Pullback attractors for non-autonomous reaction–diffusion equation with infinite delays in $C_{\gamma,L^{r}(\Omega)}$ or $C_{\gamma,W^{1,r}(\Omega)}$

Abstract

In this paper, the well-posedness for the non-autonomous reaction–diffusion equation with infinite delays on a bounded domain is established. The existence of pullback attractors for the process in $C_{\gamma,L^{r}(\Omega)}$ and $C_{\gamma,W^{1,r}(\Omega)}$ is proved, respectively. The noncompact Kuratowski measure is applied to check the asymptotic compactness.

1 Introduction

Let $\Omega \subseteq \mathbb{R}^{n}$ be a smooth bounded domain. Consider the long-time behavior of the following non-autonomous nonlinear reaction–diffusion equation:

$$ \textstyle\begin{cases} \frac{\partial u}{\partial t}-\Delta u+\lambda u=f(t,u_{t})+g(t,x), & \text{in } [\tau ,+\infty ]\times \Omega , \\ u|_{\partial \Omega} =0, & t>\tau , \\ u(t,x)=\phi (t-\tau ,x),& t\in (-\infty ,\tau ], x\in \Omega , \end{cases} $$

(1)

where $\lambda \geq 0$, and we have the nonlinear term

$$ f\bigl(t,u_{t}(t,x)\bigr)=F\bigl(t,u\bigl(t-\rho (t),x\bigr)\bigr)+ \int^{0}_{-\infty } G\bigl(t,z,u(t+z,x)\bigr)\,dz. $$

Suppose there exist two positive constants $k_{1}$, $k_{2}$, and three positive scalar functions $m_{0}(\cdot )$, $e^{-r\gamma \rho (t)}m _{1}(t)$, $m_{2}(\cdot )e^{-\gamma z}$ which are all in $L^{1}((- \infty ,0],\mathbb{R}^{+})$ such that the functions $F\in C( \mathbb{R}\times \mathbb{R};\mathbb{R})$, $\rho \in C(\mathbb{R};[0,+ \infty ))$, and $G\in C(\mathbb{R} \times (-\infty ,0]\times \mathbb{R};\mathbb{R})$ satisfy

$$\begin{aligned}& \bigl\vert F(t,\upsilon ) \bigr\vert ^{r}\leq \vert k_{1} \vert ^{r}+k_{2}^{r}e^{-r\gamma \rho (t)} \vert \upsilon \vert ^{r}, \quad \forall t,\upsilon \in \mathbb{R}, \end{aligned}$$

(2)

$$\begin{aligned}& \bigl\vert G(t,z,\upsilon ) \bigr\vert \leq m_{0}(z)+m_{1}(z) \vert v \vert , \quad \forall t,\upsilon \in \mathbb{R},z\in (-\infty ,0], \end{aligned}$$

(3)

$$\begin{aligned}& \bigl\vert F(t,\upsilon )-F(t,\nu ) \bigr\vert \leq C_{1}e^{-\gamma \rho (t)} \vert \upsilon - \nu \vert , \quad \forall t, \upsilon ,\nu \in \mathbb{R}, z\in (-\infty ,0], \end{aligned}$$

(4)

$$\begin{aligned}& \bigl\vert G(t,z,\upsilon )-G(t,z,\nu ) \bigr\vert \leq C_{2}m_{2}(z) \vert \upsilon -\nu \vert , \quad \forall t, \upsilon ,\nu \in \mathbb{R},z\in (-\infty ,0], \end{aligned}$$

(5)

and the non-autonomous term $g\in L^{r}_{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))$ ($r>1$) satisfies

$$ \sup_{\tau \leq t}e^{-\delta \tau } \int^{\tau }_{-\infty } \bigl\Vert g(s) \bigr\Vert ^{r}_{X} e^{\delta s}\,ds< \infty ,\quad \forall t\in \mathbb{R}, $$

(6)

for each $\delta \in \{\alpha ,\alpha -L,r(\delta -\eta )\}$, where α, L, δ, η will be given in Lemma 4.1, the local r-power integral is the Bochner integral. We will denote $m_{0}=\int^{0}_{-\infty }m_{0}(s)\,ds$, $m_{1}=\int^{0} _{-\infty }e^{-\gamma s}m_{1}(s)\,ds$, and $m_{2}=\int^{0}_{-\infty }e ^{-\gamma s}m_{2}(s)\,ds$.

Let $C_{\gamma ,X}$ denote the Banach space $C((-\infty ,0];X)$ endowed with the norm

$$ \Vert \phi \Vert _{C_{\gamma ,X}}=\sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert \phi (z) \bigr\Vert _{X}, \quad \gamma >0, $$

where X is $L^{r}{(\Omega )}$ or $W^{1,r}(\Omega )$.

Given $\tau \in \mathbb{R}$, $T>\tau$ and a function $u:(-\infty ,T]\rightarrow X$. For each $t\in [\tau ,T]$, $u_{t}:(- \infty ,0]\rightarrow X$ denotes the function defined by $u_{t}(z)=u(t+z)$ for $z\in (-\infty ,0]$. We are interested in the initial condition $\phi \in C_{\gamma ,X}$.

Retarded differential equations have been used to research many physical systems with non-instant transmission phenomena such as internet data transmission, other memory processes, and specially biological motivations (e.g. species growth or incubating time on disease models [1, 2]). For autonomous systems with delays, the existence of solutions or global attractors has been studied widely in [3–5] and their qualitative theory has also been well-established. For autonomous systems with variable bounded or unbounded delays, the classical theory extended in [6–13] has been applied to deal with the existence of solution and special attractors. In fact, autonomous systems with variable delays are non-autonomous in essence. Except that time-periodic equations can be dealt with classic theory relatively straightforward manner, the qualitative properties or asymptotic behavior of many general non-autonomous systems are analyzed by new ideas and methods. In recent years, non-autonomous diffusion equations have attracted much attention in mathematical literature. Duong [14] considered a class of flux-limited diffusions with external force and established the comparison and maximum principles. Jung et al. [15] considered the nonlinear singularly perturbed reaction–diffusion problems in the polygonal domain and proposed a boundary layer analysis which fits a domain with corners.

For the reaction–diffusion systems with finite delays, there are also a sires of work [11, 16, 17]. More recently, Wang et al. [10] proved the existence of pullback attractors in the weighted space $C_{\gamma ,H^{1}(\Omega )}$ for the multi-value process generated by (1) based on the concept of the Kuratowski measure of the noncompactness of a bounded set, where the growth of nonlinear term $F(x,v)$ and $G(x,s,v)$ are both linear, and the non-autonomous term $g(t,x)\in L^{2}_{\mathrm {loc}}(\mathbb{R};L^{2}( \Omega ))$ satisfies

$$ \sup_{\tau \leq t}e^{-\eta \tau } \int^{\tau }_{-\infty } \bigl\Vert g(s) \bigr\Vert ^{2} _{L^{2}{(\Omega )}}e^{\eta s}\,ds< +\infty ,\quad \forall \eta \in \mathbb{R}, \eta >0. $$

(7)

In the present paper, we will prove the existence of solution and the pullback attractors of (1) in the bounded domain of $C_{\gamma , L^{r}{(\Omega )}}$ or $C_{\gamma , W^{1,r}(\Omega )}$ under the conditions (2)–(6) for ${r\geq 2}$.

The main work of this paper contains three issues. Since the space $L^{r}{(\Omega )}$ ($r>2$) loses the inner product and orthogonality, canonical projector and approximation methods [10] are both ineffective to prove the existence of solutions and pullback attractors of (1). In order to overcome this difficulty, we adopt the idea of [17] and decompose (1) into two equations to separate the non-autonomous term to establish well-posedness (see Theorem 3.7 and Theorem 3.10). In addition we investigate the existence of pullback absorbing set by using the approximation technique of [9, 10] to overcome difficulties stemming from infinite delays and infinite dimensions. Consequently, for verifying the asymptotic compactness of (1) in $C_{\gamma ,L^{r}{(\Omega )}}$ ($r>2$), we employ the weak continuous semigroup theory and finite dimensional approximation method in [16, 18] to construct compact embedding results (see Theorem 5.6). Moreover, by improving smooth effect of the semigroup $e^{At}$, we prove the dissipativity and the existence of pullback attractors for (1) in $C_{\gamma ,W^{1,r}(\Omega )}$ (see Lemma 6.1).

The paper is organized as follows. Section 2 gives some preliminaries concerning the definitions of processes and the pullback attractors of non-autonomous dynamical systems. We also give the definition of ω-limit compact and a suitable non-autonomous frameworks for the discussion of attractors in the future. In Sect. 3, we consider the well-posedness of (1) in $C_{\gamma ,L^{r}{(\Omega )}}$ and $C_{W^{1,r}(\Omega )}$, respectively. In Sects. 4 and 6, we prove the existence of bounded absorbing sets in both spaces above. In Sects. 5 and 7, the existence of pullback attractors in $C_{\gamma ,L^{r}{( \Omega )}}$ and $C_{\gamma ,W^{1,r}(\Omega )}$ is proved.

2 Preliminaries

Let X be a complete metric space with metric $d_{X}(\cdot ,\cdot )$. Denote by $H^{*}_{X}(\cdot ,\cdot )$ the Hausdorff semi-distance between two nonempty subsets of a complete metric space X, which is defined by

$$ H^{*}_{X}(A,B)=\sup_{a\in A}\inf _{b\in B}d_{X}(a,b). $$

Definition 2.1

A mapping $U(t,\tau ):X\rightarrow X$, $t> \tau$ in $\mathbb{R}$, is called a process if

• (1) $U(\tau ,\tau )x=x$, $\forall \tau \in \mathbb{R}$, $x\in X$;

• (2) $U(t,s)U(s,\tau )x=U(t,\tau )x$, $\forall \tau \leq s\leq t \in \mathbb{R}$, $x\in X$.

Definition 2.2

The Kuratowski measure $k(A)$ of noncompactness of the set A is defined by

$$ k(A)=\inf\{\delta >0\mid A \text{ admits a finite cover by sets whose diameter} \leq \delta \}. $$

Definition 2.3

Let $\{U(t,\tau )\}$ be a process on X. We say that $\{U(t,\tau )\}$ is

• (1) pullback dissipative, if there exists a family of bounded sets $\mathcal{D}=\{{D(t)}\}_{t\in \mathbb{R}}$ in X so that, for any bounded set $B\subset X$ and each $t\in \mathbb{R}$, there exists a $S_{0}=S_{0}(B,t)\in \mathbb{R}^{+}$ such that

  $$ U(t,t-s)B\subset D(t), \quad \forall s\geq S_{0}; $$

• (2) $\mathcal{{D}}$-pullback ω-limit compact with respect to each $t\in \mathbb{R}$, if, for any $\varepsilon >0$, there exists a $S_{1}=S_{1}(\mathcal{{D}},t,\varepsilon )\in \mathbb{R}^{+}$ such that

  $$ k \biggl(\bigcup_{s\geq S_{1}}U(t,t-s)D(t-s) \biggr)\leq \varepsilon . $$

Proposition 2.4

If the process $\{U(t,\tau )\}$ is $\mathcal{D}$-pullback ω-limit compact in X, then $\{U(t,\tau )\}$ is pullback ω-limit compact for any bounded subset B of X.

It follows from Theorem 3 of [10].

Definition 2.5

A family of nonempty compact subsets $A=\{A(t)\}_{t\in \mathbb{R}}$ of X is called to be a pullback attractor for the process $\{U(t,\tau )\}$ if

• (1) $\mathcal{A}=\{A(t)\}_{t\in \mathbb{R}}$ is invariant, i.e.,

  $$ U(t,\tau )A(t)=A(t), \quad \forall t\geq \tau ,\tau \in \mathbb{R} ; $$

• (2) $\mathcal{A}$ is pullback attracting, i.e., for every bounded set B of X and any fixed $t\in \mathbb{R}$,

  $$ \lim_{s\rightarrow +\infty } H^{*}_{X} \bigl(U(t,t-s)B,A(t)\bigr)=0. $$

Definition 2.6

Let $\{U(t,\tau )\}$ be a process on X. We say that ${U(t,\tau )\zeta }$ is norm-to-weak continuous in ζ for any fixed $t\geq \tau$, $\tau \in \mathbb{R}$, if there exists a sequence $\zeta_{n}\rightarrow \zeta$ in X and $t_{n} \rightarrow t$ such that ${U(t_{n},\tau )\zeta_{n}}\rightharpoonup {U(t,\tau )\zeta }$ (weak convergence).

The general existence of pullback attractors has been given as follows [10].

Proposition 2.7

Let X be a Banach space, and let $\{U(t,\tau )\}$ be a process on X. Let $U(t,\tau ) \zeta$ is norm-to-weak continuous in x for fixed $t\geq \tau$, $\tau \in {\mathbb{R}}$. If, for any fixed $t\in \mathbb{R}$, $\forall T\in \mathbb{R}^{+}$, $\bigcup_{t\geq T}D(t)$ is bounded, the process $\{U(t,\tau )\}$ is pullback dissipative and $\mathcal{{D}}$-pullback ω-limit compact with respect to each $t\in \mathbb{R}$, then $\{U(t,\tau )\}$ possesses a pullback attractor in $\mathcal{A}=\{A(t)\}_{t\in \mathbb{R}}$ in X given by

$$ A(t)=\bigcap_{T\in \mathbb{R}^{+}} \overline{{\bigcup _{s\geq T}U(t,t-s)D(t-s)}}\subset D(t). $$

3 Existence of solutions

By a solution $u\in C((-\infty ,T];X^{1})$ of (1), we mean that, for any $T>0$, $z\in (-\infty ,0]$, $\tau < t\leq T$,

$$\begin{aligned} u(t) =&e^{\Delta (t-\tau )}u(\tau )+ \int^{t}_{\tau }e^{\Delta (t-s)}\bigl[- \lambda u+f(x,u_{s})+g(x,s)\bigr]\,ds, \\ =&e^{\Delta (t-\tau )}u(\tau )+ \int^{t}_{\tau }e^{\Delta (t-s)}\bigl[- \lambda u +f \bigl(x,u{(s+z)}\bigr)+g(x,s)\bigr]\,ds, \end{aligned}$$

(8)

where $u(t)=\phi (t-\tau ,x)$, $u(\tau )=\phi (0,x)$, $t\in (-\infty , \tau ]$.

