Well-posedness results for a class of semilinear time-fractional diffusion equations

Abstract

In this paper, we discuss an initial value problem for the semilinear time-fractional diffusion equation. The local well-posedness (existence and regularity) is presented when the source term satisfies a global Lipschitz condition. The unique continuation of solution and finite time blowup result are presented when the reaction terms are logarithmic functions (local Lipschitz types).

1 Introduction

Let $\Omega \subset {\mathbb {R}}^N, ~ ( N \ge 1)$ be a bounded open set with boundary $\Omega ^c$. The main aim of the paper is to study the properties of the solutions of a class of time-fractional diffusion equations involving the so-called Riemann–Liouville (R–L) time-fractional derivative. More precisely, we consider the following initial value problem:

where $T>0$, $\alpha \in (0,1)$ is real number and $\dfrac{\partial ^{1-\alpha } }{\partial t} u$ denotes the R–L time-fractional derivative of order $1-\alpha$ of the function u formally given by

$$\begin{aligned} \frac{\partial ^{1-\alpha }}{\partial t} f (t):= \frac{\mathrm {d}}{\mathrm {d}}t \left( {\mathcal {J}}^\alpha f\right) (t), \quad t >0, \end{aligned}$$

(1.1)

where the Riemann–Liouville fractional integral operator $\mathcal J^\alpha : L^2(0,T) \rightarrow L^2(0,T)$ is defined by the formula (see, e.g., [1])

$$\begin{aligned} ({\mathcal {J}}^\alpha f) (t) := {\left\{ \begin{array}{ll} \dfrac{1}{\Gamma (\alpha )} \displaystyle \int \limits _0^t \tau ^{\alpha -1} f(t-\tau ) \mathrm {d}\tau , &{}\quad 0< \alpha <1, \\ f(t), &{}\quad \alpha =0, \end{array}\right. } \end{aligned}$$

(1.2)

and $\Gamma (\cdot )$ is the Gamma function. The operator ${\mathcal {A}}$ is a linear, positive definite, self-adjoint operator with compact inverse in $L^2(\Omega )$, $u = u(x, t)$ is the state of the unknown function and $u_0(x)$ is a given function. The function F is a nonlinear source term which appears in some physical phenomena [2,3,4].

When $\alpha = 1$ and ${\mathcal {A}} = -\Delta$ problem ${{\mathbb {P}}}$ describes the nonlinear heat Eq. [2, 5,6,7]

$$\begin{aligned} \dfrac{\partial }{\partial t} u(x,t) -\Delta u(x,t)= F(u). \end{aligned}$$

(1.3)

If $\alpha \in (0,1)$, Problem ${{\mathbb {P}}}$ is called an initial value problem for the semilinear time-fractional diffusion equation; we refer the reader to [3, 8,9,10] and the references therein. Many important physical models and practical problems require one to consider the diffusion model with a fractional derivative rather than a classical one, like physical models considering memory effects [2,3,4, 11,12,13] and some corresponding engineering problems [2, 3, 14, 15] with power-law memory (non-local effects) in time [4, 8, 16,17,18,19,20]. For nonlinearities of power-type $F(u)=|u|^{p-1}u$ for $p \ge 1$, Bruno de Andrade et al. [3] considered the fractional reaction–diffusion equation to discuss the global well-posedness and asymptotic behavior of solutions; see also [7, 21] and the references therein. Studies of logarithmic nonlinearity have a long history in physics as they occur naturally in inflation cosmology, quantum mechanics, and nuclear physics [22] and PDEs with logarithmic nonlinearity have attracted many authors; see [23,24,25,26] and the references therein.

Results on initial value problems for R–L time-fractional diffusion equation with logarithmic nonlinearity are quite limited. The solution operator of our problem $E_{\alpha ,1}\left( - {\mathcal {A}} t^{\alpha } \right)$ brings some difficulties in estimating and analyzing the solution (existence and regularity estimate of the solutions). We consider the model with the source terms $F_p(u) = \eta V_p(u) \log |u|$ and $V_p(u) = |u|^{p-2} u, ~ p\ge 2, \eta >0$ (locally Lipschitz type). To present the properties of the solutions in $W^{s,q}(\Omega )$, we need to consider the Lipschitz properties of the source function (both global Lipschitz property and local Lipschitz property). Based on the conditions of the constants s, q depending on the dimensions $N \ge 1$ and the constant $s>0$, we set up the Sobolev embeddings ${\mathbb {X}}^{s}(\Omega ) \hookrightarrow W^{s,q}(\Omega ) \hookrightarrow L^p(\Omega )$ (see the definition of the spaces $W^{s,q}(\Omega )$ and ${\mathbb {X}}^{s}(\Omega )$ in (2.3) and (2.7) below).

In Sect. 2, we present some basic definitions and the setting for our work. Moreover, we obtain a precise representation of solutions using Mittag–Leffler operators. In Sect. 3, we first present local well-posedness results when the source term satisfies a global Lipschitz condition. Also local existence, continuation of solutions and finite time blowup results are presented when the source terms are logarithmic functions.

2 Notations and preliminaries

2.1 Relevant notations and the functional spaces

Given two positive quantities y, z, we write $y \lesssim z$ if there exists a constant $C > 0$ such that $y \le Cz$. Let us recall that the spectral problem

$$\begin{aligned} {\left\{ \begin{array}{ll} {\mathcal {A}} \phi _j(x) =\lambda _j \phi _j(x), &{}\quad x \in \Omega , \quad \sigma \in (0,1],\\ \phi _j(x)=0, &{}\quad x \in \partial \Omega , \end{array}\right. } \end{aligned}$$

(2.1)

admits a family of eigenvalues

$$\begin{aligned} 0 < \lambda _1 \le \lambda _2 \le \lambda _3 \le \cdots \le \lambda _j \le \cdots \nearrow \infty . \end{aligned}$$

Given a Banach space B, let C([0, T]; B) be the set of all continuous functions which map [0, T] into B. The norm of the function space $C^k ([0, T ]; B),$ for $0 \le k \le \infty$ is denoted by

$$\begin{aligned} \Vert v\Vert _{C^k ([0, T ]; B)} = \sum _{i=0}^k \sup _{t\in [0, T ]} \big \Vert v^{(i)} (t) \big \Vert _{B} < \infty . \end{aligned}$$

(2.2)

For any real numbers $s>0$ and $1 \le p < \infty$, we recall the fractional Sobolev-type spaces $W^{s,p}(\Omega )$ via the Gagliardo approach (also called Aronszajn or Slobodeckij spaces). Fix a number $s \in (0,1)$ and for any $p \in [1,\infty )$, define $W^{s,p}(\Omega )$ as follows

$$\begin{aligned} W^{s,p}(\Omega ) = \left\{ v \in L^p(\Omega ) \quad \hbox {s.t.} \quad \frac{\left| v(x) - v(y)\right| }{\left| x-y\right| ^{\frac{N+ps}{p}}} \in L^p(\Omega \times \Omega )\right\} . \end{aligned}$$

(2.3)

For $0<s<1$, it can be said that $W^{s,p}(\Omega )$ is an intermediate Banach space between $L^p(\Omega )$ and $W^{1,p}(\Omega )$, endowed the corresponding norm

$$\begin{aligned} \big \Vert v\big \Vert _{W^{s,p}(\Omega )} = \left( \int \limits _{\Omega } |v|^p\mathrm {d}x + \int \limits _{\Omega }\int \limits _{\Omega } \frac{\left| v(x) - v(y)\right| ^p}{\left| x-y\right| ^{N+sp}} \mathrm {d}x \mathrm {d}y \right) ^{\frac{1}{p}}, \end{aligned}$$

(2.4)

where the seminorm

$$\begin{aligned} \big |v\big |_{W^{s,p}(\Omega )} = \left( \int \limits _{\Omega }\int \limits _{\Omega } \frac{\left| v(x) - v(y)\right| ^p}{\left| x-y\right| ^{N+sp}} \mathrm {d}x \mathrm {d}y \right) ^{\frac{1}{p}}, \end{aligned}$$

(2.5)

denotes the Gagliardo (semi)norm of v. For $p=2$ in (2.3), together with the norm $\left\| \cdot \right\| _{W^{s,2}(\Omega )}$ the space becomes a Hilbert space. Let us also set $W_0^{s,2} (\Omega ) = \overline{C_c^\infty (\Omega )}^{W^{s,2} (\Omega )}.$ It is well known that if $\Omega$ is bounded, then we have the following continuous embeddings:

$$\begin{aligned} W_0^{s,2} (\Omega ) \hookrightarrow {\left\{ \begin{array}{ll} L^{\frac{2N}{N-2s}}(\Omega ), &{}\quad \hbox {if} \quad s< \frac{N}{2}, \\ L^{p}(\Omega ), &{}\quad \hbox {if} \quad s = \frac{N}{2}, \\ C^{0,s-\frac{N}{2}}(\Omega ), &{}\quad \hbox {if} \quad s > \frac{N}{2}; \end{array}\right. } \end{aligned}$$

(2.6)

for more details on fractional Sobolev spaces see [17, 27] and the references therein.

For each number $s\ge 0$, we define

$$\begin{aligned} {\mathbb {X}}^{s}( \Omega ):= \left\{ v = \sum _{j=1}^\infty v_j \phi _j \in L^2(\Omega ): \left\| v\right\| _{{\mathbb {X}}^s(\Omega )}^2 = \sum _{j=1}^\infty v_j^2 \lambda _j^{s} <\infty \right\} , \quad v_j = \int \limits _\Omega v(x) \phi _j(x)\mathrm{d}x. \end{aligned}$$

(2.7)

Let us denote by $H^s(\Omega )$ the Sobolev–Slobodecki space $W^{s,p}(\Omega )$ when $p=2$, and by $H^s_0(\Omega )$ the closure of $C_c^\infty (\Omega )$ in $H^s(\Omega )$. Throughout this paper, $\Omega$ is assumed to be smooth enough such that $C_c^\infty ( \Omega )$ is dense in $H^s( \Omega )$ for $0<s< \frac{1}{2}$. This guarantees $H_0^s(\Omega )=H^{s}(\Omega )$. Moreover, it is well-known that

$$\begin{aligned} {\mathbb {X}}^{s}( \Omega ) = \left\{ \begin{array}{lllccc} H_0^{s}( \Omega ), &{}&{} \text {for } 0 \le s<\frac{1}{2},\\ H_{00}^{1/2} ( \Omega ) \subsetneqq H_0^{1/2}( \Omega ), &{}&{} \text {for } s= \frac{1}{2}, \\ H_0^{s}( \Omega ), &{}&{} \text {for } \frac{1}{2}< s \le 1,\\ H_{0}^{1} ( \Omega )\cap H^s( \Omega ), &{}&{} \text {for } 1 < s \le 2, \end{array} \right. \end{aligned}$$

where we denote by $H_{00}^{1/2} ( \Omega )$ the Lions–Magenes space. Let ${\mathbb {X}} ^{-s}( \Omega )$ be the duality of ${\mathbb {X}} ^{s}$ which corresponds to the dual inner product $\left( \cdot ,\cdot \right) _{-s,s}$. Then, the operator ${\mathcal {A}}^s: {\mathbb {X}}^{s}( \Omega ) \rightarrow {\mathbb {X}} ^{-s} (\Omega )$ of the fractional power s can be defined by

$$\begin{aligned} {\mathcal {A}}^s v := \sum _{j=1}^\infty \lambda _j^s \left( v, \phi _j \right) _{-s,s} \phi _j, \quad \forall v \in {\mathbb {X}} ^{s}. \end{aligned}$$

The above settings can be found in [28] (Sect. 3) and [29] (Sect. 2). In the next lemmas, we present some useful embeddings between the spaces mentioned above.

Lemma 2.1

Given $1 \le p , p' < \infty$, $0 \le s \le s' <\infty$ and $s' - \frac{N}{p'} \ge s - \frac{N}{p}$. Then

$$\begin{aligned} W^{s',p'}(\Omega ) \hookrightarrow W^{s,p}(\Omega ). \end{aligned}$$

(2.8)

Lemma 2.2

Let $0\le s \le s' \le 2$ and let $H^{-s}( \Omega )$ be the dual space of $H_0^s( \Omega )$. Then the following embeddings hold

$$\begin{aligned} {\mathbb {X}} ^s( \Omega ) \hookrightarrow L^2( \Omega ) \hookrightarrow {\mathbb {X}} ^{-s}( \Omega ), \end{aligned}$$

(2.9)

and

$$\begin{aligned} {\mathbb {X}} ^{s'} ( \Omega ) \hookrightarrow {\mathbb {X}} ^s ( \Omega ) \hookrightarrow H^s( \Omega ) \hookrightarrow L^2(\Omega ) \hookrightarrow H^{-s}( \Omega ) \hookrightarrow {\mathbb {X}} ^{-s} ( \Omega ) \hookrightarrow {\mathbb {X}} ^{-s'}( \Omega ). \end{aligned}$$

(2.10)

2.2 Properties of Mittag–Leffler functions and some related results

The Mittag–Leffler function is defined by (see [30])

$$\begin{aligned} E_{\alpha ,\alpha '} (z)= \sum _{j=0}^\infty \frac{z^j}{\varGamma (\alpha j + \alpha ')},\quad z \in {\mathbb {C}}, \end{aligned}$$

(2.11)

where $\alpha >0$ and $\alpha ' \in {\mathbb {R}}$ are arbitrary constants, $\varGamma$ is the usual gamma function.

