int _\varOmega |u_n|^{q-2}u_n\mathcal {D}_{0|t}^{\alpha }u_{n}dx, \end{aligned}\end{aligned}$$

(4.12)

with

$$\begin{aligned} P(u_{n},\mathcal {D}_{0|t}^{\alpha }u_{n}(t))&=\int _{\varOmega }\int _{\varOmega }\frac{|u_{n}(x,t)-u_{n}(y,t)|^{p-2}}{|x-y|^{N+sp}}(u_{n}(x,t)-u_{n}(y,t))\nonumber \\&\times \mathcal {D}_{0|t}^{\alpha }[u_{n}(x,t)-u_{n}(y,t)]dxdy. \end{aligned}$$

(4.13)

Due to Lemma 5 it follows that

$$\begin{aligned}(u_{n}(x,t)-u_{n}(y,t))\mathcal {D}_{0|t}^{\alpha }[u_{n}(x,t)-u_{n}(y,t)]\ge \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }[u_{n}(x,t)-u_{n}(y,t)]^2 .\end{aligned}$$

Moreover the identity (4.13) becomes

$$\begin{aligned} \begin{aligned} P(u_{n},\mathcal {D}_{0|t}^{\alpha }u_{n}(t))&\ge \frac{1}{2}\int _{\varOmega }\int _{\varOmega }\frac{|u_{n}(x,t)-u_{n}(y,t)|^{p-2}}{|x-y|^{N+sp}}\\ {}&\times \mathcal {D}_{0|t}^{\alpha }[u_{n}(x,t)-u_{n}(y,t)]^2dxdy. \end{aligned}\end{aligned}$$

(4.14)

At this stage, we consider the function

$$\begin{aligned}H(\omega )(t)=\frac{2}{p}|\omega (t)|^\frac{p}{2},\,\,p\ge 2,\end{aligned}$$

which is convex. By differentiating respect to $\omega$ we have $H'(\omega )(t)=|\omega (t)|^\frac{p-2}{2}$. From Lemma 4 for the function $H(\omega )(t)$ we obtain the following inequality

$$\begin{aligned}|\omega (t)|^\frac{p-2}{2}\mathcal {D}^\alpha _{0|t}\omega (t)\ge \frac{2}{p}\mathcal {D}^\alpha _{0|t}|\omega |^\frac{p}{2}(t).\end{aligned}$$

Denote $\omega (t)=|u_{n}(x,t)-u_{n}(y,t)|^2$. Then, we obtain

$$\begin{aligned}\begin{aligned} |u_{n}(x,t)-u_{n}(y,t)|^{p-2}\mathcal {D}^\alpha _{0|t}|u_{n}(x,t)-u_{n}(y,t)|^2\ge \frac{1}{p} \mathcal {D}^\alpha _{0|t}|u_{n}(x)-u_{n}(y)|^{p}. \end{aligned}\end{aligned}$$

Therefore, using (4.14) and the last inequality we get

$$\begin{aligned}\begin{aligned} \left| P(u_{n},\mathcal {D}_{0|t}^{\alpha }u_{n}(t))\right| \ge&\frac{1}{p}\int _{\varOmega }\int _{\varOmega }\frac{1}{|x-y|^{N+sp}} \mathcal {D}_{0|t}^{\alpha }|u_{n}(x,t)-u_{n}(y,t)|^{p}dxdy. \end{aligned}\end{aligned}$$

Since the operator $\mathcal {D}_{0|t}^{\alpha }$ is with respect to the variable t it follows that

$$\begin{aligned}\begin{aligned} \left| P(u_{n},\mathcal {D}_{0|t}^{\alpha }u_{n}(t))\right|&\ge \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }\int _{\varOmega }\int _{\varOmega }\frac{|u_{n}(x,t)-u_{n}(y,t)|^{p}}{|x-y|^{N+sp}} dxdy \\ {}&=\frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}. \end{aligned}\end{aligned}$$

Finally, the identity (4.12) can be rewritten as

$$\begin{aligned} \begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le \gamma \int _\varOmega |u_n|^{m-1}u_n\mathcal {D}_{0|t}^{\alpha }u_{n}dx+\mu \int _\varOmega |u_n|^{q-2}u_n\mathcal {D}_{0|t}^{\alpha }u_{n}dx. \end{aligned}\end{aligned}$$

(4.15)

At this stage, we should study the different cases of the coefficients $\gamma$ and $\mu$.

$\bullet$ The case $\gamma ,\mu >0$. Using the Hölder and $\varepsilon$-Young inequalities

$$\begin{aligned}XY\le \frac{\varepsilon }{p} X^p+C(\varepsilon )Y^{p'},\,\, \frac{1}{p}+\frac{1}{p'}=1,\,\, X,Y\ge 0,\end{aligned}$$

where $\displaystyle C(\varepsilon )=\frac{1}{p'\varepsilon ^{p'-1}}$ for the right hand side of (4.15), respectively, we get

$$\begin{aligned} \begin{aligned} \gamma \int _{\varOmega }|u_{n}|^{m-1}&u_{n}\mathcal {D}_{0|t}^{\alpha }u_{n}dx \le \gamma \left( \int _{\varOmega }|u_{n}|^{2m}dx\right) ^\frac{1}{2}\left( \int _{\varOmega }\left| \mathcal {D}_{0|t}^{\alpha }u_{n}\right| ^2dx\right) ^\frac{1}{2} \\ {}&\le \gamma \left\| u_{n} (\cdot , t) \right\| ^m _{L^{2m}(\varOmega ) } \left\| \mathcal {D}_{0|t}^{\alpha } u_{n}(\cdot , t) \right\| _{L^2(\varOmega ) } \\ {}&\le \frac{\varepsilon }{2}\gamma ^2\left\| u_{n}(\cdot , t)\right\| _{L^{2m}(\varOmega )}^{2m}+\frac{1}{2\varepsilon } \left\| \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{2} \end{aligned}\end{aligned}$$