Let $A=\Delta$. $X^{\alpha }$ is the fractional power space associated to the operator Δ. The linear operator $A=\Delta$ with Dirichlet boundary conditions in a bounded and smooth domain Ω can be seen as an unbounded operator in $L^{r}(\Omega )$, $1< r< \infty$, with domain $D(A)=W^{2,r}(\Omega )\cap W^{1,r}_{0}(\Omega )$. In this situation, $-A=-\Delta$ is a sectorial operator and generates an analytic semigroup $e^{At}$ in $L^{r}({\Omega })$. Denote by $\{E^{\alpha }_{r}\}_{\alpha \in \mathbb{R}}$ the fractional power spaces associated to A with the norm $\Vert u \Vert _{E^{\alpha }_{r}}= \Vert (-A)^{ \alpha }u \Vert _{L^{r}({\Omega })}$, $u\in E^{\alpha }_{r}$. Notice that $E^{0}_{r}=L^{r}(\Omega )$ and $E^{1}_{r}=W^{2,r}(\Omega )\cap W^{1,r} _{0}(\Omega )$. It follows from [19] that the semigroup $e^{At}$ has the following smooth effect:

$$\begin{aligned} \bigl\Vert e^{At}x \bigr\Vert _{E^{\beta }_{r}}\leq t^{-(\beta -\alpha )} \Vert x \Vert _{E^{ \alpha }_{r}},\quad x\in E^{\beta }_{r}, t>0, 0\leq \alpha \leq \beta . \end{aligned}$$

(9)

Since the embedding $E^{1}_{r}\hookrightarrow E^{0}_{r}$ is compact, we know from Remark 6.1 of [20] that the resolvent of −A is compact, and the embedding $E^{\alpha }_{r}\hookrightarrow E^{\beta } _{r}$ is continuous and compact for $\forall \alpha >\beta$.

3.1 Local existence of solutions for (1) in $C_{\gamma,L^{r}(\Omega)}$ ($1< r<\infty$)

In order to apply Theorem 1 [18] to prove the existence of a solution for (1), we decompose system (1) into a linear system and a non-autonomous nonlinear system as follows, respectively:

$$ \textstyle\begin{cases} \frac{\partial v}{\partial t}-\Delta v =g(t,x) & \text{in } [\tau ,+\infty ]\times \Omega , \\ v|_{\partial \Omega} =0, & t>\tau , \\ v(t,x)=0,& \tau \in \mathbb{R}, t\in (-\infty ,\tau ], x\in \Omega , \end{cases} $$

(10)

and

$$ \textstyle\begin{cases} \frac{\partial w}{\partial t}-\Delta w=\tilde{f}(x,w_{t})+f_{1}(w) & \text{in } [\tau ,+\infty ]\times \Omega , \\ w|_{\partial \Omega} =0, & t>\tau , \\ w(t,x)=\phi (t-\tau ,x),& \tau \in \mathbb{R}, t\in (-\infty ,\tau ], x\in \Omega , \end{cases} $$

(11)

where $\tilde{f}(x,w_{t})=f(x,w_{t}+v_{t})$, $f_{1}(w)=-\lambda (w+v)$, $u _{t}=v_{t}+w_{t}$.

Lemma 3.1

([21])

For any $\tau \leq t_{1}< t_{2}$, $\frac{1}{p}+\frac{1}{q}=1$,

$$\begin{aligned} \biggl\Vert \int^{t_{2}}_{t_{1}}e^{A(t_{2}-s)}g(x,s)\,ds \biggr\Vert _{L^{r}( \Omega )}\leq \bigl\Vert g(x,t) \bigr\Vert _{L^{p}_{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))} (t _{2}-t_{1})^{\frac{1}{q}}. \end{aligned}$$

Furthermore, Eq. (10) has a unique solution $v(t)$ in the sense of (8) such that

$$\begin{aligned} v(t)\in C\bigl([\tau ,T_{0}+\tau ];L^{r}(\Omega )\bigr) \end{aligned}$$

satisfies

$$\begin{aligned} v(t)= \int^{t}_{\tau }e^{A(t-s)}g(x,s)\,ds, \end{aligned}$$

(12)

where $T_{0}$ is chosen in Lemma 3.6 later.

Proof

$$\begin{aligned}& \biggl\Vert \int^{t_{2}}_{t_{1}}e^{A(t_{2}-s)}g(x,s)\,ds \biggr\Vert _{L^{r}( \Omega )} \\& \quad \leq \int^{t_{2}}_{t_{1}} \bigl\Vert g(x,t) \bigr\Vert _{L^{p}_{\mathrm {loc}}(\mathbb{R};L^{r}( \Omega ))}\,ds \\& \quad \leq \biggl( \int^{t_{2}}_{t_{1}} \,ds \biggr)^{\frac{1}{q}} \biggl( \int^{t_{2}}_{t_{1}} \bigl\Vert g(x,t) \bigr\Vert _{L^{p}_{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))}\,ds \biggr)^{\frac{1}{p}} \\& \quad \leq \bigl\Vert g(x,t) \bigr\Vert _{L^{p}_{\mathrm {loc}}({t_{1}}, {t_{2}};L^{r}(\Omega ))} ( {t_{2}}-{t_{1}})^{\frac{1}{q}}. \end{aligned}$$

Note that we can choose $0<{t_{2}}-{t_{1}}\leq 1$. □

Lemma 3.2

Assuming (2)–(5) hold, we have

$$\begin{aligned}& \bigl\Vert \tilde{f}(t,w_{t})+f_{1}(w)\bigr\Vert _{X^{1} }\leq C_{3}(\lambda +1) \bigl(1+ \Vert w_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}} \bigr), \end{aligned}$$

(13)

$$\begin{aligned}& \bigl\Vert \tilde{f}(t,w_{t})-\tilde{f}(t, \nu_{t})+f_{1}(w)-f_{1}(\nu ) \bigr\Vert _{X ^{1}}\leq C_{4}(\lambda +1) \Vert w_{t}- \nu_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}, \end{aligned}$$

(14)

where $w,\nu \in C((-\infty ,T_{0}+\tau ];L^{r}(\Omega ))$, $t\in (\tau ,T_{0}+\tau ]$.

Proof

Denote $X^{\alpha }_{r}:=E^{\alpha -1}_{r}$, $\alpha \in \mathbb{R}$. Especially, $X^{1}_{r}:=L^{r}(\Omega )$. For any $u,\psi \in C((-\infty ,T_{0}+\tau ];L^{r}(\Omega ))$ and any $t\in (\tau ,T_{0}+\tau ]$ we get

$$\begin{aligned} \bigl\Vert F(t,u_{t}) \bigr\Vert _{X^{1}} \leq & C_{5}\bigl( \bigl\Vert k_{1}+k_{2}e^{-\gamma \rho (t)}u _{t} \bigr\Vert _{X^{1}}\bigr) \\ \leq & C_{5}\bigl(k_{1} \vert \Omega \vert +k_{2} \Vert u_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\bigr) \\ \leq & C_{5} \bigl(1+ \Vert u_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\bigr) \end{aligned}$$

(15)

and

$$\begin{aligned}& \biggl\Vert \int^{0}_{-\infty }G\bigl(t,z,u(t+z)\bigr) \,dz \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq \biggl\Vert \int^{0}_{-\infty }\bigl( \bigl\vert m_{0}(z) \bigr\vert +m_{1}(z) \bigl\vert u(t+z) \bigr\vert \bigr)\,dz \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq m_{0} \vert \Omega \vert +m_{1} \Vert u_{t} \Vert _{C_{\gamma ,L^{r}( \Omega )}} \\& \quad \leq C_{6} \bigl(1+ \Vert u_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}} \bigr). \end{aligned}$$

(16)

Combining with (15) and (16), for any $u,\psi \in C(( \tau ,T_{0}+\tau ];X^{1 })$, we have

$$\begin{aligned} \bigl\Vert f(t,u_{t}) \bigr\Vert _{X^{1}} \leq & C_{3} \bigl(1+ \Vert u_{t} \Vert _{C_{\gamma ,L ^{r}(\Omega )}} \bigr). \end{aligned}$$

(17)

By (4) and (5), we find

$$\begin{aligned}& \bigl\Vert f(t,u_{t})-f(t,\psi_{t}) \bigr\Vert _{X^{1}} \\& \quad \leq C_{1}e^{-\gamma \rho (t)} \bigl\Vert u\bigl(t-\rho (t)\bigr)-v \bigl(t-\rho (t)\bigr) \bigr\Vert _{L ^{r}(\Omega )}+ C_{2} \biggl\Vert \int^{0}_{-\infty }m_{1}(z) \vert u_{t}-\psi_{t} \vert \,dz \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq C_{3} \Vert u_{t}-\psi_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}, \end{aligned}$$

(18)

where $C_{3}$ and $C_{4}$ depend on $(k_{1},k_{2},m_{0},m_{1},m_{2})$. From (17) and (18), we obtain

$$\begin{aligned} \bigl\Vert \tilde{f}(t,w_{t}) \bigr\Vert _{{X^{1}}} \leq & C'_{3} \bigl( \Vert w_{t} \Vert _{C_{ \gamma ,L^{r}(\Omega )}}+1\bigr), \end{aligned}$$

(19)

and

$$\begin{aligned} \bigl\Vert \tilde{f}(t,w_{t})-\tilde{f}(t, \nu_{t}) \bigr\Vert _{L^{r}(\Omega )} =& \bigl\Vert f(t,w _{t}+\nu_{t})-f(t,w_{t}+\nu_{t}) \bigr\Vert _{L^{r}(\Omega )} \\ \leq & C'_{4} \Vert w_{t}- \nu_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$

(20)

Hence, (13) and (14) are obvious. □

Lemma 3.3

If $u\in C((-\infty ,T_{0}+\tau ],L^{r}(\Omega ))$, then, for all $t\in (\tau ,T_{0}+\tau ]$, $z\in (-\infty ,0]$, we have

$$\begin{aligned} \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\bigl(f_{1}(w)+ \tilde{f}(t,w_{s})\bigr)\,ds \biggr\Vert _{L^{r}(\Omega )}\leq C( \lambda +1) (t-\tau ) \bigl(\omega (t)+1\bigr), \end{aligned}$$

(21)

where

$$\begin{aligned} \omega (t)=\Bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + \sup _{\theta \in (\tau ,t]} \bigl\Vert w(\theta )+v(\theta ) \bigr\Vert _{{L ^{r}(\Omega )}}\Bigr). \end{aligned}$$

Proof

By (9), it is not difficult to see that

$$\begin{aligned}& \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\tilde{f}(t,w_{s})\,ds \biggr\Vert _{L ^{r}(\Omega )} \\& \quad \leq C(\lambda +1) \int^{t}_{\tau } \bigl(1+ \Vert w_{s}+v_{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\bigr)\,ds \\& \quad \leq C (\lambda +1) \int^{t}_{\tau } \Bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}( \Omega )}} + \sup_{\theta \in (\tau ,s]} \bigl\Vert w(\theta )+v(\theta ) \bigr\Vert _{{L^{r}(\Omega )}}\Bigr)\,ds +C(\lambda +1) (t-\tau ) \\& \quad \leq C(\lambda +1) (t-\tau )\omega (t)+C(\lambda +1) (t-\tau ). \end{aligned}$$

(22)

 □

Lemma 3.4

For any $t\in (\tau ,T_{0}+\tau ]$, $z\in (-\infty ,0]$ and any $w,\nu \in C((-\infty ,T_{0}+\tau ],L^{r}(\Omega ))$ be such that $(t-\tau ) \Vert w_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\leq \mu$, $(t-\tau ) \Vert \nu_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\leq \mu$, for some $\mu >0$. Then we have

$$\begin{aligned}& \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\bigl[\bigl( \tilde{f}(s,w_{s})-\tilde{f}(s, \nu_{s})\bigr)+ \bigl(f_{1}\bigl(w(s)\bigr)- f_{1}\bigl(\nu (s)\bigr)\bigr) \bigr]\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq C(1+\lambda ) (t-\tau ) \sup_{\theta \in (\tau ,t]} \bigl\Vert w( \theta )-\nu (\theta ) \bigr\Vert _{L^{r}(\Omega )} . \end{aligned}$$

(23)

Proof

$$\begin{aligned}& \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\bigl[\bigl( \tilde{f}(s,w_{s})-\tilde{f}(s, \nu_{s})\bigr)+ \bigl(f_{1}\bigl(w(s)\bigr)- f_{1}\bigl(\nu (s)\bigr)\bigr) \bigr]\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq C (1+\lambda ) \int^{t}_{\tau } \Vert w_{s}- \nu_{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\,ds \\& \quad \leq C (1+\lambda ) (t-\tau ) \sup_{\theta \in (\tau ,t]} \bigl\Vert w( \theta ) -\nu (\theta ) \bigr\Vert _{ { L^{r}(\Omega )}}. \end{aligned}$$

(24)

 □

Lemma 3.5

([22])

Assume $u:(-\infty ,T_{0})\rightarrow X$ is continuous and $u_{\tau }=\phi$. If there exists a nondecreasing function $m(t)\geq 0$ such that

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{X}\leq \bigl\Vert \phi (\tau ) \bigr\Vert _{X}+m(t), \quad \textit{for all } -\infty < t\leq T_{0}, \end{aligned}$$

then

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u(t+z) \bigr\Vert _{X}\leq \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert \phi (t+z) \bigr\Vert _{X}+m(t),\quad -\infty < t\leq T_{0}. \end{aligned}$$

(25)

Lemma 3.6

Assume (2)–(6) hold. Let $1< r<\infty$, $z\in (- \infty ,0]$. For any $\chi_{\tau }\in C( (-\infty ,0]; L^{r}(\Omega ))$, there exist $R(\chi_{\tau })>0$ and $T_{0}=T_{0}(\chi_{\tau })$ with the property that, for any $\phi \in B_{C_{\gamma ,L^{r}(\Omega )}}( \chi_{\tau },R)$, there exists a continuous function $w(\cdot ;\phi (0))$ with $w_{\tau }=\phi$:

$$\begin{aligned} w\in C\bigl([\tau ,T_{0}+\tau ];L^{r}(\Omega )\bigr) \end{aligned}$$

(26)

such that, for any $t\in [\tau ,T_{0}+\tau ]$, w is the unique solution of Eq. (11) in the sense of (8). This solution is a classical solution and for any $t\in (\tau ,T_{0}+ \tau ]$, satisfies

$$\begin{aligned} w_{t}\in C\bigl((-\infty ,0];L^{r}(\Omega ) \bigr) \end{aligned}$$

(27)

and

$$\begin{aligned} \lim_{t\rightarrow \tau^{+}}(t-\tau )\sup_{z\in (-\infty ,0]}e^{ \gamma z} \bigl\Vert w(t+z,\phi ) \bigr\Vert _{L^{r}(\Omega )}=0, \end{aligned}$$

(28)

and, moreover, if $\phi_{1},\phi_{2}\in B_{C_{\gamma ,L^{r}(\Omega )}}( \chi_{\tau },R)$ then

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert w(t+z, \phi_{1})-\nu (t+z,\phi _{2}) \bigr\Vert _{L^{r}(\Omega )} \leq M_{1}e^{M_{2}(t-\tau )} \Vert \phi_{1}-\phi _{2} \Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$

(29)

Furthermore, the time of existence is uniform on any bounded set (respectively, compact set) S of $C_{\gamma ,L^{r}(\Omega )}$.