Next, we give some properties of the Mittag–Leffler function. Let $\alpha '\in {\mathbb {R}},$ and $\alpha \in (0,2)$, we have:

$$\begin{aligned} \left| E_{\alpha ,\alpha '} (-z)\right| \le \frac{C}{1+|z|}, \quad \tau \le \arg (z) \le \pi , \end{aligned}$$

where $C>0$ depends on $\alpha , \alpha ',\tau$ and $\frac{\pi \alpha '}{2}< \tau < \min \{\pi ,\pi \alpha '\}$ (see e.g. [30]).

Lemma 2.3

(See [30, 31]) For $0< \alpha _1<\alpha _2 <1$ and $\alpha \in [\alpha _1,\alpha _2],$ there exist positive constants ${\underline{C}}, {\overline{C}},$ such that

$$\begin{aligned}&\mathrm{(a)}\quad E_{\alpha ,1} (-z)>0, \quad \hbox {for any} \quad z>0; \end{aligned}$$

(2.12a)

$$\begin{aligned}&\mathrm{(b)}\quad \dfrac{{\underline{C}}}{1+z} \le E_{\alpha ,\alpha '}(-z) \le \dfrac{{\overline{C}}}{1+z}, \quad \hbox {for} \quad \alpha ' \in \mathbb R,~~ z>0. \end{aligned}$$

(2.12b)

Lemma 2.4

(See [31]) Let $\alpha , \lambda , \gamma$ are positive constants, and for every $t>0, n \in {\mathbb {N}}$, we have

$$\begin{aligned}&\mathrm{(a)} \quad \frac{\mathrm {d}^{n}}{\mathrm {d} t^n} \left[ E_{\alpha ,1}(-\lambda t^\alpha )\right] =-\lambda t^{\alpha - n} E_{\alpha ,\alpha - n +1} (-\lambda t^\alpha ); \end{aligned}$$

(2.13a)

$$\begin{aligned}&\mathrm{(b)} \quad \left| \lambda ^{\gamma } t^{\alpha -1} E_{\alpha ,\alpha '} (-\lambda t^\alpha ) \right| \le C t^{\alpha -1- \alpha \gamma }. \end{aligned}$$

(2.13b)

Lemma 2.5

(See [32]) The following equality holds

$$\begin{aligned} E_{\alpha ,1}(-z) = \int \limits _0^\infty {\mathcal {M}}_\alpha (s) e^{-zs} \mathrm {d}s, \quad \hbox {for}\quad z \in {\mathbb {C}}, \end{aligned}$$

(2.14)

where we recall the definition of the Wright-type function

$$\begin{aligned} {\mathcal {M}}_\alpha (s) := \sum _{j=0}^\infty \frac{s^j}{j!\varGamma (-\alpha j + 1-\alpha )}, \quad 0< \alpha <1. \end{aligned}$$

(2.15)

Moreover, ${\mathcal {M}}_\alpha (s)$ is a probability density function, that is,

$$\begin{aligned} {\mathcal {M}}_\alpha (s) \ge 0, \quad \hbox {for}~ s>0; \quad \hbox {and} \quad \int \limits _0^\infty {\mathcal {M}}_\alpha (s) \mathrm {d} s =1. \end{aligned}$$

(2.16)

Lemma 2.6

(See [3], expression (6), for ${\mathcal {A}} = -\Delta$) The function u is a mild solution of ${{\mathbb {P}}}$ if $u \in C([0, T];L^2(\Omega ))$ and satisfies the following integral equation

$$\begin{aligned} u(t)=E_{\alpha ,1}(-t^\alpha {\mathcal {A}} )u_0 + \int \limits _0^t E_{\alpha ,1}(-(t-\tau )^\alpha {\mathcal {A}} ) F(u)(\tau ) \mathrm {d}\tau \end{aligned}$$

(2.17)

for all $t <T$, and $\alpha \in (0,1)$.

Lemma 2.7

(a) (Weakly singular Grönwall’s inequality, see [33], Theorem 1.2, page 2) Let $a,b, \beta , \beta '$ be non-negative constants and $\beta , \beta ' <1$. Assume that $\varphi \in L^1[0,T]$ satisfies

$$\begin{aligned} \varphi (t) \le a t^{-\beta } + b \int \limits _0^t (t-s)^{-\beta '} \varphi (s) \mathrm {d} s, \quad \hbox {for a.e.}\quad t \in (0,T]. \end{aligned}$$

(2.18)

Then there exists a constant $C(b, \beta ', T )$ such that

$$\begin{aligned} \varphi (t) \le C(b,\beta ', T ) \frac{at^{-\beta }}{1-\beta }, \quad \hbox {for a.e.}\quad t \in (0,T]. \end{aligned}$$

(2.19)

(b) (Fractional Grönwall’s inequality, see [34], Corollary 2) Assume $\beta >0$, $\varphi$ is nonnegative, locally integrable, and

$$\begin{aligned} \varphi (t) \le a + b \int \limits _0^t (t-\tau )^{\beta -1} \varphi (s) \mathrm {d}s, \end{aligned}$$

on (0, T), where a, b are positive constants. Then,

$$\begin{aligned} \varphi (t) \le a E_{\beta ,1} \big (b\Gamma (\beta )t^\beta \big ), \quad \text {on } (0,T). \end{aligned}$$

Lemma 2.8

(a) For $z >0$, then there exists a constant $C>0$ depending on $\theta$ such that

$$\begin{aligned} {\left\{ \begin{array}{ll} \big |\log z \big | \le C z^{-\theta }, &{}\quad \hbox {for}\quad \theta>0,\quad 0< z <1, \\ \big |\log z \big | \le Cz^{\theta }, &{} \quad \hbox {for}\quad \theta >0,\quad z\ge 1. \end{array}\right. } \end{aligned}$$

(2.20)

(b) (Hölder’s inequality for negative exponents) (see [35]) Let $k'<0$, and $k \in {\mathbb {R}}$ be such that $\frac{1}{k'} + \frac{1}{k} = 1$ and $f(x),g(x) \ge 0, ~ \forall x \in \Omega$ are Lebesgue measurable functions. Then

$$\begin{aligned} \int \limits _{\Omega } fg \mathrm {d}x \ge \left( \int \limits _{\Omega }|f|^{k'} \mathrm {d}x \right) ^{\frac{1}{k'}} \left( \int \limits _{\Omega } |g|^{k} \mathrm {d}x\right) ^{\frac{1}{k}}. \end{aligned}$$

(2.21)

Proof

The proof of inequalities (2.20) and (2.21) are elementary, so we omit them here. $\square$

Lemma 2.9

(See [17, 27]) Let $\Omega \subset {\mathbb {R}}^N$, $k,m \in {\mathbb {N}}$ with $k \ge m$ satisfying $(k-m)p <N$ and $1\le p <\infty$. Then we have the following Sobolev embeddings

$$\begin{aligned} \left. \begin{array}{llll} (SE1): &{} W^{k,p}(\Omega ) \hookrightarrow W^{m,q}(\Omega ),&{}\quad \hbox {for}&{}\quad 1\le q< p^*_{k,m}, \\ (SE2): &{} {\mathbb {X}}^{s}(\Omega ) \hookrightarrow H^{s}(\Omega ),&{}\quad \hbox {for}&{}\quad s>0,\\ (SE3): &{} L^p(\Omega ) \hookrightarrow {\mathbb {X}}^{s} (\Omega ),&{}\quad \hbox {for}&{}\quad -\frac{N}{2}< s \le 0, \quad p \ge 2^*_s, \\ (SE4): &{}{\mathbb {X}}^{s}(\Omega ) \hookrightarrow L^p(\Omega ), &{}\quad \hbox {for}&{}\quad 0\le s < \frac{N}{2}, \quad p \le 2^*_s, \end{array}\right\} \end{aligned}$$

(2.22)

where $p^*_{k,m}, 2^*_s$ are the so-called fractional Sobolev exponents, given by

$$\begin{aligned} \frac{1}{p^*_{k,m}} = \frac{1}{p} + \frac{m}{N} - \frac{k}{N}, \quad \hbox {and}\quad \frac{1}{2^*_s} = \frac{1}{2} - \frac{s}{N}. \end{aligned}$$

(2.23)

3 Main results

3.1 The case when the source terms are globally Lipchitz functions

In this section, we will study the existence and uniqueness of mild solutions to problem ${{\mathbb {P}}}$. First we assume the global Lipschitz continuity and the time Hölder continuity on the nonlinear term. More precisely, we suppose that $F:{\mathbb {X}}^{p}(\Omega ) \rightarrow {\mathbb {X}}^{q}(\Omega )$, $F(0)=0$, and

$$\begin{aligned} \big \Vert F( v_1)-F( v_2) \big \Vert _{{\mathbb {X}}^{q}(\Omega )} \le K \big \Vert v_1- v_2 \big \Vert _{{\mathbb {X}}^{p}(\Omega )}, \end{aligned}$$

(3.1)

where $K:[0,T] \rightarrow {\mathbb {R}}_+$ and p, q are real numbers.

Our results in this section present the local well-posedness of the problem. Here, ${\mathbb {Z}}_{\beta ,d}((0,T];{\mathbb {X}}^q(\Omega ))$ denotes the weighted space of all functions $v\in C ((0,T];{\mathbb {X}}^q(\Omega ))$ such that

$$\begin{aligned} \Vert v\Vert _{{\mathbb {Z}}_{\beta ,d}((0,T];{\mathbb {X}}^q(\Omega ))}: = \sup _{t\in (0,T]} t^{\beta } e^{-d t} \Vert v(t,\cdot )\Vert _{{\mathbb {X}}^q(\Omega )} <\infty \end{aligned}$$

where $\beta>0,~d>0$. First we state the following lemma which will be useful in our main results. (This lemma can be found in [36], Lemma 8, page 9.)

Lemma 3.1

Let $a>-1$, $b>-1$ such that $a+b \ge -1$, $h>0$ and $t\in [0,T]$. For $\mu >0$, the following limit holds

$$\begin{aligned} \lim \limits _{\mu \rightarrow \infty } \left( \sup _{t\in [0,T]} t^{h} \int \limits _0^1 s^{a} (1-s)^{b} e^{ -\mu t (1-s)} \mathrm {d}s \right) = 0. \end{aligned}$$

Now, we are in the position to introduce the main contributions of this work. Our main results address the existence and regularity of the mild solution.

Theorem 3.1

Let $0<\beta <1$. Assume that $q-p <\min \left\{ \frac{2(1-\beta )}{\alpha },\frac{2\beta }{\alpha } \right\}$. Let $u_0 \in {\mathbb {X}}^{q-2\gamma }(\Omega ))$ for any $0<\gamma <\min \left\{ \frac{\beta }{\alpha };1\right\}$. Then Problem ${{\mathbb {P}}}$ has a unique solution u in ${\mathbb {Z}}_{\beta ,d_0}((0,T];{\mathbb {X}}^q(\Omega ))$ with some $d_0>0$. Moreover, there exist positive constant C independently of t, x and for $1/2<\beta <1$, $1-\beta<\alpha <1/2$ such that

$$\begin{aligned} \big \Vert u(\cdot ,t) \big \Vert _{ {\mathbb {X}}^p(\Omega )) } \le C t^{-\beta } e^{ d t} \big \Vert u_0 \big \Vert _{ {\mathbb {X}}^{q-2\gamma }(\Omega ) }. \end{aligned}$$

(3.2)

Proof

Define the mapping ${\mathfrak {B}}: \mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega )) \rightarrow \mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega ))$, $d>0$, by

$$\begin{aligned} {\mathfrak {B}} w (t)&:=E_{\alpha ,1}(-t^\alpha {\mathcal {A}} )u_0 + \int \limits _0^t E_{\alpha ,1}(-(t-\tau )^\alpha {\mathcal {A}} ) F(w)(\tau )\mathrm {d}\tau . \end{aligned}$$

(3.3)

In what follows, we shall prove the existence of a unique solution of Problem ${{\mathbb {P}}}$. This is based on the Banach principal argument. First, since $0<\gamma <1$, we have

$$\begin{aligned} \Big \Vert E_{\alpha ,1}(-t^\alpha {\mathcal {A}} )u_0 \Big \Vert _{{\mathbb {X}}^p(\Omega ))}^2&= \sum _{j=1}^\infty \left( u_0, \phi _j\right) ^2 \left( E_{\alpha ,1}\left( - \lambda _j t^{\alpha } \right) \right) ^2 \lambda _j^{p}\nonumber \\&\le \sum _{j=1}^\infty \left( u_0, \phi _j\right) ^2 \frac{C^2}{(1+\lambda _j t^\alpha )^{2\gamma }} \lambda _j^{p} \le C^2 t^{-2\alpha \gamma } \sum _{j=1}^\infty \left( u_0, \phi _j\right) ^2 \lambda _j^{p-2\gamma }. \end{aligned}$$