(4.16)

and

$$\begin{aligned} \begin{aligned} \mu \int _\varOmega |u_n|^{q-2}&u_n\mathcal {D}_{0|t}^{\alpha }u_{n}dx\le \mu \left( \int _{\varOmega }|u_{n} |^{2(q-1)} dx \right) ^{\frac{1}{2} } \left( \int _{\varOmega }|\mathcal {D}_{0|t}^{\alpha } u_{n} |^{2} dx \right) ^{\frac{1}{2} } \\ {}&\le \mu \left\| u_{n} (\cdot , t) \right\| ^{q-1}_{L^{2(q-1)}(\varOmega ) } \left\| \mathcal {D}_{0|t}^{\alpha } u_{n}(\cdot , t) \right\| _{L^2(\varOmega ) } \\ {}&\le \frac{\varepsilon _1}{2} \mu ^2\left\| u_{n} (\cdot , t) \right\| ^{2(q-1)} _{L^{2(q-1)}(\varOmega ) }+\frac{1}{2\varepsilon _1} \left\| \mathcal {D}_{0|t}^{\alpha } u_{n}(\cdot , t) \right\| _{L^2(\varOmega ) }^{2}. \end{aligned}\end{aligned}$$

(4.17)

From Lemma 2 we obtain

$$\begin{aligned} \begin{aligned} \frac{\varepsilon }{2}\gamma ^2&\left\| u_{n}(\cdot , t)\right\| _{L^{2m}(\varOmega )}^{2m}\le \frac{\varepsilon }{2}\gamma ^2 C[u_{n}(\cdot , t)]^{2ma}_{W^{s,p}(\varOmega )}\left\| u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{2m(1-a)} \\ {}&\le \mathcal {C}(\gamma ,\varepsilon ,{\tilde{\varepsilon }}, C)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+ \mathcal {C}({\tilde{\varepsilon }})\left\| u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{\frac{2mp(1-a)}{p-2ma}} \end{aligned} \end{aligned}$$

(4.18)

and

$$\begin{aligned} \begin{aligned}&\frac{\varepsilon _1}{2}\mu ^2\left\| u_{n} (\cdot , t) \right\| ^{2(q-1)} _{L^{2(q-1)}(\varOmega ) }\\ {}&\le \frac{\varepsilon _1}{2} \mu ^2C_1[u_{n}(\cdot , t)]^{2(q-1)a}_{W^{s,p}(\varOmega )}\left\| u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{2(q-1)(1-a)} \\ {}&\le \mathcal {C}(\mu ,\varepsilon _1,{\tilde{\varepsilon }}_1, C_1)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+ \mathcal {C}({\tilde{\varepsilon }}_1)\left\| u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{\frac{2p(q-1)(1-a)}{p-2(q-1)a}}. \end{aligned} \end{aligned}$$

(4.19)

Hence, from the last inequalities (4.15) we obtain

$$\begin{aligned}\begin{aligned} \frac{1}{2}\Vert \mathcal {D}_{0|t}^{\alpha }&u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )}+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le \mathcal {C}(\gamma ,\varepsilon ,{\tilde{\varepsilon }}, C)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+ \mathcal {C}({\tilde{\varepsilon }})\left\| u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{\frac{2mp(1-a)}{p-2ma}}\\ {}&+C(\varepsilon ) \left\| \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{2}+\mathcal {C}(\mu ,\varepsilon _1,{\tilde{\varepsilon }}_1, C_1)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}\\ {}&+ \mathcal {C}({\tilde{\varepsilon }}_1)\left\| u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{\frac{2p(q-1)(1-a)}{p-2(q-1)a}}+C(\varepsilon _1) \left\| \mathcal {D}_{0|t}^{\alpha } u_{n}(\cdot , t) \right\| _{L^2(\varOmega ) }^{2}. \end{aligned}\end{aligned}$$

After choosing the constants $\varepsilon , \varepsilon _1$ such that $\displaystyle 1>\frac{1}{\varepsilon }+\frac{1}{\varepsilon _1}$, and from the estimate (4.11) it follows that

$$\begin{aligned} \begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le \mathcal {C_*}[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+B(T), \end{aligned}\end{aligned}$$

(4.20)

where $\mathcal {C_*}:=\mathcal {C}(\gamma ,\varepsilon ,{\tilde{\varepsilon }}, C)+\mathcal {C}(\mu ,\varepsilon _1,{\tilde{\varepsilon }}_1, C_1)$ and $B(T):=A({\tilde{\varepsilon }},T)+A({\tilde{\varepsilon }}_1,T).$ Therefore,

$$\begin{aligned} \begin{aligned}&\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \le \mathcal {C_*}(p)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+ B(p,T). \end{aligned}\end{aligned}$$

(4.21)

Define $y(t):=[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}$ and using the left Riemann-Liouville integral $I_{0|t}^\alpha$ to (4.21), according to Property 1, we arrive at

$$\begin{aligned}\begin{aligned}&y(t)\le y(0)+\frac{1}{\varGamma (\alpha )}\int _0^t (t-s)^{\alpha -1}\left[ \mathcal {C_*}(p)y(s)+B(p,T)\right] ds, \end{aligned}\end{aligned}$$

which satisfies (see [33], Lemma 3.1)

$$\begin{aligned} y(t)\le y(0)E_{\alpha ,1} (\mathcal {C_*}(p)t^{\alpha } )+\frac{B(T)}{\mathcal {C_*}}[E_{\alpha ,1}(\mathcal {C_*}(p)t^\alpha )-1]:=E(p,T). \end{aligned}$$

Finally, we have

$$\begin{aligned}{}[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}\le E(p,T)\,\,\,\text {for all}\,\,\,t\in [0,T]. \end{aligned}$$

(4.22)

From the inequalities (4.20) and (4.22), we obtain

$$\begin{aligned} \begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )}&+ \mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le \mathcal {C_*}E(p,T)+B(p,T):=L(p,T). \end{aligned}\end{aligned}$$

(4.23)

Integrating both sides of (4.23) by the left Riemann-Liouville integral $I_{0|t}^\alpha$ and using Property 1, the last inequality becomes

$$\begin{aligned} \begin{aligned} I_{0|t}^\alpha \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )}&+ [u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le [u_{n}(\cdot , 0)]^p_{W^{s,p}(\varOmega )}+I_{0|t}^\alpha \left[ L(p,T)\right] . \end{aligned}\end{aligned}$$

(4.24)

Consequently, applying the left Caputo derivative $\mathcal {D}_{0|t}^\alpha$ due to Property 1, also noting the facts that $[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}$ is bounded and $\mathcal {D}_{0|t}^\alpha [u_{n}(\cdot , 0)]_{W^{s,p}(\varOmega )}=0$, we can establish

$$\begin{aligned} \begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )} \le L(p,T), \end{aligned}\end{aligned}$$

(4.25)

where L(p, T) does not dependent to n.