Proof

Fix $\mu >0$ and for any $\tau \in \mathbb{R}$, $\forall t\in (-\infty ,\tau ]$, let $\Vert \phi \Vert _{C_{\gamma ,L^{r}( \Omega )}}\leq \mu$. We will use the contraction mapping principle to establish the existence of a solution for (11).

Let

$$\begin{aligned} K(T_{0})=\Bigl\{ w\in C\bigl((-\infty ,T_{0}+\tau ];L^{r}(\Omega )\bigr),t\in (\tau ,T _{0}+\tau ]: \sup _{t\in (\tau ,T_{0}+\tau ]} \bigl\Vert w(t) \bigr\Vert _{ { L^{r}(\Omega )}}\leq \mu +1\Bigr\} , \end{aligned}$$

with the norm

$$\begin{aligned} \Vert w \Vert _{K(T_{0})}=\sup_{t\in (\tau ,T_{0}+\tau ]} \bigl\Vert w(t) \bigr\Vert _{{L^{r}(\Omega )}}, \end{aligned}$$

where $T_{0}$ is determined later. So that $(K, \Vert \cdot \Vert )$ is a nonempty complete metric space. For each $t\in (\tau ,T_{0}+\tau ]$, we introduce the mapping

$$\begin{aligned} \begin{aligned} & \Phi :K(T_{0})\rightarrow C\bigl((-\infty ,T_{0}+\tau ];X^{1}\bigr), \\ &\Phi (w) (t)= \textstyle\begin{cases} e^{\Delta (t-\tau )}w(\tau )+\int^{t}_{\tau }e^{\Delta (t-s)}[f_{1}(w)+ \tilde{f}(s,w_{s})]\,ds, &t> \tau , \\ w(t,x)=\phi (t-\tau ,x),&t\in (-\infty ,\tau ]. \end{cases}\displaystyle \end{aligned} \end{aligned}$$

(30)

Let us first prove that Φ is a well-defined map and $\Phi (K(T _{0}))\subset K(T_{0})$. We start by showing that

$$\begin{aligned} \text{if } w \in K(T_{0}), \text{then } \Phi (w) \in C\bigl( (-\infty ,T_{0}+\tau ];L^{r}(\Omega )\bigr). \end{aligned}$$

(31)

Fixing $t_{2}\in (\tau ,T_{0}+\tau ]$, and letting $T_{0}+ \tau \geq t_{1}>t_{2}$, then we have

$$\begin{aligned}& \bigl\Vert (\Phi w) (t_{1} )-(\Phi w) (t_{2}) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl\Vert \bigl( e^{-A(t_{1})}-e^{-A(t_{2})}\bigr)w(\tau ) \bigr\Vert _{L^{r}( \Omega )} + \biggl\Vert \int^{t_{1}}_{t_{2}}e^{A(t_{1}-s)}\tilde{f}(s,w_{s})\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad\quad {} + \biggl\Vert \int^{t_{1}}_{t_{2}}e^{A(t_{1}-s)}f_{1} \bigl(w(s)\bigr)\,ds \biggr\Vert _{L ^{r}(\Omega )} + \biggl\Vert \bigl[I-e^{-A(t_{1}-t_{2})}\bigr] \int^{t_{2}}_{\tau }e ^{A(t_{2}-s)} \tilde{f}(s,w_{s})\,ds \biggr\Vert _{L^{r}(\Omega )}. \end{aligned}$$

In the above, the first and fourth term trivially go to zero as $t_{1}\rightarrow t_{2}$. Let us consider the second term. For this term we have

$$\begin{aligned}& \biggl\Vert \int^{t_{1}}_{t_{2}}e^{A(t_{1}-s)}\tilde{f}(s,w_{s})\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq C \int^{t_{1}}_{t_{2}} \bigl(1+ \Vert w_{s}+v_{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}\bigr)\,ds \\& \quad \leq C\Bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}}+ \sup_{s\in (\tau ,t_{1}]} \bigl\Vert w(s)+v(s) \bigr\Vert _{ { L^{r}(\Omega )}}\Bigr) ( {t_{1}}-{t_{2}}) +C({t_{1}}-{t_{2}}) \\& \quad \leq C\omega (t) ({t_{1}}-{t_{2}}) +C({t_{1}}-{t_{2}}), \end{aligned}$$

which goes to zero as $t_{1}\rightarrow t_{2}^{+}$. Similarly, the third term also goes to zero as $t_{1}\rightarrow t_{2}^{+}$. The case $t_{1}< t_{2}$ is similar.

Let us now show that $\Vert \Phi (w)(t) \Vert _{L^{r}(\Omega )}\leq \mu +1$, for all $t\in (\tau ,T_{0}+\tau ]$. For $\chi_{\tau } \in C((-\infty ,0]; L^{r}(\Omega ))$ fixed, choose $r\ll 1$ and $T_{0}\leq \frac{1-r}{C(\lambda +1)(1+\omega (t))}$ such that, for any $t\in (\tau ,T_{0}+\tau ]$, by (9), we have $\Vert e^{ A(t- \tau )}\chi_{\tau } \Vert _{L^{r}(\Omega )}\leq \mu$, and $\Vert e ^{ A(t-\tau )}r \Vert _{L^{r}(\Omega )}\leq r$.

Based on the above fact, we have

$$\begin{aligned}& \bigl\Vert \Phi (w) (t) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl\Vert e^{-A(t-\tau )}w(\tau ) \bigr\Vert _{L^{r}(\Omega )} +C ( \lambda +1) (t-\tau ) +C(1+\lambda ) (t-\tau ) \int^{t}_{\tau } \Vert w _{s} \Vert _{C_{L^{r}(\Omega )}}\,ds \\& \quad \leq \bigl\Vert e^{-A(t-\tau )}r \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}}+ \bigl\Vert e^{-A(t-\tau )}\chi_{\tau } \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}} +C ( \lambda +1) (t-\tau ) \bigl(1+\omega (t)\bigr) \\& \quad \leq r+ \Vert \chi_{\tau } \Vert _{C_{\gamma ,L^{r}(\Omega )}} +C ( \lambda +1) (t-\tau ) \bigl(1+\omega (t)\bigr) \\& \quad \leq \mu +r+C (\lambda +1) (t-\tau ) \bigl(1+\omega (t)\bigr). \end{aligned}$$

On the other hand, it follows from Lemma 3.3 that Φ is a strict contraction in $K(T_{0})$ and that

$$\begin{aligned} \bigl\Vert \Phi (w)-\Phi (\nu ) \bigr\Vert _{K(T_{0})}\leq C(\lambda +1) (t-\tau )\omega (t) \Vert w- \nu \Vert _{K(T_{0})}, \quad t\in [\tau ,T_{0}+\tau ]. \end{aligned}$$

The simple computations above suggest that we can choose $T_{0}$ small enough so that the map Φ is contraction from $K(T_{0})$ into itself. By the Banach contraction principle we see that Φ has a unique fixed point in $K(T_{0})$. We will denote this fixed point by $w(t,\phi )$ for $t\in (\tau ,T_{0}+\tau ]$, $\phi \in C((-\infty ,0],L ^{r}(\Omega ))$, and it is defined for $\Vert \phi -\chi_{\tau } \Vert _{C_{ \gamma ,L^{r}(\Omega )}}\leq \rho$. Note that from (31) $w(t,\phi )\in C((-\infty ,T_{0}+\tau ];L^{r}(\Omega ))$.

Let us prove that $(t-\tau ) \Vert w_{t} \Vert _{C_{\gamma ,L ^{r}(\Omega )}} \rightarrow 0$ as $t\rightarrow \tau^{+}$.

From Lemma 3.3,

$$\begin{aligned}& (t-\tau ) \bigl\Vert w(t) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq (t-\tau ) \bigl\Vert e^{A(t-\tau )}\phi (0) \bigr\Vert _{L^{r}(\Omega )} + (t-\tau ) \int^{t}_{\tau } \bigl\Vert e^{A(t-s)} \bigl(f_{1}(w)+\tilde{f}(s,w _{s})\bigr) \bigr\Vert _{L^{r}(\Omega )}\,ds \\& \quad \leq (t-\tau ) \bigl\Vert \phi (0) \bigr\Vert _{L^{r}(\Omega )} +C(1+ \lambda ) (t-\tau ) \int^{t}_{\tau }\bigl(1+ \Vert w_{s} \Vert _{C_{\gamma ,L^{r}( \Omega )}}\bigr)\,ds \\& \quad\quad {} +C(1+\lambda ) (t-\tau ) \Vert v_{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$

By Lemma 3.5, we obtain

$$\begin{aligned}& (t-\tau ) \Vert w_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}} \\& \quad \leq (t-\tau ) \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}}+C (1+ \lambda ) (t-\tau ) \int^{t}_{\tau } \Vert w_{s} \Vert _{C_{\gamma ,L ^{r}(\Omega )}}+C(1+\lambda ) (t-\tau ). \end{aligned}$$

Thus by the Gronwall inequality, we have

$$\begin{aligned}& (t-\tau ) \Vert w_{t} \Vert _{C_{\gamma ,L^{r}(\Omega )}} \\& \quad \leq (t-\tau ) \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + C (1+ \lambda ) (t-\tau ) \\& \quad\quad {} + \bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + C(1+ \lambda ) \bigr) (t-\tau )C(1+\lambda ) \int^{t}_{\tau }\exp {\bigl(C (1+ \lambda )\bigr) (t-s)}\,ds \\& \quad \leq \bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + C (1+ \lambda ) \bigr) (t-\tau ) \\& \quad\quad {} +C(1+\lambda ) \bigl( \Vert \phi \Vert _{C_{\gamma ,L^{r}(\Omega )}} + C(1+\lambda ) \bigr) (t-\tau )^{2}\exp {\bigl(C(1+\lambda ) (t-\tau )\bigr)} \overset{t\rightarrow \tau ^{+}}{\rightarrow }0. \end{aligned}$$

Moreover, if $\forall \phi_{1},\phi_{2}\in B_{{C_{\gamma ,L ^{r}(\Omega )}} }(\chi_{\tau },r)$, taking into account the estimates of Lemma 3.3 and our choice of $T_{0}$, we have

$$\begin{aligned}& \bigl\Vert w\bigl(t,\phi_{1}(0)\bigr)-\nu \bigl(t, \phi_{2}(0)\bigr) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl\Vert e^{A(t-\tau )}\bigl(\phi_{1}(0)- \phi_{2}(0)\bigr) \bigr\Vert _{L^{r}( \Omega )} \\& \quad\quad {} + \biggl\Vert \int^{t}_{\tau } e^{A(t-s)}\bigl[ \tilde{f}(s,w_{s})-\tilde{f}(s, \nu_{s})+f_{1}(w)-f_{1}( \nu )\bigr]\,ds \biggr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl\Vert (\phi_{1}-\phi_{2}) \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}} +C(1+\lambda ) \int^{t}_{\tau } \Vert w_{s}- \nu_{s} \Vert _{C_{L^{r}( \Omega )}}\,ds \\& \quad \leq \bigl\Vert (\phi_{1}-\phi_{2}) \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}} +C(1+\lambda ) (t-\tau ) \bigl\Vert (\phi_{1}- \phi_{2}) \bigr\Vert _{C_{\gamma ,L ^{r}(\Omega )}} \\& \quad\quad {} +C(1+\lambda ) \int^{t}_{\tau } \sup_{\theta \in (\tau ,s]} \bigl\Vert w( \theta )-\nu (\theta ) \bigr\Vert _{{L^{r}(\Omega )}}\,ds. \end{aligned}$$

By Lemma 3.5, we have

$$\begin{aligned}& \sup_{\theta \in (\tau ,t]} \bigl\Vert w\bigl(t, \phi_{1}(0)\bigr)-\nu \bigl(t,\phi_{2}(0)\bigr) \bigr\Vert _{L^{r}(\Omega )} \\& \quad \leq \bigl(1+C(1+\lambda ) (t-\tau )\bigr)e^{C(1+\lambda )(t-\tau )} \bigl\Vert ( \phi_{1}-\phi_{2}) \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$

Furthermore,

$$\begin{aligned}& \bigl\Vert w_{t}(\cdot,\phi_{1})-\nu_{t}(\cdot, \phi_{2}) \bigr\Vert _{C_{\gamma ,L ^{r}(\Omega )}} \\& \quad \leq \bigl(1+C(1+\lambda ) (t-\tau )\bigr) \bigl\Vert (\phi_{1}- \phi_{2}) \bigr\Vert _{C_{\gamma ,L^{r}(\Omega )}}e^{C(1+\lambda )(t-\tau )} \\& \quad \leq M_{1}(t-\tau ) \bigl\Vert (\phi_{1}- \phi_{2}) \bigr\Vert _{C_{\gamma ,L ^{r}(\Omega )}}e^{M_{1}(t-\tau )}, \end{aligned}$$

where $M_{1}=1+C(1+\lambda )$.