(3.4)

It follows from the condition $\beta >\alpha \gamma$ that

$$\begin{aligned} t^\beta e^{-dt}\Big \Vert E_{\alpha ,1}(-t^\alpha {\mathcal {A}} )u_0 \Big \Vert _{{\mathbb {X}}^p(\Omega ))} \le C t^{\beta -\alpha \gamma } \Vert u_0\Vert _{{\mathbb {X}}^{p-2\gamma }(\Omega ))} \le C T^{\beta -\alpha \gamma } \Vert u_0\Vert _{{\mathbb {X}}^{p-2\gamma }(\Omega ))}. \end{aligned}$$

(3.5)

From the latter inequality, we deduce that $u_0 \in \mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega ))$. Indeed, for $w_1,w_2\in {\mathbb {Z}}_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega ))$, we have

$$\begin{aligned}&\big \Vert {\mathfrak {B}} w_1 - {\mathfrak {B}} w_2 \big \Vert _{\mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega )) } \nonumber \\&\quad = \sup _{t\in (0,T]} t^{\beta } e^{-d t} \Big \Vert \int \limits _0^t E_{\alpha ,1}( - (t-\tau )^\alpha {\mathcal {A}}) \big [ F(w_1(\tau )) - F(w_2 (\tau )) \big ] \mathrm {d}\tau \Big \Vert _{{\mathbb {X}}^p(\Omega )} \nonumber \\&\quad \le \sup _{t\in (0,T]} t^{\beta } e^{-d t} \int \limits _0^t \Big \Vert E_{\alpha ,1}( - (t-\tau )^\alpha {\mathcal {A}}) \big [ F(w_1(\tau )) - F(w_2 (\tau )) \big ] \Big \Vert _{{\mathbb {X}}^p(\Omega )} \mathrm {d}\tau \nonumber \\&\quad \le C \sup _{t\in (0,T]} t^{\beta } e^{-d t} \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \Big \Vert F(w_1(\tau )) - F(w_2(\tau )) \Big \Vert _{{\mathbb {X}}^q(\Omega )} \mathrm {d}\tau \nonumber \\&\quad \le CK \big \Vert v_1 - v_2 \big \Vert _{\mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega )) } \sup _{t\in (0,T]} t^{\beta } \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \tau ^{-\beta } e^{ -d (t-\tau )} \mathrm {d}\tau . \end{aligned}$$

(3.6)

We derive the estimate

$$\begin{aligned} \big \Vert {\mathfrak {B}} w_1 - {\mathfrak {B}} w_2 \big \Vert _{\mathbb Z_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega )) }&\le {\mathscr {L}}_d \big \Vert v_1 - v_2 \big \Vert _{{\mathbb {Z}}_{\beta ,d}((0,T];{\mathbb {X}}^p(\Omega )) }, \end{aligned}$$

where

$$\begin{aligned} {\mathscr {L}}_d = t^{\beta } \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \tau ^{-\beta } e^{ -d (t-\tau )} \mathrm {d}\tau . \end{aligned}$$

From the conditions of $\alpha ,\beta , p, q$, we find that

$$\begin{aligned} \beta -\frac{\alpha (q-p)}{2}>0,\quad -\frac{\alpha (q-p)}{2}>-1,\quad -\beta>-1, \quad -\frac{\alpha (q-p)}{2}-\beta >-1. \end{aligned}$$

Applying Lemma 3.1, we obtain that

$$\begin{aligned} \lim \limits _{d\rightarrow \infty } {\mathscr {L}}_d&:= K \lim \limits _{\mu \rightarrow \infty } \left( \sup _{t\in (0,T]} t^{\beta } \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \tau ^{-\beta } e^{ -(t-\tau )d } \mathrm {d}\tau \right) \nonumber \\&= K \lim \limits _{d\rightarrow \infty } \left( \sup _{t\in (0,T]} t^{\beta -\frac{\alpha (q-p)}{2}} \int \limits _0^1 (1-\tau )^{-\frac{\alpha (q-p)}{2}} \tau ^{-\beta } e^{ - t(1-\tau )d} \mathrm {d}\tau \right) \nonumber \\&= 0. \end{aligned}$$

(3.7)

Hence, there exists a positive $d>0$ such that ${\mathfrak {B}}$ is a contraction mapping on $\mathbb Z_{\beta ,d_0}((0,T];{\mathbb {X}}^p(\Omega ))$. This together with (3.5) leads to ${\mathfrak {B}} w \in \mathbb Z_{\beta ,d_0}((0,T];{\mathbb {X}}^p(\Omega ))$ if $w \in \mathbb Z_{\beta ,d_0}((0,T];{\mathbb {X}}^p(\Omega ))$. Hence, we conclude that ${\mathfrak {B}}$ has a fixed point u in $\mathbb Z_{\beta ,d_0}((0,T];{\mathbb {X}}^p(\Omega ))$, i.e, u is a unique mild solution of Problem ${{\mathbb {P}}}$.

This and the technique in (3.6) yields

$$\begin{aligned} \big \Vert u(t,\cdot ) \big \Vert _{ {\mathbb {X}}^{p}(\Omega ) }&\le \Big \Vert E_{\alpha ,1}(-t^\alpha {\mathcal {A}} )u_0 \Big \Vert _{{\mathbb {X}}^p(\Omega ))} + \int \limits _0^t \Big \Vert E_{\alpha ,1}( - (t-\tau )^\alpha {\mathcal {A}}) F(u(\tau )) \Big \Vert _{ {\mathbb {X}}^{p}(\Omega )} \mathrm {d}\tau \nonumber \\&\le C t^{-\alpha \gamma } \Vert u_0\Vert _{{\mathbb {X}}^{p-2\gamma }(\Omega ))} + CK \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \big \Vert u(\tau )) \big \Vert _{ {\mathbb {X}}^{p}(\Omega )} \mathrm {d}\tau . \end{aligned}$$

(3.8)

Multiplying both sides to $t^{\beta } e^{-dt}$, we find that

$$\begin{aligned} t^{\beta } e^{-dt} \big \Vert u(t,\cdot ) \big \Vert _{ {\mathbb {X}}^{p}(\Omega ) } \le C e^{-dt} t^{\beta -\alpha \gamma } \Vert u_0\Vert _{{\mathbb {X}}^{p-2\gamma }(\Omega ))}+ CKt^{\beta } e^{-dt} \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \big \Vert u(\tau )) \big \Vert _{ {\mathbb {X}}^{p}(\Omega )} \mathrm {d}s. \end{aligned}$$

(3.9)

By applying the Hölder inequality, and then using $e^{-2d(t-\tau )}<1$, we can find some positive constant ${\mathscr {M}}$ such that

$$\begin{aligned}&t^{\beta } e^{-dt} \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \big \Vert u(\tau )) \big \Vert _{ {\mathbb {X}}^{p}(\Omega )} \mathrm {d}\tau \nonumber \\&\quad \le \left( t^{2\beta } \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \tau ^{-2\beta } e^{-2d(t-\tau )} \mathrm {d}\tau \right) ^{\frac{1}{2}} \left( \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \Big ( \tau ^{ \beta } e^{-d \tau } \big \Vert u(s,\cdot )) \big \Vert _{ {\mathbb {X}}^{p}(\Omega )} \Big )^2 \mathrm {d}\tau \right) ^{\frac{1}{2}} \nonumber \\&\quad \le \left( t^{2\beta } t^{-\frac{\alpha (q-p)}{2}} \int \limits _0^1 (1-\tau )^{-\frac{\alpha (q-p)}{2}} \tau ^{-2\beta } \mathrm {d}\tau \right) ^{\frac{1}{2}} \left( \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \Big ( \tau ^{ \beta } e^{-d \tau } \big \Vert u(\tau ,\cdot )) \big \Vert _{ {\mathbb {X}}^{p}(\Omega )} \Big )^2 \mathrm {d}\tau \right) ^{\frac{1}{2}} \nonumber \\&\quad \le {\mathscr {M}} \left( \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \Big ( \tau ^{ \beta } e^{-d \tau } \big \Vert u(\tau ,\cdot )) \big \Vert _{ {\mathbb {X}}^{\sigma }(\Omega )} \Big )^2 \mathrm {d}\tau \right) ^{\frac{1}{2}}. \end{aligned}$$

(3.10)

Taking the estimate (3.8), and (3.10) together gives that

$$\begin{aligned} {\mathscr {U}}_{\beta ,d}(t) \le 2 C^2 e^{-2dt} t^{2\beta -2\alpha \gamma } \Vert u_0\Vert ^2_{{\mathbb {X}}^{p-2\gamma }(\Omega ))} + |{\mathscr {M}} |^2 C^2K^2 \int \limits _0^t (t-\tau )^{-\frac{\alpha (q-p)}{2}} \mathscr {U}_{\beta ,d}(\tau ) \mathrm {d}\tau , \end{aligned}$$

(3.11)

where

$$\begin{aligned} {\mathscr {U}}_{\beta ,d}(t):= \Big ( t^{\beta } e^{-dt} \big \Vert u(t,\cdot ) \big \Vert _{ {\mathbb {X}}^{p}(\Omega ) } \Big )^2. \end{aligned}$$

Applying Lemma 2.7(b), we deduce that

$$\begin{aligned} {\mathscr {U}}_{\beta ,d}(t) \le 2 C^2 T^{2\beta -2\alpha \gamma } \Vert u_0\Vert ^2_{{\mathbb {X}}^{p-2\gamma }(\Omega ))} E_{1-\frac{\alpha (q-p)}{2},1} \left( |{\mathscr {M}} |^2 C^2K^2 \Gamma \left( 1-\frac{\alpha (q-p)}{2}\right) t^{1-\frac{\alpha (q-p)}{2}} \right) . \end{aligned}$$

(3.12)

The proof of Theorem 3.1 is completed. $\square$

3.2 The case when the source terms are locally Lipschitz functions

Next, we shall present the results when the source terms are logarithmic nonlinearities of the following type $F_{p}(u) = \eta V_{p}(u) \log |u|$ and $V_p(u) = |u|^{p-2}u$, for $p \ge 2, \eta >0$.

Remark 3.1

For the source terms of polynomial type nonlinearities, i.e., $F_p(u) = V_p(u)$ a simpler result was considered in [2, 5].

Lemma 3.2

For $F_p(u)(x,t) = \eta V_p(u)\log |u| \in L^\infty (\Omega \times (0,T) \times {\mathbb {R}}), ~ p\ge 2, \eta >0$, there exists a positive constant C such that

$$\begin{aligned} \left| F_p(u) - F_p(w) \right|&\le C \left( \big |\log |u| \big | \left| u -w\right| + |u|^{p-2} \big |\log |u| \big | \left| u -w\right| \right) \nonumber \\&\quad + C \left( |w|^{p-2} \big |\log |u| \big | \left| u -w\right| + \left| w\right| ^{p-2} \left| u -w\right| \right) , \end{aligned}$$

(3.13)

for all $(x,t) \in \Omega \times (0,T),~ \forall u, w \in \mathbb R$.

Proof

For $(x,t) \in \Omega \times (0,T)$ and $u,w \in {\mathbb {R}}$, we have

$$\begin{aligned} \big |F_p(u) - F_p(w) \big |&= \eta \big |V_p(u)\log |u| - V_p(w)\log |w| \big | \nonumber \\&\le \eta \left( \big |V_p(u) - V_p(w) \big | \big | \log |u| \big | + \left| V_p(u)\right| \big |\log |u| - \log |w| \big | \right) . \end{aligned}$$

(3.14)

Thanks to the results in [5], we have that

$$\begin{aligned} \big |V_p(u) - V_p(w) \big |&= \Big |u|^{p-2}u - |w|^{p-2}w\Big | \nonumber \\&\le C(1 + |u|^{p-2} + |w|^{p-2})|u-w|. \end{aligned}$$

(3.15)

Using the basic inequality $\log (1+z) < z$ for $z>0$, one has

$$\begin{aligned} \big |\log |u| - \log |w|\big |&= \left| \log \left| 1+\frac{|u|-|w|}{|w|}\right| \right| \nonumber \\&< \left| \log \left( 1+\frac{|u-w|}{|w|}\right) \right| < \frac{|u-w|}{|w|}. \end{aligned}$$

(3.16)

From (3.14)–(3.16), we have the proof of Lemma 3.2. $\square$

Theorem 3.2

(Local existence) Let $\alpha \in \left( 0, 1\right)$, $N\ge 1, p\ge 2,0\le s < s_2$, for $0\le s_2 <N/2$. Let $1 \le q \le \min \left\{ 2^\star _{s_2,s};\frac{N \theta }{N+\theta s} \right\}$ with $2^\star _{s_2,s}$ satisfying $\frac{1}{2^\star _{s_2,s}} = \frac{1}{2} + \frac{s}{N} - \frac{s_2}{N}$ and $qs < N$. Let $u_0 \in {\mathbb {X}}^{s_2}(\Omega ) \cap W^{s,q}(\Omega )$, and for the nonlinearity source of logarithmic function type

$$\begin{aligned} F_p (u) = \eta V_p(u) \log |u|,\quad \hbox {for}~~ V_p(u) = |u|^{p-2}u, \quad \hbox {with}~ p \ge 2,\eta >0, \end{aligned}$$

then there is a time constant $T>0$ (depending only on $u_0$) such that Problem ${{\mathbb {P}}}$ has a unique mild solution belonging to $C([0,T];W^{s,q}(\Omega ))$.