$\bullet$ The case $\gamma >0$ and $\mu \le 0$. Accordingly, the inequality (4.15) becomes

$$\begin{aligned}\begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}\le \gamma \int _\varOmega |u_n|^{m-1}u_n\mathcal {D}_{0|t}^{\alpha }u_{n}dx. \end{aligned}\end{aligned}$$

From the estimates (4.16) and (4.18) we can rewrite the last inequality in the form

$$\begin{aligned}\begin{aligned} \frac{1}{2}\Vert \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le \mathcal {C}(\gamma ,\varepsilon ,{\tilde{\varepsilon }}, C)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+ \mathcal {C}({\tilde{\varepsilon }})\left\| u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{\frac{2mp(1-a)}{p-2ma}}\\ {}&+C(\varepsilon ) \left\| \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{2}. \end{aligned}\end{aligned}$$

By choosing $\varepsilon$ small enough such that $\displaystyle \frac{1}{2}-C(\varepsilon )>0$, and using (4.11) it follows that

$$\begin{aligned}\begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le \mathcal {C}(\gamma ,\varepsilon ,{\tilde{\varepsilon }}, C)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+\mathcal {C}({\tilde{\varepsilon }})A(T). \end{aligned}\end{aligned}$$

The conclusion can be obtained, as in the previous case.

$\bullet$ The case $\gamma \le 0$ and $\mu >0$. The inequality (4.15) becomes

$$\begin{aligned}\begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}\le \mu \int _\varOmega |u_n|^{q-2}u_n\mathcal {D}_{0|t}^{\alpha }u_{n}dx. \end{aligned}\end{aligned}$$

Using the estimates (4.17) and (4.19) we have

$$\begin{aligned}\begin{aligned} \frac{1}{2}&\Vert \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )}+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le \mathcal {C}(\mu ,\varepsilon _1,{\tilde{\varepsilon }}_1, C_1)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+ \mathcal {C}({\tilde{\varepsilon }}_1)\left\| u_{n}(\cdot , t)\right\| _{L^2(\varOmega )}^{\frac{2p(q-1)(1-a)}{p-2(q-1)a}}\\ {}&+C(\varepsilon _1) \left\| \mathcal {D}_{0|t}^{\alpha } u_{n}(\cdot , t) \right\| _{L^2(\varOmega ) }^{2}.\end{aligned} \end{aligned}$$

Taking the constant $\varepsilon _1$ small enough such that $\displaystyle \frac{1}{2}-C(\varepsilon _1)>0$, and noting (4.11) it follows that

$$\begin{aligned}\begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}(\cdot , t)\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )} \\ {}&\le \mathcal {C}(\mu ,\varepsilon _1,{\tilde{\varepsilon }}_1, C_1)[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}+\mathcal {C}({\tilde{\varepsilon }}_1)A(T). \end{aligned}\end{aligned}$$

The conclusion of this case also can be obtained, as in the first case.

$\bullet$ The case $\gamma ,\mu \le 0$. Then, the estimate (4.15) can rewritten as

$$\begin{aligned}\begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}\mathcal {D}_{0|t}^{\alpha }[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}\le 0. \end{aligned}\end{aligned}$$

Applying the left Riemann-Liouville integral $I^\alpha _{0|t}$ to the last inequality from Property (1) it follows that

$$\begin{aligned}\begin{aligned} I^\alpha _{0|t}\Vert \mathcal {D}_{0|t}^{\alpha }u_{n}\Vert ^2_{L^2(\varOmega )}&+ \frac{1}{p}[u_{n}(\cdot , t)]^p_{W^{s,p}(\varOmega )}\le \frac{1}{p}[u_{n}(\cdot , 0)]^p_{W^{s,p}(\varOmega )}. \end{aligned}\end{aligned}$$

From the estimate (4.22) we arrive at

$$\begin{aligned}\begin{aligned} I^\alpha _{0|t}\Vert \mathcal {D}_{0|t}^{\alpha }u_{n}\Vert ^2_{L^2(\varOmega )}&\le \frac{1}{p}[u_{n}(\cdot , 0)]^p_{W^{s,p}(\varOmega )}. \end{aligned}\end{aligned}$$

Next, using the left Riemann-Liouville fractional derivative for the last inequality, and from Property 3 and the identity

$$\begin{aligned}\mathbb {D}_{0|t}^{\alpha } [C]=\frac{C}{\varGamma (1-\alpha )}t^{-\alpha },\end{aligned}$$

it follows that

$$\begin{aligned}\begin{aligned} \Vert \mathcal {D}_{0|t}^{\alpha }u_{n}\Vert ^2_{L^2(\varOmega )}&\le \frac{t^{-\alpha }}{p\varGamma (1-\alpha )}[u_{n}(\cdot , 0)]^p_{W^{s,p}(\varOmega )},\,\,\,\text {for all}\,\,\,t\in [0,T]. \end{aligned}\end{aligned}$$

Passing to the limit where $n\rightarrow \infty$, from the estimates in the previous estimates, we conclude that

$$\begin{aligned} \left\{ \begin{array}{l} u_n\in W^{s,p}(\varOmega )\cap L^2(\varOmega ;L^\infty (0,T)),\\ {}\\ \mathcal {D}_{0|t}^{\alpha }u_n\in L^2(\varOmega ;L^\infty (0,T)).\end{array}\right. \end{aligned}$$

(4.26)