This concludes the existence of the theorem. Notice that, from the existence part, we see that, for any $\phi \in B_{{C_{\gamma ,L ^{r}(\Omega )}} }(\chi_{\tau },R)$, there exists a unique solution in the sense of (8), defined in $[\tau ,T_{0}+\tau ]$. The uniqueness of solutions for Eq. (11) is proved. □

Theorem 3.7

Assume (2)–(6) hold. Let $1< r<\infty$, $g\in L^{r} _{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))$ ($r>1$), $z\in (-\infty ,0]$. If $\nu_{\tau }\in C((-\infty ,0];L^{r}(\Omega ))$, there exist $0< R(\nu_{\tau })\leq R(\chi_{\tau })$ and $T_{0}(\nu_{\tau })\leq {T_{0}(\chi_{\tau })}$ with the property that, for any $\phi \in B _{C_{\gamma ,L^{r}(\Omega )}}(\nu_{\tau },R)$, there exists a continuous function $u(\cdot ;\phi (0))$ with $u_{\tau }=\phi$:

$$\begin{aligned} u\in C\bigl([\tau ,T_{0}+\tau ];L^{r}(\Omega )\bigr), \end{aligned}$$

(32)

which is the unique solution of (1) in the sense of (8). This solution is a classical solution and $\forall t \in (\tau ,T_{0}+\tau ]$ it satisfies

$$\begin{aligned} u_{t}\in C\bigl((-\infty ,0];L^{r}(\Omega ) \bigr) \end{aligned}$$

(33)

and

$$\begin{aligned} \lim_{t\rightarrow \tau^{+}}(t-\tau )\sup_{z\in (-\infty ,0]}e^{ \gamma z} \bigl\Vert u( t+z,\phi ) \bigr\Vert _{L^{r}(\Omega )}=0; \end{aligned}$$

(34)

if $\forall \phi_{1},\phi_{2}\in B_{\gamma ,L^{r}(\Omega )}( \upsilon_{\tau },r)$, then

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u_{1}(t+z,\phi_{1})-u_{2}(t+z, \phi_{2}) \bigr\Vert _{L^{r}(\Omega )}\leq M_{1}(t-\tau )e^{M_{1}(t-\tau )} \Vert \phi_{1}-\phi_{2} \Vert _{C_{\gamma ,L^{r}(\Omega )}}. \end{aligned}$$

(35)

Furthermore, the time of existence is uniform on any bounded set (respectively, compact set) S of $C_{\gamma ,L^{r}(\Omega )}$.

Proof

By Lemma 3.1 and Lemma 3.6, Eq. (1) has a unique solution $u\in C((-\infty ,T_{0}];L^{r}(\Omega ))$ satisfying (33)–(35). □

3.2 Local existence of solutions of (1) in $C_{\gamma,W^{1,r}(\Omega)}$ ($1< r< N$)

Lemma 3.8

([21])

For any $t_{1}< t_{2}$, $0< \frac{1}{q}-\frac{1}{2}$, where $\frac{1}{r}+\frac{1}{q}=1$, we have

$$\begin{aligned}& \biggl\Vert \int^{t_{2}}_{t_{1}}e^{A(t_{2}-s)}g(x,s)\,ds \biggr\Vert _{W^{1,r}( \Omega )} \\& \quad \leq \biggl(\frac{1}{1-\frac{q}{2}} \biggr)^{\frac{1}{q}} \bigl\Vert g(x,t) \bigr\Vert _{L^{r}_{b}(t_{1},t_{2};L^{r}{(\Omega )})} (t_{2}-t_{1})^{ \frac{1}{q}-\frac{1}{2}}. \end{aligned}$$

Furthermore, Eq. (10) has a unique solution $v(t)$ in the sense of (8) such that

$$\begin{aligned} v(t)\in C\bigl([\tau ,T_{0}];W^{1,r}(\Omega )\bigr)\cap C \bigl([\tau ,T_{0}+\tau ];W ^{2,r}(\Omega )\bigr) \end{aligned}$$

satisfies

$$\begin{aligned} v(t)= \int^{t}_{\tau }e^{A(t-s)}g(x,s)\,ds. \end{aligned}$$

(36)

Proof

We have

$$\begin{aligned}& \biggl\Vert \int^{t_{2}}_{t_{1}}e^{A(t_{2}-s)}g(x,s)\,ds \biggr\Vert _{W^{1,r}( \Omega )} \\& \quad \leq \biggl\Vert \int^{t_{2}}_{t_{1}}(t_{2}-s)^{-\frac{1}{2}}g(x,s)\,ds \biggr\Vert _{L^{r}{(\Omega )}} \\& \quad \leq \biggl( \int^{t_{2}}_{t_{1}}(t_{2}-s)^{-\frac{q}{2}}\,ds \biggr)^{\frac{1}{q}} \biggl( \int^{t_{2}}_{t_{1}} \bigl\Vert g(x,s) \bigr\Vert ^{r}_{L^{r}_{b}(\mathbb{R};L^{r}{( \Omega )})}\,ds\biggr)^{\frac{1}{r}} \\& \quad \leq \biggl(\frac{1}{1-\frac{q}{2}} \biggr)^{\frac{1}{q}} \bigl\Vert g(x,t) \bigr\Vert _{L^{r}_{\mathrm {loc}}(t_{1},t_{2};L^{r}{(\Omega )})} (t_{2}-t_{1})^{ \frac{1}{q}-\frac{1}{2}}. \end{aligned}$$

 □

Lemma 3.9

Assume (2)–(6) hold. Let $1< r< N$, $z\in (-\infty ,0]$. If $\chi_{\tau }\in C((-\infty ,0]; W^{1,r}(\Omega ))$, there exist $R(\chi_{\tau })>0$ and $T_{0}( \chi_{\tau })>0$ with the property that $\forall t\in (-\infty , \tau )$ for any $\phi \in B_{C_{\gamma ,W^{1,r}(\Omega )}}(\chi_{ \tau },R)$, there exists a continuous function $w(\cdot ;\phi (0))$ with $w_{\tau }=\phi$:

$$\begin{aligned} w\in C\bigl([\tau ,T_{0}+\tau ];W^{1,r}( \Omega )\bigr), \end{aligned}$$

(37)

which is the unique solution of (11) in the sense of (8). This solution is a classical solution and $\forall t \in (\tau ,T_{0}+\tau ]$, $z\in (-\infty ,0]$, satisfies

$$\begin{aligned} w_{t}\in C\bigl((-\infty ,0];W^{1,r}(\Omega ) \bigr) \end{aligned}$$

(38)

and

$$\begin{aligned} \lim_{t\rightarrow \tau^{+}}(t-\tau )\sup_{z\in (-\infty ,0]}e^{ \gamma z} \bigl\Vert w(t+z,\phi ) \bigr\Vert _{W^{1,r}(\Omega )}=0, \end{aligned}$$

(39)

and if $\phi_{1},\phi_{2}\in B_{C_{\gamma ,W^{1,r}(\Omega )}}(\chi _{\tau },R)$, then

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert w(t+z, \phi_{1})-\nu (t+z,\phi _{2}) \bigr\Vert _{W^{1,r}(\Omega )} \leq M_{1}T_{0}e^{M_{1}(t-\tau )} \Vert \phi _{1}- \phi_{2} \Vert _{C_{\gamma ,W^{1,r}(\Omega )}}. \end{aligned}$$

(40)

Furthermore, the time of existence is uniform on any bounded set (respectively, compact set) S of $C_{\gamma ,W^{1,r}(\Omega )}$.

Proof

For $\forall t\in (\tau ,T_{0}+\tau ]$, $z\in (-\infty ,0]$ and any $w,\nu \in C((-\infty ,T_{0}+\tau ];W^{1,r}(\Omega ))$, using (2),(3), we obtain (13) and (14). The remaining part of the proof is similar to Lemma 3.6. □

Theorem 3.10

Assume (2)–(6) hold. Let $1< r<\infty$, $r>1$, $z\in (-\infty ,0]$. If $\nu_{\tau }\in C((-\infty ,0]; W^{1,r}(\Omega ))$, there exist $0< R(\nu_{\tau })\leq R(\chi_{\tau })$ and $T_{0}(\nu_{\tau })\leq T_{0}(\chi_{\tau })$ with the property that for any $\phi \in B_{C_{\gamma ,W^{1,r}(\Omega )}}(\nu_{\tau },R)$, there exists a continuous function $u(\cdot ;\phi (0))$ with $u_{\tau }= \phi$:

$$\begin{aligned} u\in C\bigl([\tau ,T_{0}+\tau ];W^{1,r}( \Omega )\bigr), \end{aligned}$$

(41)

which is the unique solution of (11) in the sense of (8). This solution is a classical solution and $\forall t \in [\tau ,T_{0}+\tau ]$ it satisfies

$$\begin{aligned} u_{t}\in C\bigl((-\infty ,0];W^{1,r}(\Omega ) \bigr), \quad\quad \lim_{t\rightarrow \tau ^{+}}(t-\tau ) \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u(t+z,\phi ) \bigr\Vert _{W^{1,r}(\Omega )}=0, \end{aligned}$$

(42)

and if $\phi_{1},\phi_{2}\in B_{C_{\gamma ,W^{1,r}(\Omega )}}(\nu_{ \tau },R)$, then

$$\begin{aligned} \begin{aligned}[b] &\sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u(t+z, \phi_{1})-u(t+z,\phi_{2}) \bigr\Vert _{W^{1,r}(\Omega )}\\ &\quad \leq M_{1}(t-\tau )e^{M_{1}(t-\tau )} \Vert \phi_{1}- \phi_{2} \Vert _{C_{\gamma ,W^{1,r}(\Omega )}}. \end{aligned} \end{aligned}$$

(43)

Furthermore, the time of existence is uniform on any bounded set (respectively, compact set) S of $C_{\gamma ,W^{1,r}(\Omega )}$.

Proof

It follows from Lemmas 3.8 and 3.9. The proof is similar to Theorem 3.7. Here we omit the details. □

4 Uniform estimates in $C_{\gamma,L^{r}(\Omega)}$

Lemma 4.1

Assume that (2), (3), and (6) hold, $g\in L^{r}_{\mathrm {loc}}(\mathbb{R};L^{r}(\Omega ))$, and there exists a positive constant α such that

$$\begin{aligned} \bigl(\lambda -(\varepsilon_{2}+ {m}_{1} + \varepsilon_{4}) ({r-1})- \alpha \bigr)>0 \end{aligned}$$

(44)

and

$$\begin{aligned} L:= \biggl({m}_{1}+\frac{2^{r} k_{2}^{r}}{\lambda^{(r-1)}} \biggr)< \alpha \leq r\gamma . \end{aligned}$$

(45)

Then, for any initial data $\phi \in C_{\gamma ,L^{r}(\Omega )}$, any solution $u_{t}$ of Eq. (1) satisfies

$$\begin{aligned} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \leq & r e^{\alpha \tau }e ^{-\alpha t} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+ \frac{ \alpha }{\alpha -L}C_{\Omega }+ \varepsilon_{4}^{-(r-1)}e^{-\alpha t} \int^{t}_{-\infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds \\ & {} +r e^{(\alpha -L)\tau }e^{(L-\alpha )t} \Vert \phi \Vert ^{r}_{C_{\gamma , {L^{r}(\Omega )}}} \\ & {} + \varepsilon_{4}^{-(r-1)}e^{(L-\alpha )t} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r} \bigr)\,ds, \end{aligned}$$

(46)

where $\varepsilon_{2}$, $\varepsilon_{4}$ will be determined later on.