Remark 3.2

In Theorem 3.2, for $N\ge 1$, and $0\le s_2 <N/2$ let us choose $N = 3, s_2 =1$. From the conditions

$$\begin{aligned} {\left\{ \begin{array}{ll} s < 2s_2, s \in {\mathbb {N}},\\ 1 \le q \le \frac{2N}{N+2s-2s_2}, \end{array}\right. } \hbox {this implies that}~~ {\left\{ \begin{array}{ll} s=0, \quad q \in [1,6], \quad \hbox {or}, \\ s=1, \quad q \in [1,2]. \end{array}\right. } \end{aligned}$$

(3.17)

Then, the Problem ${{\mathbb {P}}}$ has a unique mild solution $u \in C([0,T]; L^{q}(\Omega )), ~ 1 \le q \le 6$, or $u \in C([0,T]; W^{1,q}(\Omega ))$, for $1 \le q \le 2$.

Proof

For $N\ge 1, p\ge 2$, $0< \theta \le p-1$ ($\theta$ is defined in Lemma 2.8), we put

$$\begin{aligned}&0 \le s_2 < \min \left\{ 1; \frac{(p-1)N}{2p}\right\} , \quad s_1 = p s_2 - s^*, \end{aligned}$$

(3.18)

$$\begin{aligned}&s \in {\mathbb {N}} \quad \hbox {satisfies}\quad s < s_2, \quad 1\le q \le \min \left\{ 2^*_{s_2,s}; \frac{N \theta }{N+\theta s}\right\} , \end{aligned}$$

(3.19)

$$\begin{aligned}&\max \left\{ ps_2; Z (a,b) \right\}< s^* < \min \left\{ ps_2 + \frac{N}{2}; 1+(p-1)s_2 \right\} , \end{aligned}$$

(3.20)

where $2^*_{s_2,s} = \frac{1}{2} + \frac{s}{N} - \frac{s_2}{N},$ and Z(a, b) be defined by

$$\begin{aligned} Z(a, b) = \frac{N(2pa-qb) + 2pq (s_2 b - s a)}{2qb}, \end{aligned}$$

(3.21)

with the pairs (a, b) as follows:

$$\begin{aligned}(a, b) \in \left\{ (1,1), (\theta ,1), (p-2,1), (\theta , p-1), (\theta ,p-1), (p-2, p-1)\right\} , \quad \hbox {for} ~~\theta >0.\end{aligned}$$

Let $T>0$ and $R>0$ to be chosen later, and we consider the following space

$$\begin{aligned} {\mathbb {W}}:=&\bigg \{u\in C([0,T]; W^{s,q}(\Omega )): u(\cdot ,0) = u_0, ~~\hbox {and}~~ \big \Vert u(\cdot ,t) - u_0\big \Vert _{W^{m,q}(\Omega )} \le R \bigg \}, \end{aligned}$$

(3.22)

for $0<\alpha <1$, and we define the mapping ${\mathbf {M}}$ on $\mathbb W$ by

$$\begin{aligned} {\mathbf {M}}{\mathbf {u}}(t)&= E_{\alpha ,1}(-t^\alpha {\mathcal {A}} )u_0 + \int \limits _0^t E_{\alpha ,1}(-(t-\tau )^\alpha {\mathcal {A}} ) F_p(u)(\tau ) \mathrm {d}\tau \nonumber \\&:= E_{\alpha ,1}(-t^\alpha {\mathcal {A}} )u_0 + J(F(u))(t). \end{aligned}$$

(3.23)

We show that ${\mathbf {M}}$ is invariant in ${\mathbb {W}}$ and ${\mathbf {M}}$ is a contraction.

$\bullet$ Claim I If $u_0 \in {\mathbb {X}}^{s_2}(\Omega ) \cap W^{s,q}(\Omega )$, then ${\mathbf {M}}$ is ${\mathbb {W}}$-invariant. In fact, from Lemma 2.3(b), we have

$$\begin{aligned} \left\| E_{\alpha ,1}(- {\mathcal {A}} t^\alpha )u_0 - u_0 \right\| _{{\mathbb {X}}^{s_2}(\Omega )}^2&= \sum _{j=1}^\infty \left( u_0, \phi _j\right) ^2 \left( E_{\alpha ,1}\left( - \lambda _j t^{\alpha } \right) - 1 \right) ^2 \lambda _j^{s_2} \nonumber \\&\le 2({\overline{C}}^2 +1) \sum _{j=1}^\infty \left( u_0, \phi _j\right) ^2 \lambda _j^{s_2} \nonumber \\&\le C \left\| u_0\right\| _{{\mathbb {X}}^{s_2}(\Omega )}^2, \quad \forall t \in (0,T]. \end{aligned}$$

(3.24)

From (3.19), one has $s < s_2$ and $1\le q \le 2^*_{s_2,s}$, and we have that ${\mathbb {X}}^{s_2}(\Omega ) \hookrightarrow H^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )$ and then, we conclude from (3.24) that

$$\begin{aligned} \left\| E_{\alpha ,1}\left( - {\mathcal {A}} t^{\alpha } \right) u_{0} - u_0 \right\| _{W^{s,q}(\Omega )} \le C \left\| u_0\right\| _{\mathbb X^{s_2}(\Omega )}, \quad t \in (0,T]. \end{aligned}$$

(3.25)

From (3.20), we have $ps_2< s^* < 1+ (p-1)s_2$, this implies that $0< s_2 - s_1 <1$ and for $ps_2< s^* < p s_2 + \frac{N}{2},~$ or $~- \frac{N}{2}< p s_2 - s^* < 0$ thus $-\frac{N}{2}< s_1 < 0.$ Taking $\frac{1}{2^*_{s_1} } = \frac{1}{2} - \frac{s_1}{N}$ and combine with Lemma 2.9, we obtain $L^{2^*_{s_1}}(\Omega ) \hookrightarrow {\mathbb {X}}^{s_1}(\Omega )$. Using Lemma 2.3(b), we have for $t \in (0,T]$

$$\begin{aligned} \big \Vert J(F(u))(t) \big \Vert _{{\mathbb {X}}^{s_2}(\Omega )}&\le \int \limits _0^t \left\| E_{\alpha ,1}(-(t-\tau )^\alpha {\mathcal {A}} ) F_p(u)(\tau ) \right\| _{{\mathbb {X}}^{s_2}(\Omega )} \mathrm {d}\tau \nonumber \\&\le \int \limits _0^t \left( \sum _{j=1}^\infty \lambda _j^{s_2 - s_1} \left( F_p(u)(\cdot ,\tau ), \phi _j\right) ^2 \lambda _j^{s_1} \left( \frac{{\overline{C}}}{1+\lambda _j(t-\tau )^\alpha }\right) ^2 \right) ^{\frac{1}{2}} \mathrm {d}\tau \nonumber \\&\le \frac{{\overline{C}}}{ \lambda _1^{\frac{2-s_2 + s_1}{2}}} \int \limits _0^t (t -\tau )^{-\alpha } \big \Vert F_{p}(u)(\cdot ,\tau )\big \Vert _{\mathbb X^{s_1}(\Omega )} \mathrm {d}\tau \nonumber \\&\le C \int \limits _0^t (t -\tau )^{-\alpha } \big \Vert F_{p}(u)(\cdot ,\tau )\big \Vert _{L^{2^*_{s_1}}(\Omega )} \mathrm {d}\tau . \end{aligned}$$

(3.26)

Let us set $\Omega ^-:= \{x\in \Omega : |u(x)| < 1\}$ and $\Omega ^+:= \{x\in \Omega : |u(x)| \ge 1\}$. Using Hölder’s inequality, we have

$$\begin{aligned}&\int \limits _{\Omega } \big |F_p(u)\big |^{2^*_{s_1}} \mathrm {d}x \le \eta ^{2^*_{s_1}} \int \limits _{\Omega } \left| u\right| ^{(p-1)2^*_{s_1}} \big |\log |u | \big |^{2^*_{s_1}} \mathrm {d}x \nonumber \\&\quad \le C \left( \int \limits _{\Omega } \left| \log \big |u\big | \right| ^{p 2^*_{s_1}} \mathrm {d}x \right) ^{\frac{1}{p}} \left( \int \limits _{\Omega } \big |u\big |^{p2^*_{s_1}} \mathrm {d}x \right) ^{\frac{p-1}{p}} \nonumber \\&\quad \le C \left( \int \limits _{\Omega ^-} \big |\log |u|\big |^{p 2^*_{s_1} } \mathrm {d}x + \int \limits _{\Omega ^+ } \big |\log |u|\big |^{p 2^*_{s_1}} \mathrm {d}x \right) ^{\frac{1}{p}} \left( \int \limits _{\Omega } \big |u\big |^{p 2^*_{s_1}} \mathrm {d}x\right) ^{\frac{p-1}{p}} \nonumber \\&\quad \le C \left( \left( \int \limits _{\Omega ^-} \big |\log |u|\big |^{p 2^*_{s_1} } \mathrm {d}x \right) ^{\frac{1}{p}} + \left( \int \limits _{\Omega ^+ } \big |\log |u|\big |^{p 2^*_{s_1}} \mathrm {d}x \right) ^{\frac{1}{p}} \right) \left( \int \limits _{\Omega } \big |u\big |^{p 2^*_{s_1}} \mathrm {d}x\right) ^{\frac{p-1}{p}}, \end{aligned}$$

(3.27)

where we have used the elementary inequality $(a+b)^c \le a^c+b^c$, for $0<c<1$. From the inequality (2.20) for $|u(x)|<1, ~ \forall x \in \Omega$, by applying Lemma 2.8b) for $k' = -\frac{1}{p2_{s_1}^*} <0$, we have

$$\begin{aligned} \left( \int \limits _{\Omega ^-} \big |\log |u|\big |^{p 2_{s_1}^* } \mathrm {d}x \right) ^{\frac{1}{p}}&\le C \left( \int \limits _{\Omega ^-} \big |u(x)\big |^{- p\theta 2_{s_1}^*} \mathrm {d}x \right) ^{\frac{1}{p}} \nonumber \\&\le C\left( \left( \int \limits _{\Omega ^-} \big |u(x)\big |^{-p\theta 2_{s_1}^* } \mathrm {d}x \right) ^{-\frac{1}{p 2_{s_1}^*}} \right) ^{- 2_{s_1}^*} \nonumber \\&\le C \left( \left( \int \limits _{\Omega ^-} |u(x)|^{\theta } \mathrm {d}x\right) \left( \int \limits _{\Omega ^-} 1 \mathrm {d}x \right) ^{-\frac{1 +p 2_{s_1}^*}{p 2_{s_1}^*}} \right) ^{- 2_{s_1}^*} \nonumber \\&\le C \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta 2_{s_1}^*} |\Omega |^{\frac{1 +p 2_{s_1}^*}{p }}, \end{aligned}$$

(3.28)

From the inequality (2.20) for $|u(x)| \ge 1$, we have

$$\begin{aligned} \left( \int \limits _{\Omega ^+} \big |\log |u|\big |^{p 2_{s_1}^*} \mathrm {d}x \right) ^{\frac{1}{p}} \le C \left( \int \limits _{\Omega ^+} \big |u(x)\big |^{p\theta 2_{s_1}^*} \mathrm {d}x \right) ^{\frac{1}{p}} \le C \big \Vert u\big \Vert _{L^{p\theta 2_{s_1}^*}(\Omega )}^{\theta 2_{s_1}^*}. \end{aligned}$$

(3.29)

From (3.27), (3.28) and (3.29), we conclude that

$$\begin{aligned}&\big \Vert F_{p}(u) \big \Vert _{L^{2^*_{s_1}}(\Omega )} \le C\left( \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta } + \big \Vert u\big \Vert _{L^{p\theta 2^*_{s_1}}(\Omega )}^{\theta } \right) \big \Vert u\big \Vert _{L^{p 2_{s_1}^* }(\Omega )}^{p-1}. \end{aligned}$$

(3.30)

For $s>0, qs <N$, and $q < \frac{N\theta }{N+\theta s}$, this implies that $q^*_s \le \theta$ with $q^*_s$ satisfies

$$\begin{aligned}\frac{1}{q^*_s} = \frac{1}{q} - \frac{s}{N},\end{aligned}$$

we deduce from Lemma 2.9 that the following Sobolev embedding holds $L^{\theta }(\Omega ) \hookrightarrow W^{s,q}(\Omega )$. Then we get that

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

and for $\theta >0$, we have that

$$\begin{aligned} \big \Vert u\big \Vert _{L^{\theta }(\Omega )}^{-\theta } \le C \big \Vert u\big \Vert _{W^{s,q}(\Omega )}^{-\theta }. \end{aligned}$$