Consequently, from (4.26) there exists a subsequence $\{u_{n_k}\}$ of $\{u_n\}_{n\in \mathbb {N}}$ weak star converging to some element from $W^{s,p}(\varOmega )\cap L^2(\varOmega ;L^\infty (0,T))$ such as

$$\begin{aligned} \begin{aligned} u_{n_k}\overset{*}{\rightharpoonup }\ u\,\,\,\text {in}\,\, W^{s,p}(\varOmega )\cap L^2(\varOmega ;L^\infty (0,T)),\\\mathcal {D}_{0|t}^{\alpha }u_{n}\overset{*}{\rightharpoonup }\ \mathcal {D}_{0|t}^{\alpha }u\,\,\, \text {in}\,\, L^2(\varOmega ;L^\infty (0,T)). \end{aligned}\end{aligned}$$

(4.27)

Similarly, from (4.27), we deduce that one can extract a subsequence $\{u_{n_k}\}$ of $\{u_n\}_{n\in \mathbb {N}}$ such that

$$\begin{aligned} u_{n_k}\overset{*}{\rightharpoonup }u_n \,\,\,\text {in}\,\,\, W^{s,p}(\varOmega )\cap L^2(\varOmega ; L^\infty (0,T)). \end{aligned}$$

(4.28)

Since, $W^{s,p}(\varOmega )\cap L^2(\varOmega ;L^\infty (0,T))\subset L^2(\varOmega ;L^\infty (0,T))$, from (4.26) it follows that the sequences $\{u_n\}_{n\in \mathbb {N}}$ and $\mathcal {D}_{0|t}^{\alpha }u_{n}$ are bounded in $L^2(\varOmega ;L^\infty (0,T))$. Then, it particular $\{u_n\}_{n\in \mathbb {N}}$ is bounded in $W^{s,p}(\varOmega )$. It is known by Lemma 1, that the embedding of $W^{s,p}(\varOmega )$ in $L^2(\varOmega )$ is continuous. It gives us that the subsequence $\{u_{n_k}\}$ can be chosen such that $u_{n_k}\rightarrow u$ in the norm of $L^2(\varOmega )$, converging almost everywhere. The previous argument leads us to the limit in (4.2). However, we multiply (4.2) by $\theta _k(t)\in C[0,T]$, then summing up both sides over $k=\overline{1,n}$, to get

$$\begin{aligned}\begin{aligned}&\int _\varOmega \mathcal {D}_{0|t}^{\alpha }u_{n}\cdot \varPsi dx+ P(u_{n},\varPsi ) =\gamma \int _\varOmega |u_n|^{m-1}u_n\cdot \varPsi dx+\mu \int _\varOmega |u_n|^{q-2}u_n\cdot \varPsi dx, \end{aligned}\end{aligned}$$

almost everywhere in $t\in [0,T]$, where $\displaystyle \varPsi (x,t)=\sum _{k=1}^{n}\theta _{k} (t)\omega _{k} (x)$.

Taking into account the obtained inclusions and convergence, we pass in (4.2) to the limit as $n\rightarrow \infty$ and obtain Definition 6 for $\varphi =\varPsi$. Since the set of all functions $\varPsi (x,t)$ is dense in $\varPi$, then the limit relation holds for all $\varphi =\varphi (x,t)\in W_0^{s,p}(\varOmega ;L^p(0,T)).$

$\bullet$ The case $1<q-1<m <p.$ We repeat the entire procedure described above by simply changing the condition inequality (4.7) to $1<q-1<m<p.$ $\square$

4.2 Uniqueness of a weak solution

In this subsection we discuss the uniqueness of weak solutions.

Theorem 3

Let $u_0\in W^{s,p}_0(\varOmega ), u_0\ge 0$ and $sp<N$. Then the local real-valued weak solution of (1.1) on (0, T),  $T<\infty$, is unique.

Proof

Assume that we have two real-valued weak solutions u and v for problem (1.1). Hence, by Definition 6, we obtain

$$\begin{aligned} \begin{aligned} \int _0^T\int _{\varOmega }\mathcal {D}_{0|t}^\alpha u\varphi dxdt&\!+\!\int _0^T\int _{\varOmega }\frac{|u(x)\!-\!u(y)|^{p-2}(u(x)\!-\!u(y))}{|x-y|^{N+sp}}(\varphi (x)\!-\!\varphi (y))dxdydt\\ {}&=\gamma \int _0^T\int _{\varOmega }|u|^{m-1}u\varphi dxdt+\mu \int _0^T\int _{\varOmega }|u|^{q-2}u\varphi dxdt \end{aligned} \end{aligned}$$

and

$$\begin{aligned} \begin{aligned} \int _0^T\int _{\varOmega }\mathcal {D}_{0|t}^\alpha v\varphi dxdt&+\int _0^T\int _{\varOmega }\frac{|v(x)\!-\!v(y)|^{p-2}(v(x)\!-\!v(y))}{|x-y|^{N+sp}}(\varphi (x)\!-\!\varphi (y))dxdydt\\ {}&=\gamma \int _0^T\int _{\varOmega }|v|^{m-1}v\varphi dxdt+\mu \int _0^T\int _{\varOmega }|v|^{q-2}v\varphi dxdt. \end{aligned} \end{aligned}$$

By subtracting the previous two inequalities, it follows for $t\in (0,T]$ that

$$\begin{aligned}\begin{aligned}&\int _0^t\int _{\varOmega }\mathcal {D}_{0|\tau }^\alpha [u-v]\varphi dxd\tau +\underbrace{\int _0^t\int _{\varOmega }[(-\varDelta )^s_pu-(-\varDelta )^s_pv]\varphi dxd\tau }_{\mathcal {C}}\\ {}&=\underbrace{\gamma \int _0^t\int _{\varOmega }(|u|^{m-1}u-|v|^{m-1}v)\varphi dxd\tau }_{\mathcal {A}}+\underbrace{\mu \int _0^t\int _{\varOmega }(|u|^{q-2}u-|v|^{q-2}v)\varphi dxd\tau }_{\mathcal {B}}.\end{aligned}\end{aligned}$$