Proof

Multiplying (1) by $\vert u(t) \vert ^{r-2}u(t)$ and integrating by parts, we get

$$\begin{aligned}& \frac{1}{r}\frac{d}{dt} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )}+\frac{4(r-1)}{r ^{2}} \int_{\Omega } \bigl\vert \nabla \bigl( \bigl\vert u(t) \bigr\vert ^{\frac{r}{2}}\bigr) \bigr\vert ^{2}\,dx+ \int_{ \Omega } \lambda \bigl\vert u(t) \bigr\vert ^{r}\,dx \\& \quad = \int_{\Omega } F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,dx + \int_{ \Omega } \int^{0}_{-\infty }G\bigl(t,s,u(t+s)\bigr) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,ds\,dx \\& \quad\quad {} + \int_{\Omega } g(t,x) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,dx. \end{aligned}$$

(47)

We fix two positive parameters $\varepsilon_{1}$ and $\varepsilon_{4}$ that will be chosen later. Then, by assumptions (2), (6) and Young’s inequality, we have

$$\begin{aligned}& \int_{\Omega }F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) \vert u \vert ^{r-2}u\,dx \\& \quad \leq \int_{\Omega } \bigl\vert F\bigl(t,u\bigl(x,t-\rho (t)\bigr) \bigr) \bigr\vert \bigl\vert u(t) \bigr\vert ^{(r-1)}\,dx \\& \quad \leq \frac{2^{r}\varepsilon_{1}^{-(r-1)}}{r} \vert k_{1} \vert ^{r} \vert \Omega \vert ^{r} +\frac{2^{r}\varepsilon_{1}^{-(r-1)}}{r}k_{2}^{r} \Vert u_{t} \Vert ^{r}_{C _{\gamma ,L^{r}(\Omega )}} + \varepsilon_{1}\biggl(\frac{r-1}{r}\biggr) \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \end{aligned}$$

(48)

and

$$\begin{aligned} \int_{\Omega }g(t,x) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,dx \leq & \int_{\Omega } \bigl\vert g(t,x) \bigr\vert \bigl\vert u(t) \bigr\vert ^{(r-1)}\,dx \\ \leq &\frac{\varepsilon_{4}^{-(r-1)}}{r} \bigl\Vert g(t) \bigr\Vert _{L^{r}(\Omega )} ^{r} +\varepsilon_{4} \biggl(\frac{r-1}{r} \biggr) \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}( \Omega )}. \end{aligned}$$

(49)

Therefore

$$\begin{aligned}& \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )}+\frac{4(r-1)}{r} \int_{\Omega } \bigl\vert \nabla \bigl( \bigl\vert u(t) \bigr\vert ^{\frac{r}{2}}\bigr) \bigr\vert ^{2}\,dx +\bigl(r\lambda -( \varepsilon_{1}+\varepsilon_{4}) ({r-1})\bigr) \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \,dx \\& \quad \leq \varepsilon_{1}^{-(r-1)} \bigl(k_{1} \vert \Omega \vert ^{r} +k_{2}^{r} \Vert u _{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \bigr) +r \int_{\Omega } \int^{0} _{-\infty }G\bigl(t,s,u(t+s)\bigr) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,ds\,dx \\& \quad\quad {} +\varepsilon_{4}^{-(r-1)} \bigl\Vert g(t) \bigr\Vert _{L^{r}(\Omega )}^{r}. \end{aligned}$$

(50)

Let $\alpha >0$, it will also be determined later. Then

$$\begin{aligned}& \frac{d}{dt} \bigl(e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \bigr) \\& \quad = \alpha e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} +e^{\alpha t} \frac{d}{dt} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq -\frac{4(r-1)}{r}e^{\alpha t} \int_{\Omega } \bigl\vert \nabla \bigl( \bigl\vert u(t) \bigr\vert ^{ \frac{r}{2}}\bigr) \bigr\vert ^{2}\,dx -\bigl(r\lambda -( \varepsilon_{1}+\varepsilon_{4}) ( {r-1})-\alpha \bigr)e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad\quad {} + \varepsilon_{1}^{-(r-1)}e^{\alpha t} \vert k_{1} \vert ^{r} \vert \Omega \vert ^{r}+ \varepsilon_{1}^{-(r-1)}e^{\alpha t}k_{2}^{r} \Vert u_{t} \Vert ^{r}_{C_{ \gamma ,L^{r}(\Omega )}}+ \varepsilon_{4}^{-(r-1)}e^{\alpha t} \bigl\Vert g(t) \bigr\Vert _{L^{r}(\Omega )}^{r} \\& \quad\quad {} +re^{\alpha t} \int_{\Omega } \int^{0}_{-\infty }G\bigl(t,s,u(t+s)\bigr) \bigl\vert u(t) \bigr\vert ^{r-2}u(t)\,ds\,dx. \end{aligned}$$

(51)

Integrating from τ to t, we have

$$\begin{aligned}& e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq e^{\alpha \tau } \bigl\Vert u(\tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} - \int^{t} _{\tau }\bigl(r\lambda -(\varepsilon_{1}+ \varepsilon_{4}) ({r-1})-\alpha \bigr)e ^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )} \,dx \\& \quad\quad {} + \varepsilon_{1}^{-(r-1)} \vert k_{1} \vert ^{r} \vert \Omega \vert ^{r}\frac{e^{\alpha t}}{ \alpha }+ \varepsilon_{1}^{-(r-1)}k_{2}^{r} \int^{t}_{\tau }e^{\alpha s} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}}\,ds \\& \quad\quad {} +r \int^{t}_{\tau }e^{\alpha s} \int_{\Omega } \int^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr) \bigl\vert u(s) \bigr\vert ^{r-2}u(s)\,dz\,dx\,ds \\& \quad\quad {} +\varepsilon_{4}^{-(r-1)} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L ^{r}(\Omega )}^{r}\,ds. \end{aligned}$$

(52)

By assumption (3), (6) and Young’s inequality, we obtain

$$\begin{aligned}& r \biggl\vert \int^{t}_{\tau }e^{\alpha s} \int_{\Omega } \int^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr) \bigl\vert u(s) \bigr\vert ^{r-2}u(s)\,dz\,dx\,ds \biggr\vert \\& \quad \leq r \int^{t}_{\tau }e^{\alpha s} \int_{\Omega } \int^{0}_{-\infty } \bigl\vert G\bigl(s,z,u(s+z)\bigr) \bigr\vert \bigl\vert u(s) \bigr\vert ^{r-1}\,dz\,dx\,ds \\& \quad \leq \varepsilon_{2}^{-(r-1)}m_{0}^{r} \vert \Omega \vert ^{r} \int^{t}_{ \tau }e^{\alpha s}\,ds + \varepsilon_{2}(r-1) \int^{t}_{\tau }e^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )}\,ds \\& \quad \quad {} +\varepsilon_{3}^{-(r-1)} {m}_{1} \int^{t}_{\tau }e^{\alpha s} \Vert u _{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}^{r}\,ds +\varepsilon_{3}(r-1) {m} _{1} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )}\,ds \\& \quad \leq \varepsilon_{2}^{-(r-1)}m_{0}^{r} \Vert \Omega \Vert ^{r}_{L^{r}( \Omega )}\frac{e^{\alpha t}}{\alpha } + \varepsilon_{2}(r-1) \int^{t} _{\tau }e^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )}\,ds \\& \quad\quad {} +\varepsilon_{3}^{-(r-1)} {m}_{1} \int^{t}_{\tau }e^{\alpha s} \Vert u _{s} \Vert _{C_{\gamma ,L^{r}(\Omega )}}^{r}\,ds +\varepsilon_{3}(r-1) {m} _{1} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )}\,ds, \end{aligned}$$

(53)

where $\varepsilon_{2}$ and $\varepsilon_{3}$ are other positive constants to be determined later.

Combining (52)–(53) we conclude that

$$\begin{aligned}& e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq e^{\alpha \tau } \bigl\Vert u(\tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} + \biggl(\frac{ k_{1} \vert \Omega \vert ^{r}}{\varepsilon_{1}^{(r-1)}\alpha }+ \frac{m_{0}^{r} \vert \Omega \vert ^{r}}{\varepsilon_{2}^{(r-1)}\alpha } \biggr)e^{\alpha t} \\& \quad\quad {} - \bigl(r\lambda -(\varepsilon_{1}+\varepsilon_{2}+ \varepsilon_{3} {m}_{1} +\varepsilon_{4}) ({r-1})-\alpha \bigr) \int^{t}_{\tau }e^{ \alpha s} \bigl\Vert u(s) \bigr\Vert ^{r}_{L^{r}(\Omega )} \,ds \\& \quad\quad {} + \biggl(\frac{{m}_{1}}{\varepsilon_{3}^{(r-1)}}+\frac{ k_{2}^{r}}{ \varepsilon_{1}^{(r-1)}} \biggr) \int^{t}_{\tau } e^{\alpha s} \Vert u_{s} \Vert _{C_{\gamma ,L^{r}({\Omega })}}^{r}\,ds + \frac{1}{\varepsilon_{4}^{(r-1)}} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds . \end{aligned}$$

(54)

Choosing $\varepsilon_{1}=\lambda$, $\varepsilon_{3}=1$, we now can choose positive constants $\varepsilon_{2}$ and $\varepsilon_{4}$ small enough such that $(\lambda -(\varepsilon_{2}+\bar{m}_{1} +\varepsilon _{4})({r-1})-\alpha )>0$. Then

$$\begin{aligned}& e^{\alpha t} \bigl\Vert u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq e^{\alpha \tau } \bigl\Vert u(\tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} + \biggl(\frac{ k_{1} \vert \Omega \vert ^{r}}{\lambda^{(r-1)}\alpha } + \frac{m_{0}^{r} \vert \Omega \vert ^{r}}{\varepsilon_{2}^{(r-1)}\alpha } \biggr)e^{\alpha t} \\& \quad\quad {} + \biggl({m}_{1}+\frac{ k_{2}^{r}}{\lambda^{(r-1)}} \biggr) \int^{t}_{ \tau }e^{\alpha s} \Vert u_{s} \Vert _{C_{L^{r}({\Omega })}}^{r}\,ds+\varepsilon _{4}^{-(r-1)} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )} ^{r}\,ds . \end{aligned}$$

(55)

Now set $t+\theta$ instead of t, where $\theta \in (-\infty ,0]$. By the assumption (45), we have $\alpha \leq r\gamma$. Multiplying (55) by $e^{-\alpha ( t+\theta )}$ and $e^{r\gamma \theta } e^{-r\gamma \theta }$, it follows that

$$\begin{aligned} \sup_{\theta \in (\tau -t,0]}e^{r\gamma \theta } \bigl\Vert u(t+ \theta ) \bigr\Vert ^{r} _{L^{r}(\Omega )} \leq & e^{-\alpha t}e^{\alpha \tau } \Vert \phi \Vert ^{r} _{C_{\gamma ,{L^{r}(\Omega )}}}+ C_{\Omega }+ \frac{e^{-\alpha t}}{ \varepsilon_{4}^{(r-1)}} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega )}^{r}\,ds \\ & {} + \biggl({m}_{1}+\frac{ k_{2}^{r}}{\lambda^{(r-1)}} \biggr)e^{-\alpha t} \int^{t}_{\tau }e^{\alpha s} \Vert u_{s} \Vert _{C_{\gamma ,L^{r}({\Omega })}} ^{r}\,ds, \end{aligned}$$

(56)

where

$$\begin{aligned} C_{\Omega }= \biggl(\frac{ k_{1} \vert \Omega \vert ^{r}}{\lambda^{(r-1)}\alpha } + \frac{m_{0}^{r} \vert \Omega \vert ^{r}}{\varepsilon_{2}^{(r-1)}\alpha } \biggr). \end{aligned}$$

(57)

Note that

$$\begin{aligned} e^{r\gamma \theta } \bigl\Vert u(t+\theta ) \bigr\Vert ^{r}_{L^{r}(\Omega )} =&e^{r \gamma \theta } \bigl\Vert \phi (t+\theta -\tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} =e ^{-r\gamma (t-\tau )}e^{r\gamma (t+\theta -\tau )} \bigl\Vert \phi (t+\theta - \tau ) \bigr\Vert ^{r}_{L^{r}(\Omega )} \\ \leq & e^{-r\gamma (t-\tau )} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}( \Omega )}}} \leq e^{-\alpha (t-\tau )} \Vert \phi \Vert ^{r}_{C_{\gamma , {L^{r}(\Omega )}}}, \quad \forall \theta \in (-\infty ,\tau -t]. \end{aligned}$$

Let $L:={m}_{1}+\frac{2^{r} k_{2}^{r}}{\lambda^{(r-1)}}<\alpha$. Then it yields

$$\begin{aligned} e^{\alpha t} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}} \leq & re ^{\alpha \tau } \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+C_{\Omega } e^{\alpha t}+ \varepsilon_{4}^{-(r-1)} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds \\ & {} +\biggl({m}_{1}+\frac{2^{r} k_{2}^{r}}{\lambda^{(r-1)}}\biggr) \int^{t}_{\tau }e ^{\alpha s} \Vert u_{s} \Vert _{C_{\gamma ,L^{r}({\Omega })}}^{r}\,ds \\ \leq &re^{\alpha \tau } \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+C _{\Omega } e^{\alpha t}+ \varepsilon_{4}^{-(r-1)} \int^{t}_{\tau }e ^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds \\ & {} +L \int^{t}_{\tau }e^{\alpha s} \Vert u_{s} \Vert _{C_{\gamma ,L^{r}({\Omega })}} ^{r}\,ds. \end{aligned}$$

By Fubini’s theorem and Grownwall’s lemma, we find that

$$\begin{aligned} e^{\alpha t} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}} \leq &r e ^{\alpha \tau } \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+ \varepsilon_{4}^{-(r-1)} \int^{t}_{\tau }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega )}^{r}\,ds \\ & {} +r e^{(\alpha -L)\tau }e^{Lt} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}( \Omega )}}} +\frac{\alpha }{\alpha -L}C_{\Omega }e^{\alpha t} \\ & {} + \varepsilon_{4}^{-(r-1)}e^{Lt} \int^{t}_{\tau } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r} \bigr)\,ds. \end{aligned}$$

(58)

Hence, (6) and condition (45) imply that

$$\begin{aligned} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \leq & Cr e^{-\alpha t} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+ \frac{\alpha }{\alpha -L}C _{\Omega }+ \varepsilon_{4}^{-(r-1)}e^{-\alpha t} \int^{t}_{-\infty }e ^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r}\,ds \\ & {} +r e^{(\alpha -L)\tau }e^{(L-\alpha )t} \Vert \phi \Vert ^{r}_{C_{\gamma , {L^{r}(\Omega )}}} \\ & {} + \varepsilon_{4}^{-(r-1)}e^{(L-\alpha )t} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )}^{r} \bigr)\,ds \\ \overset{\vartriangle }{=}&R_{1,C_{\gamma ,L^{r}(\Omega )}} (t, \phi ,g,\alpha ,L). \end{aligned}$$

(59)

For each $t\in \mathbb{R}$, let

$$\begin{aligned} B_{R_{1,C_{\gamma ,L^{r}(\Omega )}}}(t)=\bigl\{ u\in {C_{\gamma ,L^{r}( \Omega )}}\mid \Vert u \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}}\leq R_{1,C_{\gamma ,L^{r}( \Omega )}}(t,\phi ,g,\alpha ,L)\bigr\} , \end{aligned}$$

(60)

which implies that the family of bounded sets $B=\{B_{R_{1,C_{\gamma ,L^{r}(\Omega )}}}(t)\}_{t\in \mathbb{R}}$ is pullback absorbing for the process $\{U(t,\tau )\}$ on $C_{\gamma ,L^{r}(\Omega )}$. □

5 Existence of the pullback attractors in $C_{\gamma,L^{r}(\Omega)}$ ($r>2$)

In this section, we will discuss the case where the external forcing term g belongs only to $L^{r}_{\mathrm {loc}}(\mathbb{R},L^{r}(\Omega ))$. Inspired by the idea for proving the existence of global attractors in $L^{r}(\Omega )$, we modify Theorem 5.11 [18] to prove the existence of the pullback attractors in $C_{\gamma ,L^{r}( \Omega )}$.