From (3.21), the constant $s^* > Z(1,1),$ [for Z(1, 1) defined in (3.21)] and observe that

$$\begin{aligned} p 2^*_{s_1} = \frac{2Np}{N-2s_1} = \frac{2Np}{N+2s^* -2 p s_2} < \frac{2Np}{N+2 Z(1,1) -2 p s_2} = \frac{Nq}{N-sq} =q^*_s, \end{aligned}$$

(3.31)

then we also obtain $W^{s,q}(\Omega ) \hookrightarrow L^{p2^*_{s_1}}(\Omega )$. For $s^* > Z(\theta ,1)$ [for $Z(\theta ,1)$ defined in (3.21)], we infer that

$$\begin{aligned} p \theta 2^*_{s_1} = p \theta \frac{2N}{N-2s_1} = \frac{2Np}{N+2s^* -2 p s_2} = p \theta \frac{2N}{N+2 Z(\theta ,1) -2 p s_2} \le q^*_s, \end{aligned}$$

this implies that $W^{s,q}(\Omega ) \hookrightarrow L^{p \theta 2^*_{s_1} }(\Omega )$. This implies that

$$\begin{aligned} \left\| F_p(u) \right\| _{L^{2^*_{s_1}}(\Omega )} \le C\left( \big \Vert u\big \Vert _{W^{s,q}(\Omega )}^{-\theta } + \big \Vert u\big \Vert _{W^{s,q}(\Omega )}^{\theta } \right) \big \Vert u\big \Vert _{W^{s,q}(\Omega )}^{p-1}, \end{aligned}$$

and from (3.26), and for $0< \theta <p-1$, we have

$$\begin{aligned} {\underline{\hbox {The (RHS) of}}}\; (3.26)&\le C \int \limits _0^t (t -\tau )^{-\alpha } \left( \left\| u(\cdot ,\tau )\right\| _{W^{s,q}(\Omega )}^{-\theta } + \big \Vert u(\cdot ,\tau )\big \Vert _{W^{s,q}(\Omega )}^{\theta } \right) \big \Vert u(\cdot ,\tau )\big \Vert _{W^{s,q}(\Omega )}^{p-1} \mathrm {d}\tau \nonumber \\&\le C \int \limits _0^t (t -\tau )^{-\alpha } \left( \left\| u(\cdot ,\tau )\right\| _{W^{s,q}(\Omega )}^{p-1-\theta } + \big \Vert u(\cdot ,\tau )\big \Vert _{W^{s,q}(\Omega )}^{p-1+\theta } \right) \mathrm {d}\tau \nonumber \\&\le C \left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) \int \limits _0^t (t -\tau )^{-\alpha } \mathrm {d}\tau \nonumber \\&\le C \left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) \frac{t^{1-\alpha }}{1-\alpha }, \end{aligned}$$

(3.32)

where from (3.22), we have that $\big \Vert u(\cdot ,\tau )\big \Vert _{W^{s,q}(\Omega )} \le R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )}$, for all $\tau \in [0,T]$. From (3.26), (3.32), we obtain that for $t \in (0,T]$

$$\begin{aligned} \big \Vert J(F(u))(t) \big \Vert _{{\mathbb {X}}^{s_2}(\Omega )} \le C \left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) t^{1-\alpha }. \end{aligned}$$

(3.33)

For the constants s, q satisfying (3.19), we have that ${\mathbb {X}}^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )$ and $\alpha \in (0,1)$, and for all $t \in [0,T]$, we get

$$\begin{aligned} \big \Vert J(F(u))(t) \big \Vert _{W^{s,q}(\Omega )} \le C \left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) T^{1- \alpha }. \end{aligned}$$

(3.34)

Hence, from (3.24) and (3.34), for every $t\in (0, T]$,

$$\begin{aligned}&\left\| {\mathbf {M}}{\mathbf {u}}(t) - u_0 \right\| _{ W^{s,q}(\Omega )} \nonumber \\&\quad \le C \left\| u_0\right\| _{{\mathbb {X}}^{s_2}(\Omega )} + C\left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) T^{1-\alpha }. \end{aligned}$$

(3.35)

Therefore we see that if $R = 2 C \left\| u_0\right\| _{\mathbb X^{s_2}(\Omega )}$ and for the constant $C>0$, $\theta < p-1$ such that

$$\begin{aligned} R \ge 2C \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } T^{1- \alpha }, \end{aligned}$$

and

$$\begin{aligned} R \ge 2C\left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } T^{1- \alpha }. \end{aligned}$$

Then, we imply ${\mathbf {M}}$ is invariant in ${\mathbb {W}}$.

$\bullet$ Claim II ${\mathbf {M}}: {\mathbb {W}} \rightarrow {\mathbb {W}}$ is a contraction map. Let $u,w\in {\mathbb {W}}$, and similar to (3.26) and using Lemma 2.3(b), one has for every $t\in (0,T]$,

$$\begin{aligned} \left\| {\mathbf {M}}{\mathbf {u}}(t) - {\mathbf {M}}{\mathbf {w}}(t)\right\| _{\mathbb X^{s_2}(\Omega )}&= \left\| J(F(u))(t) -J(F(w))(t)\right\| _{\mathbb X^{s_2}(\Omega )} \nonumber \\&\le C \int \limits _0^t (t -\tau )^{-\alpha } \big \Vert F_{p}(u)(\cdot ,\tau ) - F_{p}(w)(\cdot ,\tau )\big \Vert _{{\mathbb {X}}^{s_1}(\Omega )} \mathrm {d}\tau \nonumber \\&\le C \int \limits _0^t (t - \tau )^{-\alpha } \left\| F_p(u)(\cdot ,\tau ) - F_p(w) (\cdot ,\tau )\right\| _{L^{2^*_{s_1}}(\Omega )} \mathrm {d}\tau , \end{aligned}$$

(3.36)

in which we used the Sobolev embedding $L^{2^*_{s_1}}(\Omega ) \hookrightarrow {\mathbb {X}}^{s_1}(\Omega )$, for $\frac{1}{2^*_{s_1}} = \frac{1}{2} - \frac{s_1}{N}$ and $-\frac{N}{2} < s_1 \le 0$. By recalling Lemma 3.2, we arrive at

$$\begin{aligned} \left\| F_p(u) - F_p(w) \right\| _{L^{2^*_{s_1}}(\Omega )}&\le C \Big \Vert \big |\log |u| \big | \left| u -w\right| \Big \Vert _{L^{2^*_{s_1}}(\Omega )} + C \left\| \left| u\right| ^{p-2} \big |\log |u| \big | \left| u -w\right| \right\| _{L^{2^*_{s_1}}(\Omega )} \nonumber \\&\quad + C \left\| \left| w \right| ^{p-2} \big |\log |u| \big | \left| u -w\right| \right\| _{L^{2^*_{s_1}}(\Omega )} + C \left\| \left| w\right| ^{p-2} \left| u -w\right| \right\| _{L^{2^*_{s_1}}(\Omega )}. \end{aligned}$$

(3.37)

For the constant $2^*_{s_1} \ge 1$, using Hölder’s inequality, we get

$$\begin{aligned}&\Big \Vert \big |\log |u| \big | \left| u -w\right| \Big \Vert _{L^{2^*_{s_1}}(\Omega )}^{2^*_{s_1}} = \int \limits _{\Omega } \left( \big |\log |u|\big | \big |u-w\big |\right) ^{2^*_{s_1}} \mathrm {d}x = \int \limits _{\Omega } \big |\log |u|\big |^{2^*_{s_1}} \big |u - w\big |^{2^*_{s_1}} \mathrm {d}x \nonumber \\&\quad \le \left( \int \limits _{\Omega } \big |\log |u|\big |^{\frac{p 2^*_{s_1}}{p-1}} \mathrm {d}x\right) ^{\frac{p-1}{p}} \left( \int \limits _{\Omega } \big |u-w\big |^{p 2^*_{s_1}} \mathrm {d}x\right) ^{\frac{1}{p}} \nonumber \\&\quad \le \left( \int \limits _{\Omega ^-} \big |\log |u|\big |^{\frac{p 2^*_{s_1}}{p-1}} \mathrm {d}x + \int \limits _{\Omega ^+} \big |\log |u|\big |^{\frac{p 2^*_{s_1}}{p-1}} \mathrm {d}x \right) ^{\frac{p-1}{p}} \left( \int \limits _{\Omega } \big |u - w \big |^{p 2^*_{s_1}} \mathrm {d}x\right) ^{\frac{1}{p}} \nonumber \\&\quad \le \left( \left( \int \limits _{\Omega ^-} \big |\log |u|\big |^{\frac{p2^*_{s_1}}{p-1}} \mathrm {d}x \right) ^{ \frac{p-1}{p}} + \left( \int \limits _{\Omega ^+} \big |\log |u|\big |^{\frac{p2^*_{s_1}}{p-1}} \mathrm {d}x \right) ^{ \frac{p-1}{p}} \right) \left( \int \limits _{\Omega } \big |u - w \big |^{p 2^*_{s_1}} \mathrm {d}x\right) ^{\frac{1}{p}}. \end{aligned}$$

(3.38)

From the inequality (2.20) for $|u(x)|<1, ~ \forall x \in \Omega$, we have

$$\begin{aligned}&\left( \int \limits _{\Omega ^-} \big |\log |u|\big |^{\frac{p 2^*_{s_1} }{p-1}} \mathrm {d}x \right) ^{\frac{p-1}{p}} \nonumber \\&\quad \le C \left( \int \limits _{\Omega ^-} \big |u(x)\big |^{- \frac{p\theta 2^*_{s_1}}{p-1}} \mathrm {d}x \right) ^{\frac{p-1}{p}} \le \left( \left( \int \limits _{\Omega ^-} \big |u(x)\big |^{-\frac{p\theta 2^*_{s_1}}{p-1}} \mathrm {d}x \right) ^{-\frac{p-1}{p 2^*_{s_1}}} \right) ^{- 2^*_{s_1}} \nonumber \\&\quad \le C \left( \left( \int \limits _{\Omega ^-} |u(x)|^{\theta } \mathrm {d}x\right) \left( \int \limits _{\Omega ^-} 1 \mathrm {d}x \right) ^{-\frac{p-1+p 2^*_{s_1}}{p 2^*_{s_1}}} \right) ^{- 2^*_{s_1}} \le C \big \Vert u\big \Vert _{L^{\theta }(\Omega )}^{-\theta 2^*_{s_1}} |\Omega |^{\frac{p-1+p 2^*_{s_1}}{p }}, \end{aligned}$$

(3.39)

where we have chosen $k' = -\frac{p-1}{p2^*_{s_1}} <0$ in Lemma 2.8(b). For $|u(x)| \ge 1, ~ \forall x \in \Omega$, we have

$$\begin{aligned} \left( \int \limits _{\Omega ^+} \big |\log |u|\big |^{\frac{p 2^*_{s_1}}{p-1}} \mathrm {d}x \right) ^{\frac{p-1}{p}} \le C \left( \int \limits _{\Omega ^+} \big |u(x)\big |^{\frac{p\theta 2^*_{s_1}}{p-1}} \mathrm {d}x \right) ^{\frac{p-1}{p}} \le C \big \Vert u\big \Vert _{L^{\frac{p\theta 2^*_{s_1}}{p-1}}(\Omega )}^{\theta 2^*_{s_1}}. \end{aligned}$$

(3.40)

From (3.38), (3.39) and (3.40), we conclude that

$$\begin{aligned} \Big \Vert \big |\log |u|\big | \big |u-w\big |\Big \Vert _{L^{2^*_{s_1}}(\Omega )} \le C \left( \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta } + \big \Vert u\big \Vert _{L^{\frac{p\theta 2^*_{s_1}}{p-1}}(\Omega )}^{\theta } \right) \big \Vert u - w\big \Vert _{L^{p 2^*_{s_1}}(\Omega )}. \end{aligned}$$

(3.41)

Thanks to Hölder’s inequality, we get that

$$\begin{aligned}&\int \limits _{\Omega } \left( |u|^{p-2}\big |\log |u|\big | \big |u - w\big |\right) ^{2^*_{s_1}} \mathrm {d}x = \int \limits _{\Omega } |u|^{(p-2) 2^*_{s_1}} \big |\log |u|\big |^{2^*_{s_1}} \big |u - w\big |^{2^*_{s_1}} \mathrm {d}x \nonumber \\&\quad \le \left( \int \limits _{\Omega } \big |\log |u|\big |^{\frac{p 2^*_{s_1}}{p-2}} \mathrm {d}x\right) ^{\frac{p-2}{p}} \left( \int \limits _{\Omega } |u|^{p(p-2) 2^*_{s_1}} \mathrm {d}x\right) ^{\frac{1}{p}} \left( \int \limits _{\Omega } \big |u-w\big |^{p 2^*_{s_1}} \mathrm {d}x\right) ^{\frac{1}{p}}. \end{aligned}$$

(3.42)