Using the fact that $\mathcal {C}$ is nonnegative from (3.6), and the estimates (3.8), (3.11) for $\mathcal {A}, \mathcal {B}$, respectively, we deduce that

$$\begin{aligned}\begin{aligned} \int _0^t\int _{\varOmega }\mathcal {D}_{0|\tau }^\alpha [u-v]\varphi dxd\tau&\le \gamma L(m)\int _0^t\int _{\varOmega }|u-v|\varphi dxd\tau \\ {}&+\mu L(q)\int _0^t\int _{\varOmega }|u-v|\varphi dxd\tau .\end{aligned}\end{aligned}$$

At this stage choosing the real-valued test function

$$\begin{aligned}\varphi =(u-v)_+=\max \{u-v, 0\}\end{aligned}$$

and using Lemma 5, we can rewrite the last inequality as

$$\begin{aligned}\begin{aligned}&\frac{1}{2}\int _0^t\int _{\varOmega }\mathcal {D}_{0|\tau }^\alpha (u-v)_+^2 dxd\tau \\ {}&\le \gamma L(m)\int _0^t\int _{\varOmega }(u-v)_+^2 dxd\tau +\mu L(q)\int _0^t\int _{\varOmega }(u-v)_+^2 dxd\tau .\end{aligned}\end{aligned}$$

Therefore, we should consider three cases depending on $\gamma , \mu$. By repeating the entire procedure as in the proof of Theorem 1, we obtain the main inequality

$$\begin{aligned}\begin{aligned} \int _{\varOmega }(u-v)_+^2 dx\le 0,\end{aligned}\end{aligned}$$

which is equivalent to $(u-v)_+=0$. Finally, we conclude that $u=v$. $\square$

5 Global existence and blow-up of solutions

5.1 Blow-up of solution

In this subsection we will show the blow-up of solution to (1.1) using the comparison principle.

Let $\xi (x)>0$ and $\lambda _1(\varOmega )>0$ be the first eigenfunction and the first eigenvalue [26, Theorem 5], respectively, related to the Dirichlet problem:

$$\begin{aligned} \left\{ \begin{array}{l} (-\varDelta )^s_p\xi (x)=\lambda _1(\varOmega ) |\xi (x)|^{p-2}\xi (x),\,\,\, x\in \varOmega ,\\ {}\\ \xi (x)=0,\,\,\, x\in \mathbb {R}^N\setminus \varOmega , \end{array}\right. \end{aligned}$$

(5.1)

with $\Vert \xi \Vert ^2_{L^2(\varOmega )}= 1.$

Theorem 4

Let $p\ge 2,$ $u_0>0$, and assume that one of the following conditions holds:

(a) $p=q\ge 2, m>1$ and $\lambda _1(\varOmega )\ge \mu , \gamma >0$;

(b) $p-1=m\ge 1, q>2$ and $\lambda _1(\varOmega )\ge \gamma , \mu >0$;

(c) $p\ge 2, m>1, q\ge 1$ and $\lambda _1(\varOmega ),\gamma >0, \mu \le 0$;

(d) $p\ge 2, m+1=q>2$ and $\gamma , \mu , \lambda _1(\varOmega )>0.$

Then the positive solution u(x, t) of (1.1) blows up in finite time

$$\begin{aligned}T^*=(k\varGamma (2-\alpha ))^{\frac{2}{2-2\alpha -k}},\end{aligned}$$

where $k=m-1$ in cases (a), (c), (d) and $k=q-2$ in cases (b), (d) and $\alpha \in (0,1)$, $\varGamma$ is the Euler Gamma function, namely, we have

$$\begin{aligned}\lim \limits _{t\rightarrow T^*}u(x,t)=+\infty .\end{aligned}$$

Proof

First we will prove the cases (a) and (b).

We shall prove this theorem by constructing a proper weak subsolution to (1.1). We will seek the solution $v(x,t)=\xi (x)f(t)>0$ with the initial data $v_0(x)=\xi (x)f(0),$ such that $0\le v_0(x)\le u_0(x)$ on $x\in \varOmega$. Multiplying the equation (1.1) by v(x, t), and integrating the equality over $\varOmega$, one obtains

$$\begin{aligned}\begin{aligned} f(t)\mathcal {D}_{0|t}^{\alpha }f(t)\Vert \xi \Vert ^2_{L^2(\varOmega )}&+\lambda _1(\varOmega ) f^{p}(t)\Vert \xi ^{p}\Vert ^2_{L^2(\varOmega )}\\ {}&=\gamma f^{m+1}(t)\Vert \xi ^{m+1}\Vert ^2_{L^2(\varOmega )}+\mu f^{q}(t)\Vert \xi ^{q}\Vert ^2_{L^2(\varOmega )}. \end{aligned}\end{aligned}$$

Hence, from Lemma 5, it follows that

$$\begin{aligned} \begin{aligned} \frac{1}{2}\mathcal {D}_{0|t}^{\alpha }f^2(t)&+\lambda _1 C(p) f^{p}(t)\le \gamma C(m)f^{m+1}(t)+\mu C(q)f^{q}(t). \end{aligned}\end{aligned}$$

(5.2)

At this stage, by denoting $f^2(t)=z(t)$, we have to consider the cases:

(a) If $p=q \ge 2, m>1$ and $\lambda _1(\varOmega ) \ge \mu , \gamma >0$, then (5.2) can rewritten as

$$\begin{aligned}\mathcal {D}_{0|t}^{\alpha }z(t)\le 2 C(m,\gamma ) z^\frac{m+1}{2}(t). \end{aligned}$$

Using the idea of paper [11], we set for any $t\in (0,b)$,

$$\begin{aligned}z(t)=\frac{b}{(b-t)^\frac{2}{m-1}},\,\,\,b:=b(m-1,\alpha )=\left( (m-1)\varGamma (2-\alpha )\right) ^\frac{2}{3-2\alpha -m}. \end{aligned}$$

Accordingly, we have the initial condition $z(0)=z_0>0$. We should note that the function z(t), $\lim _{t\rightarrow b^-}z(t)\rightarrow \infty ,$ diverges at $t=b$. Moreover, for any $t\in (0,b)$ and any $\tau \in (0,t)$ we can obtain

$$\begin{aligned}\frac{\partial }{\partial \tau }z(\tau ):&=\frac{2b}{(m-1)(b-\tau )^\frac{m+1}{m-1}}\\ {}&\le \frac{2b}{(m-1)(b-t)^\frac{m+1}{m-1}}\\ {}&=\frac{z^\frac{m+1}{2}(t)}{2(m-1)b^\frac{m-1}{2}}. \end{aligned}$$