Lemma 5.1

Hypotheses (2), (3), (6) hold, and $g\in C(\mathbb{R};L^{2}(\Omega ))$. Then there exists a pullback attractor $\{\mathcal{A}_{C_{\gamma ,L^{2}(\Omega )}}(t)\}_{t\in \mathbb{R}}$ for the processes $\{U(t,\tau )\}$ on ${C_{\gamma ,L^{2}( \Omega )}}$ generated by the solution of Eq. (1).

Proof

By Theorem 13 [10], the processes $\{U(t,\tau )\}$ on $C_{\gamma ,H^{1}(\Omega )}$ associated with Eq. (1) has a pullback attractor $\mathcal{A}_{C_{\gamma ,H^{1}( \Omega )}}$. From the Sobolev embedding theorem $H^{1}(\Omega )\hookrightarrow \hookrightarrow L^{2}(\Omega )$ and $C_{\gamma ,H^{1}(\Omega )}\subseteq C_{\gamma ,L^{2}(\Omega )}$, $\mathcal{A}_{C_{\gamma ,H^{1}(\Omega )}}$ is a pullback attractor for the processes $\{U(t,\tau )\}$ on $C_{\gamma ,L^{2}(\Omega )}$. □

Lemma 5.2

Let $\{U(t,\tau )\}$ associated with Eq. (1) be an evolution process on $C_{\gamma ,L^{r}(\Omega )}$ with a pullback absorbing set $\mathcal{D}=\{D(t)\}_{t\in \mathbb{R}}$ on $C_{\gamma ,L^{r}(\Omega )}$. Then, for each $t\in \mathbb{R}$, for any $\varepsilon >0$, and any pullback absorbing set $\mathcal{D}\subset C_{\gamma ,L^{r}(\Omega )}$, there exist $T =T (\mathcal{D},t,\varepsilon )>0$, $M=M(\varepsilon )>0$ such that

$$\begin{aligned} m\bigl(\Omega^{\cdot }_{t}\bigl( \bigl\vert U(t,t+z)u^{0}(t+z) \bigr\vert \geq M\bigr)\bigr)\leq \varepsilon , \quad \textit{for any } -z\leq T , \textit{and } u^{0}_{t}(\cdot )\in \mathcal{D}, \end{aligned}$$

where $m(e)$ denotes the Lebesgue measure of $e\subset \Omega$ and $\Omega^{\cdot }_{t}( \vert u_{t}(z) \vert \geq M)\overset{\vartriangle }{=} \bigcup_{z\in (-\infty ,0]}\{x\in \Omega\mid \vert u(t+z,x) \vert \geq M\}$.

Proof

From the assumption that $\{U(t,\tau )\}$ has a pullback absorbing set in $C_{\gamma ,L^{r}(\Omega )}$, we know that there exists a positive constant $M_{0}$, such that, for each $t\in \mathbb{R}$ and for any pullback absorbing set $\mathcal{D}$ of $C_{\gamma ,L^{r}( \Omega )}$, we can find a positive constant T which depends on $\mathcal{D}$, such that

$$\begin{aligned} \bigl\Vert U(t,t+z)u^{0}(t+z) \bigr\Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \leq M_{0}, \quad \text{for any } -z\geq T , \text{and } u^{0}_{t}( \cdot )\in \mathcal{D}. \end{aligned}$$

So, we have

$$\begin{aligned} 2M_{0} \geq &2\sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega } \bigl\vert U(t,t+z)u ^{0}(t+z) \bigr\vert ^{r}\,dx \\ \geq & \sup_{z\in (-\infty ,-T_{1}]}e^{\gamma z} \int_{\Omega^{\cdot }_{t}(\{ \vert u(t+z) \vert \geq M_{1}\})} \bigl\vert U(t,t+z)u^{0}(t+z) \bigr\vert ^{r}\,dx \\ & {} +\sup_{z\in (-T_{1},0]}e^{\gamma z} \int_{\Omega^{\cdot }_{t}(\{ \vert u(t+z) \vert \geq M_{1}\})} \bigl\vert U(t,t+z)u^{0}(t+z) \bigr\vert ^{r}\,dx \\ \geq &e^{-\gamma T_{1}}\biggl( \int_{\Omega^{\cdot }_{t}(\{ \vert U(t,t+z)u^{0}(t+z) \vert \geq M_{1}\})}M_{1} ^{r}\,dx + \int_{\Omega^{\cdot }_{t}(\{ \vert U(t,t+z)u^{0}(t+z) \vert \geq M_{1}\})}M _{1}^{r}\,dx\biggr) \\ \geq & 2e^{-\gamma T_{1}}M_{1}^{r}m\bigl(\Omega \bigl(\bigl\{ \bigl\vert U(t,t+z)u^{0}(t+z) \bigr\vert \geq M_{1} \bigr\} \bigr)\bigr). \end{aligned}$$

This inequality implies that $m(\Omega^{\cdot }_{t}(\{ \vert U(t,t+z)u^{0}(t+z) \vert \geq M_{1}\}))\leq \varepsilon$, if we choose $M_{1}$ large enough such that $M_{1}\geq (\frac{M_{0}}{e^{-\gamma T_{1}}\varepsilon })^{ \frac{1}{r}}$. □

Lemma 5.3

For each $t\in \mathbb{R}$, any $\varepsilon >0$, the pullback absorbing set $\mathcal{D}$ of process $\{U(t,\tau )\}$ associated with Eq. (1) on $C_{\gamma ,L^{r}(\Omega )}$ ($r>0$) has a finite ε-net in $C_{\gamma ,L^{r}(\Omega )}$, if there exists a positive constant $M=M(\varepsilon )$ which depends on ε, such that

• (i) $\mathcal{D}$ has a finite $(3M)^{(2-r)/2}(\frac{\varepsilon }{2})^{ \frac{r}{2}}$-net in $C_{\gamma ,L^{2}(\Omega )}$,

• (ii)
  $$\begin{aligned} \begin{aligned}[b]& \biggl(\sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}(\{ \vert u(t+z) \vert \geq M\})} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx \biggr)^{ \frac{1}{r}}\\&\quad < 2^{-(2r+2)/r}\varepsilon , \quad \textit{for any } u_{t}(\cdot )\in \mathcal{D}.\end{aligned} \end{aligned}$$

  (61)

Proof

For each $t\in \mathbb{R}$, any fixed $\varepsilon >0$, it follows from the assumptions that $\mathcal{D}$ has a finite $\frac{(3M)^{(2-r)}}{2\varepsilon^{r/2}}$-net in $C_{\gamma ,L^{2}( \Omega )}$, that is, there exist $u_{t}^{1},\ldots,u_{t} ^{k}\in \mathcal{D}$, such that, for each $u_{t}(\cdot )\in \mathcal{D}$, we can find some $u_{t}^{i}$ ($1\leq i\leq k$) satisfying

$$\begin{aligned} \bigl\Vert u(t+z)-u^{i}(t+z) \bigr\Vert ^{2}_{L^{2}(\Omega )} \leq & \sup_{z\in (-\infty ,0]} e^{\gamma z} \bigl\Vert u(t+z)-u^{i}(t+z) \bigr\Vert ^{2}_{L ^{2}(\Omega )} \\ =&\sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u_{t}-u_{t}^{i} \bigr\Vert ^{2} _{L^{2}(\Omega )} < (3M)^{(2-r)} \biggl(\frac{\varepsilon }{2} \biggr)^{r}. \end{aligned}$$

(62)

Then, obviously, we have

$$\begin{aligned}& \bigl\Vert u_{t}-u_{t}^{i} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}(\Omega )}} \\& \quad \leq \sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z)-u^{i}(t+z) \vert \geq 3M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad\quad {} +\sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z)-u^{i}(t+z) \vert \leq 3M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \end{aligned}$$

(63)

and

$$\begin{aligned}& \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z)-u^{i}(t+z) \vert \leq 3M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad \leq (3M)^{r-2}\sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u_{t}-u_{it} \vert \leq 3M)} \bigl\vert u_{t}-u_{t}^{i} \bigr\vert ^{2}\,dx, \\& \quad \leq (3M)^{r-2}(3M)^{2-r} \biggl(\frac{\varepsilon }{2} \biggr)^{r}= \biggl(\frac{\varepsilon }{2} \biggr)^{r}. \end{aligned}$$

(64)

On the other hand, set

$$\begin{aligned}& \Omega_{1}^{z}=\Omega^{z}_{t}\biggl( \bigl\vert u(t+z) \bigr\vert \geq \frac{3M}{2}\biggr)\cap \Omega _{t}^{z}\biggl( \bigl\vert u^{i}(t+z) \bigr\vert \leq \frac{3M}{2}\biggr), \\& \Omega_{2}^{z}=\Omega^{z}_{t}\biggl( \bigl\vert u(t+z) \bigr\vert \leq \frac{3M}{2}\biggr)\cap \Omega _{t}^{z}\biggl( \bigl\vert u^{i}(t+z) \bigr\vert \geq \frac{3M}{2}\biggr), \\ & \Omega_{3}^{z}=\Omega^{z}_{t} \biggl( \bigl\vert u(t+z) \bigr\vert \geq \frac{3M}{2}\biggr)\cap \Omega _{t}^{z}\biggl( \bigl\vert u^{i}(t+z) \bigr\vert \geq \frac{3M}{2}\biggr), \end{aligned}$$

then we have

$$\begin{aligned} \Omega^{z}_{t}\bigl( \bigl\vert u(t+z) \bigr\vert \geq 3M\bigr)\subset \Omega_{1}^{z}\cup \Omega_{2} ^{z}\cup \Omega_{3}^{z}. \end{aligned}$$

From the simple facts that $\vert u(t+z)-u^{i}(t+z) \vert \leq 2 \vert u(t+z) \vert$ in $\Omega_{1}^{z}$ and $\vert u(t+z)-u^{i}(t+z) \vert \leq 2 \vert u^{i}(t+z) \vert$ in $\Omega_{2}^{z}$, combining with (61), we have

$$\begin{aligned}& \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z)-u^{i}(t+z) \vert \geq 3M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\ & \quad \leq \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega_{1} ^{z}} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx+\sup_{z\in (-\infty ,0]} e^{ \gamma z} \int_{\Omega_{2}^{z}} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\ & \quad\quad {} + \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega_{3}^{z}} \bigl\vert u(t+z)-u ^{i}(t+z) \bigr\vert ^{r}\,dx \\ & \quad \leq 2^{r}\sup_{z\in (-\infty ,0]} e^{\gamma z} \biggl( \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx+ \int_{\Omega^{z}_{t}( \vert u^{i}(t+z) \vert \geq M)} \bigl\vert u^{i}(t+z) \bigr\vert ^{r}\,dx \\ & \quad\quad {} + \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx + \int_{\Omega_{t}^{z}( \vert u^{i}(t+z) \vert \geq M)} \bigl\vert u^{i}(t+z) \bigr\vert ^{r}\,dx \biggr) \\ & \quad \leq 2^{r+2}\cdot 2^{(2r+2)}\varepsilon^{r}= \biggl( \frac{\varepsilon }{2} \biggr)^{r}. \end{aligned}$$

(65)

Substituting (64) and (65) into (63), we can deduce that

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \bigl\Vert u(t+z)-u^{i}(t+z) \bigr\Vert _{L ^{r}(\Omega )} \leq \frac{\varepsilon }{2}+\frac{\varepsilon }{2}= \varepsilon , \end{aligned}$$

which means that $\mathcal{D}$ has a finite ε-net in $C_{\gamma ,L^{r}(\Omega )}$. □

Lemma 5.4

Let $\mathcal{D}$ be a pullback absorbing set in $C_{\gamma ,L^{r}( \Omega )}$ ($r\geq 1$). If $\mathcal{D}$ has a finite ε-net in $C_{\gamma ,L^{r}(\Omega )}$ ($r\geq 1$) then there exists a positive $M=M(B,\varepsilon )$, such that, for any $u_{t}(\cdot )\in \mathcal{D}$, $z\in (-\infty ,0]$, we can find

$$\begin{aligned} \sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx \leq 2^{r+1} \varepsilon^{r}. \end{aligned}$$

Proof

Since $\mathcal{D}$ has a finite ε-net in $C_{\gamma ,L^{r}(\Omega )}$ ($r\geq 1$), for each $t\in \mathbb{R}$, we know that there exist $u_{t}^{1},\ldots,u_{t}^{k}\in \mathcal{D}$, such that, for any $u_{t}(\cdot )\in \mathcal{D}$, we can find some $u_{t}^{i}$ ($1\leq i\leq k$) satisfying

$$\begin{aligned} \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx\leq \varepsilon^{r}. \end{aligned}$$

(66)

Simultaneously, for the fixed $\varepsilon >0$, there exists a $\delta >0$, such that, for each $u_{t}^{i}$, $1\leq i\leq k$, we have

$$\begin{aligned} \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{e} \bigl\vert u^{i}(t+z) \bigr\vert ^{r}\,dx \leq \varepsilon^{r}, \end{aligned}$$

(67)

provided that $m(e)<\delta$ ($e\subset \Omega$).