Similar to (3.39) and (3.40), we have the following estimate

$$\begin{aligned} \left( \int \limits _{\Omega } \big |\log |u|\big |^{\frac{p 2^*_{s_1}}{p-2}} \mathrm {d}x\right) ^{\frac{p-2}{p}}&\le \left( \int \limits _{\Omega ^-} \big |\log |u|\big |^{\frac{p2^*_{s_1}}{p-2}} \mathrm {d}x \right) ^{ \frac{p-2}{p}} + \left( \int \limits _{\Omega ^+} \big |\log |u|\big |^{\frac{p2^*_{s_1}}{p-2}} \mathrm {d}x \right) ^{ \frac{p-2}{p}} \nonumber \\&\le C\left( \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta 2^*_{s_1} } + \big \Vert u\big \Vert _{L^{\frac{p \theta 2^*_{s_1}}{p-2}}(\Omega )}^{\theta 2^*_{s_1}} \right) . \end{aligned}$$

(3.43)

Combining (3.42) and (3.43), we get that

$$\begin{aligned}&\big \Vert |u|^{p-2}\big |\log |u|\big | \big |u-w\big |\big \Vert _{L^{2^*_{s_1}}(\Omega )} \nonumber \\&\quad \le C\left( \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta } + \big \Vert u\big \Vert _{L^{\frac{p \theta 2^*_{s_1}}{p-2}}(\Omega )}^{\theta } \right) \big \Vert u\big \Vert _{L^{p (p-2) 2^*_{s_1}}(\Omega )}^{p-2} \big \Vert u - w \big \Vert _{L^{p 2^*_{s_1}}(\Omega )}. \end{aligned}$$

(3.44)

Similarly,

$$\begin{aligned}&\left\| |w|^{p-2}\big |\log |u|\big | \big |u-w\big |\right\| _{L^{2^*_{s_1}}(\Omega )} \nonumber \\&\quad \le C\left( \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta } + \big \Vert u\big \Vert _{L^{\frac{p\theta 2^*_{s_1}}{p-2}}(\Omega )}^{\theta } \right) \big \Vert w\big \Vert _{L^{p (p-2) 2^*_{s_1}}(\Omega )}^{p-2} \big \Vert u - w\big \Vert _{L^{p 2^*_{s_1}}(\Omega )}. \end{aligned}$$

(3.45)

We use the Hölder’s inequality to obtain that

$$\begin{aligned} \left\| |w|^{p-2} \big |u-w\big |\right\| _{L^{2^*_{s_1}}(\Omega )}^{2^*_{s_1}}&= \int \limits _{\Omega } \left( |w|^{p-2} \big |u-w\big |\right) ^{2^*_{s_1}} \mathrm {d}x \nonumber \\&\le \left( \int \limits _{\Omega } |w|^{\frac{p(p-2)2^*_{s_1}}{p-1}} \mathrm {d}x\right) ^{\frac{p-1}{p}} \left( \int \limits _{\Omega } \big |u-w\big |^{p 2^*_{s_1}} \mathrm {d}x\right) ^{\frac{1}{p}} \nonumber \\&\le \big \Vert w\big \Vert _{L^{\frac{p (p-2) 2^*_{s_1}}{p-1}}(\Omega )}^{(p-2)2^*_{s_1}} \big \Vert u - w\big \Vert _{L^{p2^*_{s_1}}(\Omega )}^{2^*_{s_1}}. \end{aligned}$$

(3.46)

Combining the results obtained in (3.37), (3.41), (3.44), (3.45) and (3.46), we have

$$\begin{aligned} \big \Vert F_p(u)(\cdot ,t) - F_p(w)(\cdot ,t)\big \Vert _{L^{2^*_{s_1}}(\Omega )}&\le C \left( \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta } + \big \Vert u\big \Vert _{L^{\frac{p\theta 2^*_{s_1}}{p-1}}(\Omega )}^{\theta } \right) \big \Vert u - w\big \Vert _{L^{p 2^*_{s_1}}(\Omega )} \nonumber \\&\quad + C\left( \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta } + \big \Vert u\big \Vert _{L^{\frac{p\theta 2^*_{s_1}}{p-2}}(\Omega )}^{\theta } \right) \big \Vert u\big \Vert _{L^{p (p-2) 2^*_{s_1} }(\Omega )}^{p-2} \big \Vert u - w\big \Vert _{L^{p 2^*_{s_1}}(\Omega )} \nonumber \\&\quad + C\left( \big \Vert u\big \Vert _{L^\theta (\Omega )}^{-\theta } + \big \Vert u\big \Vert _{L^{\frac{p \theta 2^*_{s_1}}{p-2}}(\Omega )}^{\theta } \right) \big \Vert w\big \Vert _{L^{p (p-2) 2^*_{s_1} }(\Omega )}^{p-2} \big \Vert u - w\big \Vert _{L^{p 2^*_{s_1}}(\Omega )} \nonumber \\&\quad + C~ \big \Vert w\big \Vert _{L^{\frac{p (p-2) 2^*_{s_1}}{p-1}}(\Omega )}^{p-2} \big \Vert u - w\big \Vert _{L^{p 2^*_{s_1} }(\Omega )}. \end{aligned}$$

(3.47)

From (3.18)–(3.21) we have the following:

$\triangleright$:

For $q^*_s$ satisfying $\frac{1}{q^*_s} = \frac{1}{q} - \frac{s}{N}$, for $q \le \frac{N\theta }{N+\theta s}$ and $sq < N$, then $q^*_s \le \theta$ and we deduce from Lemma 2.9 that $L^{\theta }(\Omega ) \hookrightarrow W^{s,q}(\Omega )$. This implies that

$$\begin{aligned} \big \Vert u\big \Vert _{L^{\theta }(\Omega )}^{-\theta } \le C \big \Vert u\big \Vert _{W^{s,q}(\Omega )}^{-\theta }, \quad \hbox {for}~~ 0< \theta \le p-1. \end{aligned}$$

$\triangleright$:

For $s^* > Z(1,1)$, a similar argument with (3.31) and we observe that $q^*_s \ge p 2^*_{s_1},$ and then we deduce from Lemma 2.9 that the following Sobolev embedding holds $W^{s,q}(\Omega ) \hookrightarrow L^{p2^*_{s_1}}(\Omega )$.

$\triangleright$:

For $s^*> Z(p-2,1)$, implies that

$$\begin{aligned} p(p-2) 2^*_{s_1} = \frac{2p(p-2)N}{N-2s_1} < \frac{2p(p-2)N}{N+2 Z(p-2,1) -2 p s_2} \le q^*_s, \end{aligned}$$

(3.48)

and we infer that $W^{s,q}(\Omega ) \hookrightarrow L^{p (p-2) 2^*_{s_1}}(\Omega )$.

$\triangleright$:

For $s^*> Z(\theta ,p-1)$, we observe that $\frac{p \theta 2^*_{s_1}}{p-1} \le q^*_s,$ then we get $W^{s,q}(\Omega ) \hookrightarrow L^{\frac{p \theta 2^*_{s_1}}{p-1}}(\Omega )$.

$\triangleright$:

For $s^*> Z(\theta ,p-2)$, we have $\frac{p \theta 2^*_{s_1}}{p-2} \le q_s^*,$ and this implies that $W^{s,q} (\Omega ) \hookrightarrow L^{\frac{p \theta 2^*_{s_1}}{p-2}}(\Omega )$.

$\triangleright$:

For $s^*> Z(p-2,p-1)$, implies $\frac{p (p-2) 2^*_{s_1}}{p-1} \le q^*_s,$ and we infer that $W^{s,q}(\Omega ) \hookrightarrow L^{\frac{p (p-2) 2^*_{s_1} }{p-1}}(\Omega )$.

We can now combine the results above together with (3.47) to deduce that

$$\begin{aligned} \big \Vert F_p(u)(\cdot ,t) - F_p(w)(\cdot ,t)\big \Vert _{L^{2^*_{s_1}}(\Omega )}&\le K (R,u_0) \big \Vert u(\cdot ,t) - w(\cdot ,t) \big \Vert _{W^{s,q}(\Omega )}, \end{aligned}$$

(3.49)

for all $t\in (0,T]$, we have used that $\max \left\{ \Vert u\Vert _{W^{s,q}(\Omega )}; \Vert w\Vert _{W^{s,q}(\Omega )} \right\} \le R + \Vert u_0\Vert _{W^{s,q}(\Omega )}$, and for the constant $K(R,u_0):=K(N,p,\theta ,s_1,R, \Vert u_0\Vert _{W^{s,q}(\Omega )})$ but independent of t. From this, one observes that

$$\begin{aligned} {\underline{\hbox {The (RHS) of}}}\;\ (3.36)&\le CK(R,u_0) \int \limits _0^t (t -\tau )^{-\alpha } \left\| u(\cdot ,\tau ) - w(\cdot ,\tau )\right\| _{W^{s,q}(\Omega )} \mathrm {d}\tau \nonumber \\&\le CK(R,u_0) \int \limits _0^t (t -\tau )^{-\alpha } \left( \left\| u(\cdot ,\tau ) - w(\cdot ,\tau )\right\| _{W^{s,q}(\Omega )} \right) \mathrm {d}\tau \nonumber \\&\le CK(R,u_0) \left\| u(\cdot ,\tau ) - w(\cdot ,\tau )\right\| _{C([0,T];W^{s,q}(\Omega ))} \int \limits _0^t (t -\tau )^{-\alpha } \mathrm {d}\tau \nonumber \\&\le CK(R,u_0) \left\| u - w \right\| _{C([0,T];W^{s,q}(\Omega ))} \frac{t^{1-\alpha }}{1-\alpha }. \end{aligned}$$

(3.50)

Inserting the result of (3.50) into (3.36), we obtain that

$$\begin{aligned} \left\| {\mathbf {M}}{\mathbf {u}}(t) - {\mathbf {M}}{\mathbf {w}}(t)\right\| _{\mathbb X^{s_2}(\Omega )} \le CK(R,u_0) T^{1-\alpha } \left\| u - w \right\| _{C([0,T];W^{s,q}(\Omega ))}. \end{aligned}$$

For the constants s, q satisfying (3.19), we have that ${\mathbb {X}}^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )$, and

$$\begin{aligned} \left\| {\mathbf {M}} u - {\mathbf {M}}{\mathbf {w}}\right\| _{C([0,T];W^{s,q}(\Omega ))} \le CK(R,u_0) T^{1-\alpha } \left\| u - w \right\| _{C([0,T];W^{s,q}(\Omega ))}. \end{aligned}$$

(3.51)

Choosing $T, K(R,u_0)$ small enough such that $CK(R,u_0) T^{1-\alpha } < 1,$ it follows that ${\mathbf {M}}$ is a contraction map on $\mathbb W$. So, we invoke the contraction mapping principle to conclude that the map ${\mathbf {M}}$ has a unique fixed point u in ${\mathbb {W}}$. The proof of Theorem 3.2 is completed. $\square$

Since we already know that the mild solution of ${{\mathbb {P}}}$ does exist, the question is whether it will continue (continuation to a bigger interval of existence) and in what situation it is non-continuation by blowup.

Definition 3.1

Given a mild solution $u \in C([0,T]; W^{s,q}(\Omega ))$ of ${{\mathbb {P}}}$ for $\alpha \in (0,1)$, we say that $u^\star$ is a continuation of u in $(0, T^\star ]$ for $T^\star > T$ if it is satisfies

$$\begin{aligned} {\left\{ \begin{array}{ll} u^\star \in C([T,T^\star ]; W^{s,q}(\Omega )) ~~ \text{ is } \text{ a } \text{ mild } \text{ solution } \text{ of } \text{( }{\mathbb {P}}\text{) } \text{ for } \text{ all }~~ t \in [T, T^\star ], \\ u^\star (x,t) = u(x,t) \quad \text{ whenever } \quad t \in [0,T], x \in \Omega . \end{array}\right. } \end{aligned}$$

(3.52)

Theorem 3.3

(Continuation) Suppose that the assumptions of Theorem 3.2 are satisfied. Then, the mild solution (unique) on (0, T] of Problem ${{\mathbb {P}}}$ can be extended to the interval $(0, T^\star ],$ for some $T^\star > T$, so that, the extended function is also the mild solution (unique) of Problem ${{\mathbb {P}}}$ on $(0, T^\star ].$

Proof

Let $u:[0, T] \rightarrow W^{s,q}(\Omega )$ be a mild solution of Problem ${{\mathbb {P}}}$ (T is the time from Theorem 3.2). Fix $R^\star >0,$ and for $T^\star > T,$ ($T^\star$ depending on $R^\star$), we shall prove that $u^\star : [0,T^\star ] \rightarrow W^{s,q}(\Omega )$ is a mild solution of Problem ${{\mathbb {P}}}$. Assume the following estimates hold:

$$\begin{aligned}&C T^{-\alpha } (T^\star )^{\alpha } \left\| u_{0}\right\| _{\mathbb X^{s_2}(\Omega )} \le \frac{R^\star }{4}, \end{aligned}$$

(3.53)

$$\begin{aligned}&C \left( R^\star + \big \Vert u(\cdot ,T)\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } (T^\star )^{1- \alpha } \le \frac{R^\star }{8}, \end{aligned}$$

(3.54)

$$\begin{aligned}&C \left( R^\star + \big \Vert u(\cdot ,T)\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } (T^\star )^{1- \alpha } \le \frac{R^\star }{8}, \end{aligned}$$

(3.55)

$$\begin{aligned}&C \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } (T^\star )^{\frac{\alpha s_1 - \alpha s_2 +2}{2} } < \frac{R^\star }{8}, \end{aligned}$$

(3.56)

$$\begin{aligned}&C \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } (T^\star )^{\frac{\alpha s_1 - \alpha s_2 +2}{2} } < \frac{R^\star }{8}, \end{aligned}$$

(3.57)

$$\begin{aligned}&CK(R,u_0) (T^\star )^{\alpha + \alpha \gamma - 2 \alpha \gamma } \le \frac{R^\star }{4}, \end{aligned}$$

(3.58)

where $0<\theta \le p-1$ and $K(R,u_0)$ is defined in the proof of Theorem 3.2. For $T^\star \ge T >0$ and $R^\star >0$, let us define

$$\begin{aligned} {\mathbb {W}}^\star :=\left\{ u^\star \in C([0,T^\star ]; W^{s,q}(\Omega )): \left| \right. \begin{array}{lll} &{}u^\star (\cdot ,t) =u(\cdot ,t),&{}\quad \forall t\in (0,T], \\ &{}\left\| u^\star (\cdot ,t) - u (\cdot ,T)\right\| _{C([T, T^\star ]; W^{s,q}(\Omega ))} \le R^\star , &{} \quad \forall t\in [ T, T^\star ]. \end{array} \right\} \end{aligned}$$

(3.59)

$\bullet$ Step I We show that ${\mathbf {M}}$ defined as in (3.23) is the operator on ${\mathbb {W}}^\star$. Let $u^\star \in {\mathbb {W}}^\star$ and we consider two cases.