From Definition 3 it follows for all $t\in (0,b)$,

$$\begin{aligned}\begin{aligned} \mathcal {D}^\alpha _{0|t}z(t)&=\frac{2 C(m,\gamma )}{\varGamma (1-\alpha )}\int _0^t\frac{z'(\tau )}{(t-\tau )^\alpha }d\tau \\ {}&\le \frac{2 C(m,\gamma ) z^\frac{m+1}{2}(t)}{2(m-1)b^\frac{m-1}{2}\varGamma (1-\alpha )}\int _0^t\frac{d\tau }{(t-\tau )^\alpha }\\ {}&=\frac{C(m,\gamma )t^{1-\alpha }z^\frac{m+1}{2}(t)}{(m-1)b^\frac{m-1}{2}\varGamma (2-\alpha )} \\ {}&\le \frac{ C(m,\gamma )b^{1-\alpha }z^\frac{m+1}{2}(t)}{(m-1)b^\frac{m-1}{2}\varGamma (2-\alpha )} \\ {}&=\frac{ C(m,\gamma )b^\frac{3-2\alpha -m}{2}z^\frac{m+1}{2}(t)}{(m-1)\varGamma (2-\alpha )} \\ {}&= C(m,\gamma )z^\frac{m+1}{2}(t). \end{aligned} \end{aligned}$$

Therefore, z(t) diverges at $t=b$ yielding that

$$\begin{aligned}T_*\le b=b(m-1,\alpha ).\end{aligned}$$

(b) If $p-1=m\ge 1, q>2$ and $\lambda _1(\varOmega )\ge \gamma , \mu >0$, then from (5.2) we obtain

$$\begin{aligned}\mathcal {D}_{0|t}^{\alpha }z(t)\le 2C(q,\mu )z^\frac{q}{2}(t). \end{aligned}$$

We can argue as the previous case by choosing for any $t\in (0,b)$ the function

$$\begin{aligned}z(t):=\frac{b}{(b-t)^\frac{2}{q-2}},\,\,\,b:=b(q-2,\alpha )=\left( (q-2)\varGamma (2-\alpha )\right) ^\frac{2}{4-2\alpha -q}.\end{aligned}$$

Similarly, for any $t\in (0,b)$ and any $\tau \in (0,t)$, we obtain

$$\begin{aligned}\frac{\partial }{\partial \tau }z(\tau ):&=\frac{2b}{(q-2)(b-\tau )^\frac{q}{q-2}}\\ {}&\le \frac{2b}{(q-2)(b-t)^\frac{q}{q-2}}\\ {}&=\frac{z^\frac{q}{2}(t)}{2(q-2)b^\frac{q-2}{2}}. \end{aligned}$$

Finally, z(t) diverges at $t=b$ for

$$\begin{aligned}T_*\le b=b(q-2,\alpha ).\end{aligned}$$

(c) For $p\ge 2, m>1, q\ge 1$ and $\lambda _1(\varOmega ),\gamma >0, \mu <0$, inequality (5.2) yields

$$\begin{aligned}\mathcal {D}_{0|t}^{\alpha }z(t)\le 2 C(m,\gamma ) z^\frac{m+1}{2}(t). \end{aligned}$$

(d) For $p\ge 2, m+1=q>2$ and $\gamma , \mu , \lambda _1(\varOmega )>0$, using (5.2) we have

$$\begin{aligned}\mathcal {D}_{0|t}^{\alpha }z(t)\le 2 \left[ C(m,\gamma )+C(p,\mu )\right] z^\frac{q}{2}(t). \end{aligned}$$

Proof of (c) and (d) can be derived from the previous cases. We just omit it. The proof is complete. $\square$

5.2 Global solution

In this subsection, we prove the existence of global solutions of problem (1.1).

Theorem 5

Assume that $u_0\in W^{s,p}_0(\varOmega )\cap L^\infty (\varOmega ),\,s\in (0,1),\, u_0\ge 0$, and let $p, q, m, \gamma , \mu$ satisfy one of the following conditions:

(a) $p=m+1=q>2$ and $0<\gamma +\mu \le \lambda _1(\varOmega );$

(b) $p=q$ or $p=m+1$ and $0\le \gamma , \mu \le \lambda _1(\varOmega );$

(c) $p\le m+q$ and $\gamma ,\mu \in \mathbb {R};$

(d) $p\ge 2, m>1, q\ge 1$ and $\gamma ,\mu \le 0;$

(e) $p=q, m>1$ and $\gamma \le 0,\,\mu >0$.

Then the problem (1.1) admits a global in time positive solution.

Remark 1

Note that in the limiting case $\alpha \rightarrow 1$ and $s\rightarrow 1,$ the results of Theorem 5 coincides with the results obtained in [25].

Proof of Theorem 5

(a) Let $\varOmega ^*\subset \mathbb {R}^N$ be a smooth domain such that $\varOmega \subset \subset \varOmega ^*$.

Define $\psi$ and $\lambda _1(\varOmega ^*)$ to be the first eigenfunction and the first eigenvalue related to the Dirichlet problem:

$$\begin{aligned} \left\{ \begin{array}{l} (-\varDelta )^s_p\psi (x)=\lambda _1(\varOmega ^*)|\psi (x)|^{p-2}\psi (x),\,\,\, x\in \varOmega ^*,\\ {}\\ \psi (x)=0,\,\,\, x\in \mathbb {R}^N\setminus \varOmega ^*, \end{array}\right. \end{aligned}$$

with $\displaystyle \int _{\varOmega ^*}|\psi (x)|^{p}dx=1$, for more details see [26, Lemma 15]. Then, from Lemma 6 we have $\lambda _1(\varOmega ^*)\le \lambda _1(\varOmega )$, where $\lambda _1(\varOmega )$ is the first eigenvalue of (2.2). Moreover, in view of [26, Theorem 16], we can choose a suitable $\varOmega ^*$ and $\theta >0$ which satisfies $\theta \le \lambda _1(\varOmega ^*)\le \lambda _1(\varOmega )$ . Therefore, let K be so large such that