On the other hand, since $\mathcal{D}$ is bounded in $C_{\gamma ,L^{r}(\Omega )}$ ($r\geq 1$), for the fixed $\delta >0$ above, there exists $M>0$, such that $m(\Omega^{\cdot }_{t}( \vert u(t+z) \vert \geq M))< \delta$ holds for each $u_{t}\in B$. So, $m(\Omega^{z}_{t}( \vert u(t+z) \vert \geq M))<\delta$ also holds for each $u_{t}\in B$.

Therefore,

$$\begin{aligned}& \sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx \\& \quad = \sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z)-u^{i}(t+z)+u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad \leq 2^{r}\sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z)-u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad\quad {} + 2^{r}\sup_{z\in (-\infty ,0]}e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u^{i}(t+z) \bigr\vert ^{r}\,dx \\& \quad \leq 2^{r+1}\varepsilon^{r}. \end{aligned}$$

(68)

 □

Lemma 5.5

For each $t\in \mathbb{R}$, for any $\varepsilon >0$ and any pullback absorbing set $\mathcal{D}\in C_{\gamma ,L^{2}(\Omega )}$, there exist two positive constants $T_{3}=T_{3}(B,\varepsilon )=\max \{T_{1},T _{2}\}$ and $M=M(\varepsilon )$, such that

$$\begin{aligned} \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z} _{t}( \vert u(t+z) \vert \geq M)} \bigl\vert u(t+z) \bigr\vert ^{r}\,dx< C \varepsilon , \quad \textit{for any } -z\geq T_{3},u^{0}_{t}( \cdot )\in \mathcal{D}, \end{aligned}$$

(69)

where the constant C is independent of ε and $\mathcal{D}$.

Proof

For each $t\in \mathbb{R}$, any fixed $\varepsilon >0$, there exists $\delta >0$ such that if $e\subset \Omega$ and $m(e)\leq \delta$, then

$$\begin{aligned} \int_{e} \bigl\vert \phi (x) \bigr\vert ^{r} \,dx \leq C\varepsilon , \end{aligned}$$

(70)

where $\phi (x),g(x)\in L^{r}{(\Omega )}$. Moreover, from Lemmas 5.1, 5.2 and 5.4, we know that there exist $T=T (\mathcal{D},\varepsilon )>0$ and $M=M(\varepsilon )$, for each $-z\geq T$, $u_{t}(\cdot )\in D$, we have

$$\begin{aligned} m\bigl(\Omega^{z}_{t}\bigl( \bigl\vert u(t+z) \bigr\vert \geq M\bigr)\bigr)< \min \{\varepsilon ,\delta \}, \quad \text{for each } t\in \mathbb{R}, \end{aligned}$$

(71)

and

$$\begin{aligned} \sup_{z\in (-\infty ,0]} e^{\gamma z} \int_{\Omega^{z}_{t}( \vert u(t+z) \vert )\geq M} \bigl\vert u(t+z) \bigr\vert ^{2}< 8 \varepsilon . \end{aligned}$$

(72)

Thus, we also have

$$\begin{aligned} \int_{\Omega^{0}_{t}( \vert u(t) \vert \geq M)} \bigl\vert u(t) \bigr\vert ^{2}< 8 \varepsilon , \quad \text{for } t\in [T,+\infty ]. \end{aligned}$$

(73)

Multiplying (1) by $(u-M)_{+}^{r-1}$ and integrating over $\Omega^{0}_{t}=\Omega^{0}_{t}(u >M)$, we have

$$\begin{aligned}& \int_{\Omega^{0}_{t}(u >M)}\frac{\partial u}{\partial t}(u-M)_{+} ^{r-1} \,dx - \int_{\Omega^{0}_{t}(u >M)}\Delta u(u-M)_{+}^{r-1} \,dx \\& \quad\quad {} + \int_{\Omega^{0}_{t}(u >M)}\lambda u(u-M)_{+}^{r-1} \,dx \\& \quad = \int_{\Omega^{0}_{t}(u >M)}f(t,u_{t}) (u-M)_{+}^{r-1} \,dx + \int_{\Omega^{0}_{t}(u >M)}g(t,x) (u-M)_{+}^{r-1} \,dx. \end{aligned}$$

(74)

After integrating over $\Omega^{0}_{t}(u >M)$, (74) becomes

$$\begin{aligned}& \frac{1}{r}\frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} - \int_{\Omega^{0}_{t}(u >M)}\Delta u(u-M)_{+}^{r-1} \,dx + \lambda \int_{\Omega^{0}_{t}(u >M)}u(u-M)_{+}^{r-1} \,dx \\& \quad = \int_{\Omega^{0}_{t}(u >M)}F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) (u-M)_{+}^{r-1} \,dx+ \int_{\Omega^{0}_{t}( \vert u \vert >M)}g(t,x) (u-M)_{+}^{r-1} \,dx \\& \quad\quad {} + \int_{\Omega^{0}_{t}(u >M)} \int^{0}_{-\infty } \bigl\vert G\bigl(s,z,u(s+z)\bigr) \bigr\vert (u-M)_{+} ^{r-1}\,dz \,dx, \end{aligned}$$

(75)

where

$$ (u-M)_{+}= \textstyle\begin{cases} u-M,& u\geq M, \\ 0,& u\leq M. \end{cases} $$

Let $\Omega^{0}_{1,t}=\Omega^{0}_{t}(u >M)$, then we have

$$\begin{aligned}& \frac{1}{r}\frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} - \int_{ \Omega^{0}_{1,t}}\Delta u(u-M)_{+}^{r-1} \,dx+\lambda \int_{ \Omega^{0}_{1,t}}u(u-M)_{+}^{r-1} \,dx \\& \quad = \int_{ \Omega^{0}_{1,t}}F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) (u-M)_{+}^{r-1} \,dx+ \int_{ \Omega^{0}_{1,t}}g(t,x) (u-M)_{+}^{r-1} \,dx \\& \quad\quad {} + \int_{ \Omega^{0}_{1,t}} \int^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr) (u-M)_{+} ^{r-1} \,dz \,dx. \end{aligned}$$

We now estimate every term of (75). First, we obtain

$$\begin{aligned} - \int_{ \Omega^{0}_{1,t}}\Delta u(u-M)_{+}^{r-1} \,dx=(r-1) \int_{\Omega^{0}_{1}}\nabla u \bigl\vert (u-M)_{+} \bigr\vert ^{r-2}\nabla u \,dx\geq 0 \end{aligned}$$

(76)

and

$$\begin{aligned} \lambda \int_{ \Omega^{0}_{1,t}}u(u-M)_{+}^{r-1}\,dx\geq \lambda \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )}. \end{aligned}$$

(77)

By the assumption (2), (3), (6) and Young’s inequality, we have

$$\begin{aligned}& \int_{ \Omega^{0}_{1,t}}F\bigl(t,u\bigl(x,t-\rho (t)\bigr)\bigr) (u-M)_{+}^{r-1}\,dx \\ & \quad \leq \frac{\varepsilon^{-(r-1)}_{1}}{r} \int_{ \Omega^{0}_{1,t}} \bigl\vert F\bigl(x,u\bigl(x,t- \rho (t)\bigr) \bigr) \bigr\vert ^{r}\,dx +\frac{(r-1)\varepsilon_{1}}{r} \int_{ \Omega^{0} _{1,t}}(u-M)_{+}^{r}\,dx \\ & \quad \leq \frac{ \varepsilon^{-(r-1)}_{1}}{r} \int_{ \Omega^{0}_{1,t}} \vert k _{1} \vert ^{r}\,dx + \frac{ k^{r}_{2}\varepsilon^{-(r-1)}_{1}}{r} \int_{ \Omega^{0}_{1,t}}e^{-r\gamma \rho (t)} \bigl\vert u\bigl(x,t-\rho (t) \bigr) \bigr\vert ^{r}\,dx \\ & \quad\quad {} +\frac{(r-1)\varepsilon_{1}}{r} \int_{ \Omega^{0}_{1,t}}(u-M)_{+} ^{r}\,dx \\ & \quad \leq \frac{ \varepsilon^{-(r-1)}_{1}}{r} \vert k_{1} \vert ^{r}| _{L^{r}( \Omega ^{0}_{1,t})} +\frac{ k^{r}_{2}\varepsilon^{-(r-1)}_{1}}{r} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1})}}+ \frac{(r-1)\varepsilon_{1}}{r} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}( \Omega^{0} _{1,t})} , \end{aligned}$$

(78)

$$\begin{aligned}& \int_{ \Omega^{0}_{1,t}} \int^{0}_{-\infty }G\bigl(x,z,u(s+z)\bigr) (u-M)_{+} ^{r-1} \,dz \,dx \\ & \quad \leq \int_{ \Omega^{0}_{1,t}} \int^{0}_{-\infty } \bigl\vert m_{0}(z) \bigr\vert \bigl\vert (u-M)_{+} \bigr\vert ^{r-1}\,dz \,dx + \int_{ \Omega^{0}_{1,t}} \int^{0}_{-\infty }m_{1}(z)| u(t+z ) (u-M)_{+} ^{r-1}\,dz\,dx \\ & \quad \leq \frac{ \varepsilon^{-(r-1)}_{2}}{r} \int_{ \Omega^{0}_{1,t}} \vert m _{0} \vert ^{r}\,dx + \frac{(r-1)\varepsilon_{2}}{r} \int_{\Omega_{1}}(u-M)_{+} ^{r}\,dx \\ & \quad\quad {} +\frac{\bar{m}_{1}\varepsilon^{-(r-1)}_{3}}{r} \int_{ \Omega^{0}_{1,t}} \bigl\vert u(t+z ) \bigr\vert ^{r}\,dx + \frac{\bar{m}_{1}(r-1)\varepsilon _{3}}{r} \int_{ \Omega^{0}_{1,t}}(u-M)_{+}^{r}\,dx \\ & \quad \leq \frac{ \varepsilon^{-(r-1)}_{2}}{r} \vert m_{0} \vert ^{r}_{L^{r}( \Omega ^{0}_{1,t})}+ \frac{(r-1)\varepsilon_{2}}{r} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})} \\ & \quad\quad {} +\frac{\bar{m}_{1}(r-1)\varepsilon_{3}}{r} \bigl\Vert u(t+z ) \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}, \end{aligned}$$

(79)

and

$$\begin{aligned} \int_{ \Omega^{0}_{1,t}}g(t,x) (u-M)_{+}^{r-1} \,dx &\leq \int_{\Omega^{0}_{1,t}} \bigl\vert g(t,x) \bigr\vert (u-M)_{+}^{r-1}\,dx \\ & \leq \frac{\varepsilon^{-(r-1)}_{4}}{r} \int_{\Omega^{0}_{1,t}} \bigl\vert g(t,x) \bigr\vert ^{r}\,dx + \frac{(r-1)\varepsilon_{4}}{r} \int_{\Omega^{0}_{1,t}}(u-M)_{+}^{r}\,dx \\ & \leq \frac{\varepsilon^{-(r-1)}_{4}}{r} \bigl\Vert g(t,x) \bigr\Vert ^{r}_{L^{r}( \Omega ^{0}_{1,t})} +\frac{(r-1)\varepsilon_{4}}{r} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}. \end{aligned}$$

(80)

Combining with (76)–(80), we can conclude that

$$\begin{aligned}& \frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} +r(r-1) \int_{ \Omega^{0}_{1,t}}\nabla u(u-M)_{+}^{r-2}\nabla u \,dx \\ & \quad\quad {} +r\lambda \int_{ \Omega^{0}_{1,t}}u(u-M)_{+}^{r-1}\,dx \\ & \quad \leq \varepsilon^{-(r-1)}_{1} \int_{ \Omega^{0}_{1,t}} \vert k_{1} \vert ^{r}\,dx + \varepsilon^{-(r-1)}_{2} \int_{ \Omega^{0}_{1,t}} \vert m_{0} \vert ^{r}\,dx \\ & \quad\quad {} +(r-1) (\varepsilon_{1}+\varepsilon_{2}+ {m}_{1}\varepsilon_{3}+ \varepsilon_{4}) \int_{ \Omega^{0}_{1,t}}(u-M)_{+}^{r}\,dx \\ & \quad\quad {} + k^{r}_{2}\varepsilon^{-(r-1)}_{1}e^{-r\gamma \rho (t)} \int_{ \Omega^{0}_{1,t}} \bigl\vert u\bigl(x,t-\rho (t)\bigr) \bigr\vert ^{r}\,dx \\ & \quad\quad {} + {m}_{1}\varepsilon^{-(r-1)}_{3} \int_{ \Omega^{0}_{1,t}}e^{\gamma z} \bigl\vert u(t+z ) \bigr\vert ^{r}\,dx +\varepsilon^{-(r-1)}_{4} \int_{ \Omega^{0}_{1,t}} \bigl\vert g(t,x) \bigr\vert ^{r}\,dx. \end{aligned}$$

(81)

We also have

$$\begin{aligned}& \frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad \leq -r\lambda \bigl\Vert (u-M)_{+} \bigr\Vert _{r}^{r}+ \varepsilon^{-(r-1)}_{1} \vert k _{1} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}+\varepsilon^{-(r-1)}_{2} \vert m_{0} \vert ^{r} _{L^{r}( \Omega^{0}_{1,t})} \\& \quad\quad {} +(r-1) (\varepsilon_{1}+\varepsilon_{2}+ {m}_{1}\varepsilon_{3}+ \varepsilon_{4}) \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}+ k^{r} _{2}\varepsilon^{-(r-1)}_{1} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega ^{0}_{1,t})}} \\& \quad\quad {} + {m}_{1}\varepsilon^{-(r-1)}_{3} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{0}_{1,t})}}+\varepsilon^{-(r-1)}_{4} \bigl\Vert g(t,x) \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}. \end{aligned}$$

(82)