$*$ If $t\in (0,T]$, then by virtue of Theorem 3.2, we have the Problem ${{\mathbb {P}}}$ has a unique solution and we also have $u^\star (\cdot ,t)=u(\cdot ,t)$. Thus ${\mathbf {M}} u^\star (t) = {\mathbf {M}} u(t) = u(\cdot ,t)$ for all $t \in (0, T]$.

$*$ If $t\in [ T, T^\star ]$, we have

$$\begin{aligned}&\left\| {\mathbf {M}} u^\star (t)-u(\cdot , T) \right\| _{W^{s,q}(\Omega )} \nonumber \\&\quad \le \left\| \left( E_{\alpha ,1}\left( - t^{\alpha } {\mathcal {A}} \right) - E_{\alpha ,1}\left( - T^{\alpha } {\mathcal {A}}\right) \right) u_{0} \right\| _{W^{s,q}(\Omega )} \nonumber \\&\qquad + \int \limits _T^t \left\| E_{\alpha ,1}\left( - (t-\tau )^{\alpha } {\mathcal {A}} \right) F_p(u^\star )(\tau ) \right\| _{W^{s,q}(\Omega )} \mathrm {d}\tau \nonumber \\&\qquad + \int \limits _0^T \left\| \left( E_{\alpha ,1}\left( - (t-\tau )^{\alpha } {\mathcal {A}} \right) -E_{\alpha ,1}\left( - (T-\tau )^{\alpha } {\mathcal {A}} \right) \right) F_p(u^\star )(\tau ) \right\| _{W^{s,q}(\Omega )}\mathrm {d}\tau \nonumber \\&\quad =: \left\| J_2(u_0)( t)\right\| _{W^{s,q}(\Omega )} + \left\| J_3(u^\star )( t)\right\| _{W^{s,q}(\Omega )} + \left\| J_4(u^\star )( t)\right\| _{W^{s,q}(\Omega )}. \end{aligned}$$

(3.60)

Estimating the term $\left\| J_2(u_0)( t)\right\| _{W^{s,q}(\Omega )}$, using Lemma 2.5, we have for all $t \in [T,T^\star ]$,

$$\begin{aligned} \left\| J_2(u_0)( t)\right\| _{{\mathbb {X}}^{s_2}(\Omega )}^2&= \sum _{j=1}^\infty \lambda _j^{ s_2}\left( u_{0},\phi _j \right) ^2 \left( E_{\alpha ,1}\left( - \lambda _j t^{\alpha }\right) - E_{\alpha ,1}\left( - \lambda _j T^{\alpha } \right) \right) ^2 \nonumber \\&= \sum _{j=1}^\infty \lambda _j^{s_2} \left( u_{0},\phi _j \right) ^2 \left( \int \limits _0^\infty {\mathcal {M}}_\alpha (z) \left| e^{- z \lambda _j t^{\alpha }} - e^{- z \lambda _j T^{\alpha } } \right| \mathrm {d} z \right) ^2 \nonumber \\&\le \sum _{j=1}^\infty \lambda _j^{ s_2} \left( u_{0},\phi _j \right) ^2 \left( \int \limits _0^\infty {\mathcal {M}}_\alpha (z)e^{- z \lambda _j T^{\alpha } } \left| e^{- z \lambda _j (t^\alpha - T^{\alpha }) } - 1 \right| \mathrm {d} z \right) ^2. \end{aligned}$$

For $z>0$, using the inequality $1-e^{-z}\le z, ~ \hbox { and} ~~ ze^{-z} \le 1,$ one obtains

$$\begin{aligned} \left\| J_2(u_0)( t)\right\| _{{\mathbb {X}}^{s_2}(\Omega )}^2&\le \sum _{j=1}^\infty \lambda _j^{s_2} \left( u_{0},\phi _j \right) ^2\left( \lambda _j (t^\alpha - T^{\alpha }) \int \limits _0^\infty {\mathcal {M}}_\alpha (z) \left( z \lambda _j T^{\alpha } \right) ^{-1} z \mathrm {d} z \right) ^2 \nonumber \\&\le \sum _{j=1}^\infty \lambda _j^{s_2} \left( u_{0},\phi _j \right) ^2 \left( \left( t^{\alpha } - T^{\alpha }\right) T^{-\alpha } \int \limits _0^\infty {\mathcal {M}}_\alpha (z) \mathrm {d} z \right) ^2 \nonumber \\&\le \left( t - T\right) ^{2\alpha } T^{-2\alpha } \left\| u_{0}\right\| _{{\mathbb {X}}^{s_2}(\Omega )}^2, \end{aligned}$$

(3.61)

where we have use the inequalities

$$\begin{aligned}a^{c} - b^{c} \le (a-b)^{c},\quad \hbox {for}~~ a>b>0, c \in (0,1),\quad \hbox {and}~~ \int \limits _0^\infty {\mathcal {M}}_\alpha (z) \mathrm {d} z = 1.\end{aligned}$$

For the constants s, q satisfying (3.19), we have that $\mathbb X^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )$. Hence, we get that

$$\begin{aligned} \left\| J_2(u_0)(t)\right\| _{W^{s,q}(\Omega )} \le C \left( t - T\right) ^{\alpha } T^{-\alpha } \left\| u_{0}\right\| _{\mathbb X^{s_2}(\Omega )} \le C \left( T^\star \right) ^{\alpha } T^{-\alpha } \left\| u_{0}\right\| _{{\mathbb {X}}^{s_2}(\Omega )}. \end{aligned}$$

(3.62)

From (3.53), this implies that the following estimate holds

$$\begin{aligned} \left\| J_2(u_0)\right\| _{C([0,T^\star ];W^{s,q}(\Omega ))} \le C T^{-\alpha } (T^\star )^{\alpha } \left\| u_{0}\right\| _{\mathbb X^{s_2}(\Omega )}\le \frac{R^\star }{4}. \end{aligned}$$

(3.63)

Similar to (3.32), we have the following estimate for all $t \in [T,T^\star ]$ (note that we can choose $T^\star >T$ and close enough to T)

$$\begin{aligned}&\Vert J_3(u^\star )(t) \Vert _{W^{s,q}(\Omega )} \le C \big \Vert J_3(u)(t) \big \Vert _{{\mathbb {X}}^{s_2}(\Omega )} \nonumber \\&\quad \le C \left( \left( R^\star + \big \Vert u(\cdot ,T)\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R^\star + \big \Vert u(\cdot ,T)\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) (t-T)^{1- \alpha }, \end{aligned}$$

(3.64)

where from (3.59), for all $t \in [T,T^\star ]$, we have used that

$$\begin{aligned} \Vert u^\star (\cdot ,t) \Vert _{W^{s,q}(\Omega )} \le R^\star + \big \Vert u(\cdot ,T)\big \Vert _{W^{s,q}(\Omega )}.\end{aligned}$$

Using (3.54) and (3.55), we infer that

$$\begin{aligned}&\left\| J_3(u^\star ) \right\| _{C([0,T^\star ]; W^{s,q}(\Omega ))} \nonumber \\&\quad \le C \left( \left( R^\star + \big \Vert u(\cdot ,T)\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R^\star + \big \Vert u(\cdot ,T)\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) (T^\star )^{1- \alpha } \le \frac{R^\star }{4}. \end{aligned}$$

(3.65)

We continue with the estimate on the third term of (3.60), and using Lemma 2.3(b) and Lemma 2.4, we obtain for all $t \in [ T, T^\star ]$

$$\begin{aligned} \left| E_{\alpha ,1}\left( - \lambda _j\left( t - \tau \right) ^{\alpha } \right) - E_{\alpha ,1}\left( - \lambda _j \left( T - \tau \right) ^{\alpha } \right) \right|&= \left| \int \limits _{T - \tau }^{t - \tau } -\lambda _j z^{\alpha -1} E_{\alpha ,\alpha }\left( - \lambda _j z^{\alpha } \right) \mathrm {d} z\right| \nonumber \\&= \left| \int \limits _{T - \tau }^{t - \tau } \lambda _j^{1+ \frac{s_2 - s_1}{2}} z^{\alpha -1} E_{\alpha ,\alpha }\left( - \lambda _j z^{\alpha } \right) \mathrm {d} z\right| \lambda _j^{\frac{s_1-s_2}{2}} \nonumber \\&\le C \left| \int \limits _{T - \tau }^{t - \tau } z^{\frac{\alpha s_1 - \alpha s_2}{2} -1} \mathrm {d} z\right| \lambda _j^{\frac{s_1-s_2}{2}} \le C \lambda _j^{\frac{s_1-s_2}{2}} (T-\tau )^{\frac{\alpha s_1 - \alpha s_2}{2}}. \end{aligned}$$

(3.66)

For the constant $s_1$ satisfying $-\frac{N}{2} < s_1 \le 0$ and $\frac{1}{2^\star _{s_1} } = \frac{1}{2} - \frac{s_1}{N}$, from Lemma 2.9, we obtain $L^{2^\star _{s_1}}(\Omega ) \hookrightarrow {\mathbb {X}}^{s_1}(\Omega ).$ Hence, we deduce that

$$\begin{aligned}&\left\| J_4(u^\star )( t) \right\| _{{\mathbb {X}}^{s_2}(\Omega )} \nonumber \\&\quad \le \int \limits _0^T \Bigg (\sum _{j=1}^{\infty } \lambda _j^{s_2} \left( F_p(u^\star )(\cdot ,\tau ),\phi _j\right) ^2 \left| E_{\alpha ,1}\left( - \lambda _j\left( t - \tau \right) ^{\alpha } \right) - E_{\alpha ,1}\left( - \lambda _j \left( T - \tau \right) ^{\alpha } \right) \right| ^2 \Bigg )^{\frac{1}{2}} \mathrm {d}\tau \nonumber \\&\quad \le C \int \limits _0^T \Bigg (\sum _{j=1}^{\infty } \lambda _j^{s_1} (T-\tau )^{\alpha s_1- \alpha s_2} \left( F_p(u^\star )(\cdot ,\tau ),\phi _j\right) ^2 \Bigg )^{\frac{1}{2}} \mathrm {d}\tau \nonumber \\&\quad \le C \int \limits _0^T \left( T - \tau \right) ^{\frac{\alpha s_1 - \alpha s_2}{2} } \left\| F_p(u^\star )(\cdot ,\tau )\right\| _{\mathbb X^{s_1}(\Omega )} \mathrm {d}\tau \nonumber \\&\quad \le C \int \limits _0^T \left( T - \tau \right) ^{\frac{\alpha s_1 - \alpha s_2}{2} } \left\| F_p(u^\star )(\cdot ,\tau )\right\| _{ L^{2^\star _{s_1}}(\Omega )} \mathrm {d}\tau , \end{aligned}$$

(3.67)

In the same way as in (3.32), and from (3.59), one obtains

$$\begin{aligned}&{\underline{\hbox {The (RHS) of}}}\;\; (3.67) \nonumber \\&\quad \le C \int \limits _0^T (T - \tau )^{\frac{\alpha s_1 - \alpha s_2}{2} } \left( \left\| u(\cdot ,\tau )\right\| _{W^{s,q}(\Omega )}^{-\theta } + \big \Vert u(\cdot ,\tau )\big \Vert _{W^{s,q}(\Omega )}^{\theta } \right) \big \Vert u(\cdot ,\tau )\big \Vert _{W^{s,q}(\Omega )}^{p-1} \mathrm {d}\tau \nonumber \\&\quad \le C \left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) \int \limits _0^T (T - \tau )^{\frac{\alpha s_1 - \alpha s_2}{2} } \mathrm {d}\tau \nonumber \\&\quad \le C \left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) T^{\frac{\alpha s_1 - \alpha s_2 +2}{2} }, \end{aligned}$$

(3.68)

for $\theta < p-1$. From (3.67), we have that $\mathbb X^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )$ and obtain

$$\begin{aligned} \big \Vert J_4(u^\star )(t) \big \Vert _{W^{s,q}(\Omega )} \le C \left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) T^{\frac{\alpha s_1 - \alpha s_2 +2}{2} }. \end{aligned}$$

(3.69)

Thus, for $t\in (T,T^\star ]$, we obtain

$$\begin{aligned} \big \Vert J_4(u^\star )(t) \big \Vert _{W^{s,q}(\Omega )} \le C \left( \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) (T^\star )^{\frac{\alpha s_1 - \alpha s_2 +2}{2} }. \end{aligned}$$

(3.70)

From (3.56) and (3.57), we get

$$\begin{aligned} \left\| J_4(u^\star ) \right\| _{C([0,T^\star ];W^{s,q}(\Omega ))} \le \frac{R^\star }{4}. \end{aligned}$$

(3.71)

It follows from (3.63), (3.65), (3.71) that, for every $t\in [ T, T^\star ]$

$$\begin{aligned} \left\| {\mathbf {M}} u^\star - u(\cdot ,T) \right\| _{C([0,T^\star ];W^{s,q}(\Omega ))} \le \frac{R^\star }{4} + \frac{R^\star }{4} + \frac{R^\star }{4} \le R^\star . \end{aligned}$$

We have shown that ${\mathbf {M}}$ is a map ${\mathbb {W}}^\star$ into ${\mathbb {W}}^\star$.