$$\begin{aligned}w=K\psi \ge K\beta \ge \Vert u_0\Vert _{L^\infty (\varOmega )},\end{aligned}$$

where $\beta =\inf _{\varOmega }\psi >0,$ which we note that $\psi >0$ in $\varOmega$ from the results of Lindgren and Lindqvist in [26, Theorem 5]. Following that, a simple calculation shows that for each nonnegative test-function $\varphi =\varphi (x,t)\in \varPi \cap W_0^{s,p}(\varOmega ;L^\infty (0,T))$, we have

$$\begin{aligned} \begin{aligned} \int _0^T\int _{\varOmega }\mathcal {D}_{0|t}^{\alpha }w\varphi dxdt&+\int _0^T\langle (-\varDelta )^s_pw,\varphi \rangle dt \\ {}&=\gamma \int _0^T\int _{\varOmega }w^m\varphi dxdt+\mu \int _0^T\int _{\varOmega }w^{q-1}\varphi dxdt,\end{aligned}\end{aligned}$$

(5.3)

where $\langle \cdot ,\cdot \rangle$ is the inner product. Hence, noting that $p=m+1=q>2,$ and choosing $\theta :=\gamma +\mu$, the last identity takes the form

$$\begin{aligned}\begin{aligned}\int _0^T\int _{\varOmega }\mathcal {D}_{0|t}^{\alpha }w\varphi dxdt+\int _0^T\langle (-\varDelta )^s_pw,\varphi \rangle dt&=\lambda _1(\varOmega )\int _0^T\int _{\varOmega }w^{p-1}\varphi dxdt\\ {}&\ge \lambda _1(\varOmega ^*)\int _0^T\int _{\varOmega }w^{p-1}\varphi dxdt \\ {}&\ge (\gamma +\mu )\int _0^T\int _{\varOmega }w^{p-1}\varphi dxdt.\end{aligned}\end{aligned}$$

It follows that $w=K\psi$ is a weak supersolution of problem (1.1). From Theorem 1, we have $0\le u\le w$ almost everywhere in $\varOmega _T$. It is also important to note that the function w is independent of t, allowing us to continue the method at any time interval $[T, T']$. As a result, we may say that the solution to (1.1) is global in time.

(b) Due to the expression (5.3) and the conditions $p=q$ or $p=m+1$, it follows that

$$\begin{aligned}\begin{aligned}&\int _0^T\int _{\varOmega }\mathcal {D}_{0|t}^{\alpha }w\varphi dxdt+\int _0^T\langle (-\varDelta )^s_pw,\varphi \rangle dt \ge \mu \int _0^T\int _{\varOmega }w^{p-1}\varphi dxdt\end{aligned}\end{aligned}$$

and

$$\begin{aligned}\begin{aligned}&\int _0^T\int _{\varOmega }\mathcal {D}_{0|t}^{\alpha }w\varphi dxdt+\int _0^T\langle (-\varDelta )^s_pw,\varphi \rangle dt \ge \gamma \int _0^T\int _{\varOmega }w^{p-1}\varphi dxdt.\end{aligned}\end{aligned}$$

Since, $0\le \gamma ,\mu \le \lambda _1(\varOmega )$, then the function $w=K\psi$ is also a weak supersolution of (1.1). The conclusion is established using the same argument as before.

(c) From Definition 5 assume that u is an eigenfunction associated to the eigenvalue $\lambda _1(\varOmega )$, which is a nonnegative [26, Theorem 5]. Then by (5.3) it follows that

$$\begin{aligned} \lambda _1(\varOmega )\int _0^T\int _{\varOmega }u^{p-1}\varphi dxdt =\gamma \int _0^T\int _{\varOmega }u^m\varphi dxdt+\mu \int _0^T\int _{\varOmega }u^{q-1}\varphi dxdt.\end{aligned}$$

(5.4)

Choosing constants $r, r'$ such as

$$\begin{aligned}\begin{aligned} \frac{1}{r}+\frac{1}{r'}=1,\,\,\,r, r'>1\,\,\,\text {and}\,\,\,p-1=\frac{m}{r}+\frac{q-1}{r'}\le m+q-1, \end{aligned}\end{aligned}$$

we obtain

$$\begin{aligned}\begin{aligned} \int _0^T\int _{\varOmega }u^{p-1}\varphi dxdt =\int _0^T\int _{\varOmega }u^{\frac{m}{r}+\frac{q-1}{r'}}\varphi ^{\frac{1}{r}+\frac{1}{r'}}dxdt.\end{aligned}\end{aligned}$$

Using the Hölder and $\varepsilon$-Young inequalities to the last expression, it follows that

$$\begin{aligned}\begin{aligned} \int _0^T\int _{\varOmega }u^{p-1}\varphi dxdt&\le \left( \int _0^T\int _{\varOmega }u^m\varphi dxdt\right) ^{\frac{1}{r}}\left( \int _0^T\int _{\varOmega }u^{q-1}\varphi dxdt\right) ^{\frac{1}{r'}} \\ {}&\le \varepsilon \int _0^T\int _{\varOmega }u^m\varphi dxdt+C(\varepsilon )\int _0^T\int _{\varOmega }u^{q-1}\varphi dxdt.\end{aligned}\end{aligned}$$

Therefore, the identity (5.4) becomes

$$\begin{aligned}\begin{aligned} \gamma \int _0^T\int _{\varOmega }u^m\varphi dxdt&+\mu \int _0^T\int _{\varOmega }u^{q-1}\varphi dxdt\\ {}&\le \lambda _1(\varOmega )\varepsilon \int _0^T\int _{\varOmega }u^m\varphi dxdt+\lambda _1(\varOmega ) C(\varepsilon )\int _0^T\int _{\varOmega }u^{q-1}\varphi dxdt.\end{aligned}\end{aligned}$$

Now, taking $\varepsilon$ small enough, such that $\lambda _1(\varOmega )\varepsilon -\gamma \ge 0$ and $\lambda _1(\varOmega ) C(\varepsilon )-\mu \ge 0$, we can get that the last inequality will be non-positive

$$\begin{aligned}\begin{aligned} (\lambda _1(\varOmega )\varepsilon -\gamma )\int _0^T\int _{\varOmega }u^m\varphi dxdt+(\lambda _1(\varOmega ) C(\varepsilon )-\mu )\int _0^T\int _{\varOmega }u^{q-1}\varphi dxdt\ge 0,\end{aligned}\end{aligned}$$

which completes our proof by the comparison principle.

(d) We proceed by multiplying each term of (1.1) by $u\ge 0$ and then integrating over $\varOmega$. Thus, we obtain

$$\begin{aligned} \begin{aligned} \int _\varOmega [\mathcal {D}_{0|t}^{\alpha }u]udx&=-\langle (-\varDelta )^s_pu,u\rangle +\gamma \int _\varOmega u^{m+1}dx+\mu \int _\varOmega u^{q}dx\end{aligned} \end{aligned}$$

(5.5)

and taking into account that $\gamma ,\mu \le 0,$ $\langle (-\varDelta )^s_pu,u\rangle \ge 0$, it follows that

$$\begin{aligned}\begin{aligned} \int _\varOmega [\mathcal {D}_{0|t}^{\alpha }u]udx&\le 0. \end{aligned}\end{aligned}$$

By Lemma 5, it implies

$$\begin{aligned}\begin{aligned} \int _\varOmega \mathcal {D}_{0|t}^{\alpha }u^2dx\le 0. \end{aligned}\end{aligned}$$

Moreover, the Caputo derivative depends on the variable t, and the last expression can be rewritten as

$$\begin{aligned} \begin{aligned} \mathcal {D}_{0|t}^{\alpha }\int _\varOmega u^2dx\le 0. \end{aligned}\end{aligned}$$

(5.6)

Hence, applying the left Riemann-Liouville integral $I_{0|t}^{\alpha }$ to the inequality (5.6) and using Property 1, we obtain

$$\begin{aligned}\begin{aligned}\int _\varOmega u^2(x,t)dx\le \int _\varOmega u^2(x,0)dx.\end{aligned}\end{aligned}$$

Finally, using $u_0\ge 0$ and Corollary 1, we get

$$\begin{aligned}\Vert u(\cdot ,t)\Vert _{L^2(\varOmega )}\le \Vert u_0\Vert _{L^2(\varOmega )},\,\,\,\text {for all}\,\,\,t\ge 0.\end{aligned}$$

(e) Without loss of generality, for $\gamma \le 0,\,\mu >0$ we can get from (5.5), by (2.3), that

$$\begin{aligned}\begin{aligned} \int _\varOmega [\mathcal {D}_{0|t}^{\alpha }u]udx&\le -C_{N,s,p}\int _{\varOmega }\int _{\varOmega }\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+sp}}dxdy+\mu \int _\varOmega u^{q}dx.\end{aligned} \end{aligned}$$

Then, by Lemma 7 for $p=q$, we obtain

$$\begin{aligned}\begin{aligned} \int _\varOmega [\mathcal {D}_{0|t}^{\alpha }u]udx&\le -C_{N,s,p}[u]^p_{W^{s,p}(\varOmega )}+ \mu \lambda _1(\varOmega )[u]^p_{W^{s,p}(\varOmega )}.\end{aligned} \end{aligned}$$

Using the fact that $\lambda _1(\varOmega )$ coincides with the sharp constant in Lemma 7 [8, page 2] we choose the domain such that $\displaystyle C_{N,s,p}\ge \mu \lambda _1(\varOmega )\ge \frac{\mu }{\mathcal {I}_{N,s,p(\varOmega )}}$ holds, which gives us

$$\begin{aligned}\begin{aligned} \int _\varOmega [\mathcal {D}_{0|t}^{\alpha }u]udx&\le 0.\end{aligned} \end{aligned}$$

Accordingly, the conclusion follows as in the previous case. $\square$

5.3 Asymptotic behavior of solution

In this subsection, we give the time-decay estimates of global solutions of problem (1.1).

Theorem 6

Assume that $u_0>0$ and that one of the following conditions holds:

(a) $m=q-1>0$ and $\gamma +\mu < 0;$

(b) $m>0, q>1$ and $\gamma < 0,\,\mu =0;$

(c) $m>0, q>1$ and $\gamma =0,\,\mu < 0$.

Then the positive global solution to problem (1.1) satisfies the estimate

$$\begin{aligned} 0<u(x,t)\le \frac{M}{1+t^\frac{\alpha }{r}},\,t\ge 0,\,x\in \varOmega , \end{aligned}$$

where M is a positive constant dependent of $u_0,$ and $r=m$ in cases (a), (b) and $r=q-1$ in cases (a), (c).

Proof

(a) Let us consider the function $v(x,t):=v(t)>0$ for all $x\in {\overline{\varOmega }}$. Then it follows that

$$\begin{aligned}\mathcal {D}_{0|t}^{\alpha }v(t)+(-\varDelta )^s_pv(t)=\gamma v^{m}(t)+\mu v^{q-1}(t).\end{aligned}$$

According to the fact that $(-\varDelta )^s_pv(t)=0$ and $m=q-1>0$, $\gamma +\mu < 0$, the last expression can be rewritten in the following form

$$\begin{aligned} \mathcal {D}_{0|t}^{\alpha }v(t)+\nu v(t)^{m}=0,\,\nu =-(\gamma +\mu )>0, \end{aligned}$$

(5.7)

which ensures that v(t) satisfies (1.1) with the initial data $0<\max \limits _{x\in \varOmega }u_0(x)\le v_0$.

It is known from the results of Zacher and Vergara in [37, Theorem 7.1], that if $v_0>0, \nu>0, m>0$, then the solution to equation (5.7) satisfies estimate $v(t)\le \frac{M}{1+t^\frac{\alpha }{r}},$ for all $t\ge 0.$ As $0<u_0(x)\le v_0,$ then v(t) is a supersolution of problem (1.1). This completes the proof.

Cases (b) and (c) are proved in a similar way, completely repeating the above calculations.

The proof is complete. $\square$