$\bullet$ Step II We show that ${\mathbf {M}}$ is a contraction on ${\mathbb {W}}^\star$. Let $u,w \in {\mathbb {W}}^\star$, and we have that for $0 \le t \le T^\star ,$

$$\begin{aligned} {\mathbf {M}} u(t)-{\mathbf {M}} w(t) = \int \limits _0^t E_{\alpha ,1}\left( - \mathcal A \left( t - \tau \right) ^{\alpha } \right) \left( F_p(u)(\tau ) - F_p(w) (\tau ) \right) \mathrm {d}\tau , \end{aligned}$$

(3.72)

where we note that ${\mathbf {M}}{\mathbf {u}}(t)-{\mathbf {M}}{\mathbf {w}}(t) = 0,$ vanishes in ${\mathbb {W}}^\star$ for all $t \in (0, T]$. Then, for all $t \in [0, T^\star ]$, proceeding as in Claim (2) of the last theorem, we have

$$\begin{aligned} \big \Vert {\mathbf {M}}{\mathbf {u}}(t)-{\mathbf {M}}{\mathbf {w}}(t) \big \Vert _{{\mathbb {X}}^{s_2}(\Omega )} \le CK(R,u_0) (T^\star )^{1-\alpha } \left\| u - w \right\| _{C([0,T^*];W^{s,q}(\Omega ))} \le \frac{R^\star }{4} \left\| u - w \right\| _{C([0,T^*];W^{s,q}(\Omega ))}. \end{aligned}$$

Thus, using the Sobolev embedding ${\mathbb {X}}^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )$ with s, q satisfying (3.19), for all $T^\star >0$, so without loss of generality, we may assume that $0\le R^\star <4$, and we infer that

$$\begin{aligned} \big \Vert {\mathbf {M}}{\mathbf {u}} -{\mathbf {M}}{\mathbf {w}} \big \Vert _{C([0,T^\star ];W^{s,q}(\Omega ))} \le \frac{R^\star }{4} \big \Vert u-v\big \Vert _{C([0,T^\star ];W^{s,q}(\Omega ))}. \end{aligned}$$

(3.73)

This implies that ${\mathbf {M}}$ is a $\frac{R^\star }{4}$-contraction. By the Banach contraction principle it follows that ${\mathbf {M}}$ has a unique fixed point $u^\star$ of ${\mathbf {M}}$ in ${\mathbb {W}}^\star$, which is a continuation of u. This finishes the proof. $\square$

The next results are on global existence or non-continuation by a blowup.

Definition 3.2

Let u(x, t) be a solution of ${{\mathbb {P}}}$. We define the maximal existence time $T_{\max }$ of u(x, t) as follows:

• (i) If u(x, t) exists for all $0 \le t < \infty$, then $T_{\max } = \infty$.

• (ii) If there exists $T \in (0,\infty )$ such that u(x, t) exists for $0 \le t < T$, but does not exist at $t = T$, then $T_{\max }= T$.

Definition 3.3

Let u(x, t) be a solution of ${{\mathbb {P}}}$. We say u(x, t) blows up in finite time if the maximal existence time $T_{\max }$ is finite and

$$\begin{aligned} \lim \;\;\sup _{t\rightarrow T_{\max }^-} \left\| u(\cdot ,t)\right\| _{W^{s,q}(\Omega )} = \infty . \end{aligned}$$

(3.74)

Theorem 3.4

(Global existence or finite time blowup) For $N\ge 1, p\ge 2,0\le s < 2s_2$, for $0\le s_2 <N/2$ and $1 \le q \le \min \left\{ 2^\star _{s_2,s};\frac{N \theta }{N+\theta s} \right\}$ with $2^\star _{s_2,s}$ satisfying $\frac{1}{2^\star _{s_2,s}} = \frac{1}{2} + \frac{s}{N} - \frac{s_2}{N}$ and $qs < N$. For $u_0 \in {\mathbb {X}}^{s_2}(\Omega ) \cap W^{s,q}(\Omega )$, there exists a maximal time $T_{\max } >0$ such that $u \in C([0,T_{\max }];W^{s,q}(\Omega ))$ is the mild solution of ${{\mathbb {P}}}$. Thus, either Problem ${{\mathbb {P}}}$ has a unique global mild solution on $[0,\infty )$ or there exists a maximal time $T_{\max } < \infty$ such that

$$\begin{aligned} \lim \;\;\sup _{t \rightarrow T_{\max }^{-}} \Vert u(\cdot ,t)\Vert _{W^{s,q}(\Omega )} = \infty . \end{aligned}$$

Proof

Let $u_0 \in {\mathbb {X}}^{s_2}(\Omega ) \cap W^{s,q}(\Omega )$ and define

$$\begin{aligned} T_{\max }:=\sup \left\{ T>0:\hbox {there exits a solution on}\;(0, T]\right\} . \end{aligned}$$

Assume that $T_{\max } < \infty$, and $\Vert u(\cdot ,t) \Vert _{\mathbb X^{s_2}(\Omega )} \le R_0,$ for some $R_0 >0$. Now suppose there exists a sequence $\{t_{n}\}_{n\in {\mathbb {N}}}\subset [0, T_{\max })$ such that $t_n \rightarrow T_{\max }$ and $\{u(\cdot ,t_{n})\}_{n\in {\mathbb {N}}} \subset \mathbb X^{s_2}(\Omega )$. Let us show that $\{u(\cdot ,t_{n})\}_{n\in \mathbb N}$ is a Cauchy sequence in ${\mathbb {X}}^{s_2}(\Omega )$. Indeed, given $\epsilon >0$, fix $N \in {\mathbb {N}}$ such that for all $n,m>N$, $0< t_n< t_{m} < T_{\max }$, we have

$$\begin{aligned}&\big \Vert u(\cdot ,t_{m})-u(\cdot ,t_n)\big \Vert _{W^{s,q}(\Omega )} \nonumber \\&\quad \le \left\| \left( E_{\alpha ,1}\left( - t_m^{\alpha }{\mathcal {A}} \right) - E_{\alpha ,1}\left( - t_n^{\alpha } {\mathcal {A}} \right) \right) u_{0} \right\| _{W^{s,q}(\Omega )} \nonumber \\&\qquad + \int \limits _{t_n}^{t_m} \left\| E_{\alpha ,1}\left( -\left( t_m - \tau \right) ^{\alpha } {\mathcal {A}} \right) F_p(u )( \tau ) \right\| _{W^{s,q}(\Omega )} \mathrm {d}\tau \nonumber \\&\qquad + \int \limits _0^{t_n} \left\| \left( E_{\alpha ,1}\left( - \left( t_n - \tau \right) ^{\alpha } {\mathcal {A}} \right) - E_{\alpha ,1}\left( - \left( T_{\max } - \tau \right) ^{\alpha } {\mathcal {A}} \right) \right) F_p(u )(\tau ) \right\| _{W^{s,q}(\Omega )} \mathrm {d}\tau \nonumber \\&\qquad + \int \limits _0^{t_m} \left\| \left( E_{\alpha ,1}\left( -\left( T_{\max } - \tau \right) ^{\alpha } {\mathcal {A}} \right) - E_{\alpha ,1}\left( - \left( t_{m} - \tau \right) ^{\alpha } \right) {\mathcal {A}} \right) F_p(u )(\tau ) \right\| _{W^{s,q}(\Omega )} \mathrm {d}\tau \nonumber \\&\quad = \left\| J_5(u_0)\right\| _{W^{s,q}(\Omega )} + \left\| J_6(u )\right\| _{W^{s,q}(\Omega )} + \left\| J_7(u )\right\| _{W^{s,q}(\Omega )} + \left\| J_8(u )\right\| _{W^{s,q}(\Omega )}. \end{aligned}$$

(3.75)

Similar to (3.61), and using the Sobolev embedding $\mathbb X^{s_2}(\Omega ) \hookrightarrow W^{s,q}(\Omega )$, we have that

$$\begin{aligned} \left\| J_5(u_0) \right\| _{W^{s,q}(\Omega )} \le C\left\| J_5(u_0)(t)\right\| _{{\mathbb {X}}^{s_2}(\Omega )} \le C \left| t_m - t_n\right| ^{ \alpha } t_n^{- \alpha } \left\| u_{0}\right\| _{{\mathbb {X}}^{s_2}(\Omega )}. \end{aligned}$$

(3.76)

In the same way as in (3.32), we get

$$\begin{aligned} \left\| J_6(u)(t)\right\| _{W^{s,q}(\Omega )} \le C \left( \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) |t_m-t_n|^{1-\alpha }. \end{aligned}$$

(3.77)

Similar to (3.68), we have

$$\begin{aligned} \left\| J_7(u) \right\| _{W^{s,q}(\Omega )} \le C \left( \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) |T_{\max }-t_n|^{\frac{\alpha s_1 - \alpha s_2 +2}{2} }, \end{aligned}$$

(3.78)

and

$$\begin{aligned} \left\| J_8(u) \right\| _{W^{s,q}(\Omega )} \le C \left( \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) |T_{\max }-t_m|^{\frac{\alpha s_1 - \alpha s_2 +2}{2} }. \end{aligned}$$

(3.79)

Thus, since $\{t_n\}_{n\in {\mathbb {N}}^*}$ is convergent we can take $N:=N(\epsilon ) \in {\mathbb {N}}^*$ with $m \ge n \ge N$ such that $|t_{m} - t_n|$ is as small as we want, and we have

$$\begin{aligned}&C \left| t_m - t_n\right| ^{ \alpha } t_n^{- \alpha } \left\| u_{n}\right\| _{{\mathbb {X}}^{s_2}(\Omega )}< \frac{\epsilon }{4},\\&C \left( \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) |t_m-t_n|^{1-\alpha }< \frac{\epsilon }{4},\\&C \left( \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) |T_{\max }-t_m|^{\frac{\alpha s_1 - \alpha s_2 +2}{2} } < \frac{\epsilon }{4}, \end{aligned}$$

and

$$\begin{aligned} C \left( \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1-\theta } + \left( R_0 + \big \Vert u_0\big \Vert _{W^{s,q}(\Omega )} \right) ^{p-1+\theta } \right) |T_{\max }-t_n|^{\frac{\alpha s_1 - \alpha s_2 +2}{2} } < \frac{\epsilon }{4}. \end{aligned}$$

Hence, given $\epsilon >0$ there exists $N \in {\mathbb {N}}$ such that

$$\begin{aligned} \big \Vert u(\cdot ,t_{m})-u(\cdot ,t_n)\big \Vert _{W^{s,q}(\Omega )} < \epsilon , \quad \hbox {for}~~m,n \ge N. \end{aligned}$$

(3.80)

It follows that $\{u(\cdot ,t_n)\}_{n\in {{\mathbb {N}}}} \subset W^{s,q}(\Omega )$ is a Cauchy sequences and for $\{t_n\}_{n\in {\mathbb {N}}^*}$ arbitrary we have proved the existence of the limit

$$\begin{aligned} \lim _{t\rightarrow T_{\max }^-} \left\| u(\cdot ,t)\right\| _{W^{s,q}(\Omega )} < \infty . \end{aligned}$$

FRom our previos result we deduce that the solution can extended to some larger interval (u can be continued beyond $T_{\max }$), and this contradict the definition of $T_{\max }$. Thus, either $T_{\max } = \infty$ or if $T_{\max }<\infty$ then $\lim _{t\rightarrow T_{\max }^-} \left\| u(\cdot ,t)\right\| _{W^{s,q}(\Omega )} = \infty$. The proof of Theorem 3.4 is finished. $\square$