Let $\alpha >0$, which will also be determined later. Then

$$\begin{aligned}& \frac{d}{dt}e^{\alpha t} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad =\alpha e^{\alpha t} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{r} +e^{\alpha t}\frac{d}{dt} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{r} \\& \quad \leq -\bigl(r\lambda -\alpha -(r-1) (\varepsilon_{1}+ \varepsilon_{2}+ {m}_{1}\varepsilon_{3}+ \varepsilon_{4})\bigr) e^{\alpha t} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega )} \\& \quad\quad {} +\bigl( \varepsilon^{-(r-1)}_{1} \vert k_{1} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}+ \varepsilon^{-(r-1)}_{2} \vert m_{0} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}\bigr) e^{ \alpha t} +\varepsilon^{-(r-1)}_{4}e^{\alpha t} \bigl\Vert g(t,x) \bigr\Vert ^{r}_{L^{r}( \Omega^{0}_{1,t})} \\& \quad\quad {} +\bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1}\varepsilon^{-(r-1)} _{3}\bigr)e^{\alpha t} \Vert u_{t} \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{0}_{1,t})}}. \end{aligned}$$

(83)

Let $A=(r\lambda -\alpha -(r-1)(\varepsilon_{1}+\varepsilon_{2}+ {m}_{1}\varepsilon_{3}+\varepsilon_{4}))$. By Gronwall’s inequality, we have

$$\begin{aligned}& e^{\alpha t} \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega_{t,0}^{1} )} \\& \quad \leq e^{-A(t-\tau )}e^{\alpha \tau } \bigl\Vert \bigl(u(\tau )-M \bigr)_{+} \bigr\Vert ^{r}_{C _{\gamma ,L^{r}(\Omega )}} + \varepsilon^{-(r-1)}_{4}e^{-At} \int^{t} _{-\infty }e^{(A+\alpha )s} \bigl\Vert g(s,x) \bigr\Vert ^{r}_{L^{r}(\Omega^{0}_{1,t})}\,ds \\& \quad\quad {} +\bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1}\varepsilon^{-(r-1)} _{3}\bigr)e^{-At} \int^{t}_{\tau }e^{(A+\alpha )s} \Vert u_{s} \Vert ^{r}_{C_{ \gamma ,L^{r}(\Omega^{0}_{1,t})}}\,ds \\& \quad\quad {} +\bigl( \varepsilon^{-(r-1)}_{1} \vert k_{1} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}+ \varepsilon^{-(r-1)}_{2} \vert m_{0} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}\bigr) \frac{e ^{\alpha t}}{A+\alpha }. \end{aligned}$$

(84)

Thanks to (46), and letting $\alpha_{1}>\alpha \geq \alpha ^{*}$, we can deduce that

$$\begin{aligned}& \bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1}\varepsilon^{-(r-1)} _{3} \bigr)e^{-At} \int^{t}_{\tau } e^{(A+\alpha )s} \Vert u_{s} \Vert ^{r}_{C _{\gamma ,L^{r}(\Omega_{t,0}^{1} )}}\,ds \\& \quad \leq \bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1} \varepsilon^{-(r-1)}_{3} \bigr) \biggl( \frac{re^{\alpha \tau }}{A} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega_{t,0}^{1})}} + \frac{\alpha C_{ \Omega_{t,0}^{1}}e^{\alpha t}}{(A+\alpha )(\alpha -L)} \\& \quad\quad {} + \varepsilon_{4}^{-(r-1)}\frac{1}{A} \int^{t}_{-\infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega_{t,0}^{1})}^{r}\,ds +\frac{r e^{(\alpha -L) \tau }e^{Lt}}{(A+L)} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega_{t,0}^{1})}} \\& \quad\quad {} +\varepsilon_{4}^{-(r-1)}\frac{e^{Lt}}{(A+L)} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega_{t,0}^{1})}^{r} \bigr)\,ds \biggr). \end{aligned}$$

(85)

Multiplying (84) by $e^{-\alpha t}$, we have

$$\begin{aligned}& \bigl\Vert (u-M)_{+} \bigr\Vert ^{r}_{L^{r}(\Omega_{t,0}^{1} )} \\& \quad \leq e^{-A(t-\tau )}e^{\alpha \tau }e^{-\alpha t} \bigl\Vert \bigl(u( \tau )-M\bigr)_{+} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}}+ \frac{\varepsilon^{-(r-1)} _{1} \vert k_{1} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}}{A+\alpha } \\& \quad\quad {} +\varepsilon^{-(r-1)}_{4}e^{-(A+\alpha )t} \int^{t}_{-\infty }e^{(A+ \alpha )s} \bigl\Vert g(s,x) \bigr\Vert ^{r}_{L^{r}(\Omega^{0}_{1,t})}\,ds+\frac{ \varepsilon^{-(r-1)}_{2} \vert m_{0} \vert ^{r}_{L^{r}( \Omega^{0}_{1,t})}}{A+ \alpha } \\& \quad\quad {} + \bigl( k^{r}_{2}\varepsilon^{-(r-1)}_{1}+ {m}_{1}\varepsilon^{-(r-1)} _{3} \bigr) \biggl( \frac{re^{\alpha \tau }e^{-\alpha t}}{A} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}} + \frac{\alpha C_{\Omega ^{0}_{1,t}}}{(A+\alpha )(\alpha -L)} \\& \quad\quad {} + \varepsilon_{4}^{-(r-1)}\frac{1}{A} e^{-\alpha t} \int^{t}_{- \infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega^{0}_{1,t})}^{r}\,ds +\frac{r e^{(\alpha -L)\tau }e^{-(\alpha -L)t}}{(A+L)} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}} \\& \quad\quad {} +\varepsilon_{4}^{-(r-1)}\frac{e^{-(\alpha -L)t}}{(A+L)} \int^{t} _{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega^{0}_{1,t})} ^{r} \bigr)\,ds \biggr) \\& \quad \leq e^{\alpha \tau }e^{-\alpha t} \bigl\Vert (\phi -M)_{+} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t} )}} +Ce^{-(A+\alpha )t} \int^{t}_{-\infty }e ^{(A+\alpha )s} \bigl\Vert g(s,x) \bigr\Vert ^{r}_{L^{r}(\Omega^{0}_{1,t})}\,ds \\& \quad\quad {} + C m\bigl({\Omega^{0}_{1,t}}\bigr) +Ce^{\alpha \tau }e^{-\alpha t} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}} +CC_{\Omega^{0}_{1,t}} \\& \quad\quad {} + C e^{-\alpha t} \int^{t}_{-\infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega^{0}_{1,t})}^{r}\,ds +Ce^{(\alpha -L)\tau }e^{-(\alpha -L)t} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}(\Omega^{0}_{1,t})}} \\& \quad\quad {} +Ce^{-(\alpha -L)t} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega^{0}_{1,t})}^{r} \bigr)\,ds. \end{aligned}$$

(86)

Now replacing t by $t+z$, similar to the arguments in Lemma 4.1, in view of (45), we have

$$\begin{aligned}& e^{r\gamma z} \bigl\Vert (u_{t}-M)_{+} \bigr\Vert ^{r}_{{L^{r}( \Omega^{z}_{1,t} )}} \\& \quad \leq e^{\alpha \tau }e^{-\alpha t}e^{(r\gamma -\alpha )z} \bigl\Vert ( \phi -M)_{+} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{z}_{1,t} )}} +Ce^{-(A+ \alpha )t} \int^{t}_{-\infty }e^{(A+\alpha )s} \bigl\Vert g(s,x) \bigr\Vert ^{r}_{L^{r}( \Omega^{z}_{1,t})}\,ds \\& \quad\quad {} + C m\bigl({ \Omega^{z}_{1,t}}\bigr)e^{(r\gamma -\alpha )z} +Ce^{\alpha \tau }e^{-\alpha t} \Vert \phi \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{z}_{1,t})}} +CC_{ \Omega^{z}_{1,t}} \\& \quad\quad {} + C e^{-\alpha t}e^{(r\gamma -\alpha )z} \int^{t}_{-\infty }e^{ \alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega^{z}_{1,t})}^{r}\,ds +Ce^{(\alpha -L) \tau }e^{-(\alpha -L)t}e^{(r\gamma +L-\alpha )z} \Vert \phi \Vert ^{r}_{C_{ \gamma ,L^{r}( \Omega^{z}_{1,t})}} \\& \quad\quad {} +Ce^{-(\alpha -L)t}e^{(r\gamma +L-\alpha )z} \int^{t}_{-\infty } \bigl(e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L^{r}( \Omega^{z}_{1,t})}^{r} \bigr)\,ds. \end{aligned}$$

(87)

Furthermore, by (57) and (70), we have

$$\begin{aligned}& \bigl\Vert (u_{t}-M)_{+} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{z}_{1,t} )}} \\& \quad \leq e^{\alpha \tau }e^{-\alpha t}\varepsilon +C\varepsilon e^{-(A+ \alpha )t} \int^{t}_{-\infty }e^{(A+\alpha )s} \,ds+ C\varepsilon +Ce ^{\alpha \tau }e^{-\alpha t}\varepsilon +C\varepsilon \\& \quad\quad {} + C e^{-\alpha t}\varepsilon \int^{t}_{-\infty }e^{\alpha s} \,ds +Ce ^{(\alpha -L)\tau }e^{-(\alpha -L)t}\varepsilon +Ce^{-(\alpha -L)t} \varepsilon \int^{t}_{-\infty } e^{(\alpha -L)s}\,ds \\& \quad \leq e^{\alpha \tau }e^{-\alpha t}\varepsilon +C\varepsilon + C \varepsilon +Ce^{\alpha \tau }e^{-\alpha t}\varepsilon +C\varepsilon + C \varepsilon +Ce^{(\alpha -L)\tau }e^{-(\alpha -L)t}\varepsilon +C \varepsilon \\& \quad \leq C\varepsilon, \end{aligned}$$

(88)

where $\alpha >L$. Repeating the same steps above, just taking $(u(t+z )-M)_{-}$ instead of $(u(t+z )-M)_{+}$, we deduce that

$$\begin{aligned} \bigl\Vert \bigl(u(t+z )-M\bigr)_{-} \bigr\Vert ^{r}_{C_{\gamma ,L^{r}( \Omega^{z}_{1,t} )}}\leq C \varepsilon . \end{aligned}$$

(89)

From (88), (89) and Lemma 5.1, we know the hypotheses of Lemma 5.3 are all satisfied. Therefore the process $\{U(t,\tau )\}$ generated by Eq. (1) is $\mathcal{D}$-pullback ω-limit compact. □

Theorem 5.6

Suppose in addition to the hypotheses in Lemma 4.1 that $g\in C(\mathbb{R},L^{r}(\Omega ))$. Then the processes $\{U(t,\tau ) \}$ on $C_{\gamma , L^{r}(\Omega )}$ generated by the solution of Eq. (1) with $u_{0}\in C_{\gamma , L^{r}(\Omega )}$ has the $\mathcal{D}$-pullback attractors $\{\mathcal{A}_{C_{\gamma , L^{r}( \Omega )}}(t)\}_{t\in \mathbb{R}}$.

Proof

From Theorem 7.1, Lemmas 4.1, 5.1 and 5.5, now for every bounded subset B in $C_{\gamma ,L^{r}(\Omega )}$, the process generated by Eq. (1) has the pullback attractors in $C_{\gamma ,L^{r}(\Omega )}$. □

6 Uniform estimates in $C_{\gamma,W^{1,r}(\Omega)}$

Let semigroup $e^{At}$ has the following higher smooth effect [19]:

$$\begin{aligned} \bigl\Vert e^{At}x \bigr\Vert _{E^{\beta }_{r}}\leq Mt^{-(\beta -\alpha )}e^{-\delta t} \Vert x \Vert _{E^{\alpha }_{r}},\quad x\in E^{\beta }_{r}, t>0, 0\leq \alpha \leq \beta , 0< \delta < \lambda_{1}. \end{aligned}$$

(90)

Lemma 6.1

Suppose the conditions of Lemma 4.1 hold and

$$\begin{aligned} \alpha < r(\delta -\eta )\leq r\gamma ,\quad r>2, \end{aligned}$$

(91)

holds, the family of processes $\{U_{g}(t,\tau )\}$ is uniformly dissipative in $C_{\gamma ,W^{1,r}(\Omega )}$, where $g(x,t)\in L^{r} _{\mathrm {loc}}{(\mathbb{R};L^{r}(\Omega ))}$, $\eta >0$ will be determined later.

Proof

Choosing $\alpha_{1}$ with $\alpha <\alpha_{1}$ and using (46), we obtain

$$\begin{aligned}& \int^{t}_{\tau }e^{-\alpha_{1}(t-s)} \Vert u_{s} \Vert _{C_{L^{r}(\Omega )}} ^{r}\,ds \\& \quad \leq \int^{t}_{\tau }e^{-\alpha_{1}(t-s)} \biggl(r e^{\alpha \tau }e^{-\alpha s} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}}+ \frac{ \alpha }{\alpha -L}C_{\Omega } \\& \quad\quad {} + \varepsilon_{4}^{-(r-1)}e^{-\alpha s} \int^{s}_{-\infty }e^{ \alpha l} \bigl\Vert g(l) \bigr\Vert _{L^{r}(\Omega )}^{r}\,dl +r e^{(\alpha -L)\tau }e^{(L- \alpha )s} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}} \\& \quad\quad {} + \varepsilon_{4}^{-(r-1)}e^{(L-\alpha )s} \int^{s}_{-\infty } e^{( \alpha -L)l} \bigl\Vert g(l) \bigr\Vert _{L^{r}(\Omega )}^{r}\,dl \biggr)\,ds \\& \quad \leq \frac{C}{\alpha_{1}-\alpha } e^{\alpha \tau } \Vert \phi \Vert ^{r} _{C_{\gamma ,{L^{r}(\Omega )}}}+ C+ \frac{C}{\alpha_{1}-\alpha }e^{- \alpha t} \int^{t}_{-\infty }e^{\alpha s} \bigl\Vert g(s) \bigr\Vert _{L^{r}(\Omega )} ^{r}\,ds \\& \quad\quad {} + \frac{C e^{(\alpha -L)\tau }e^{(L-\alpha )t}}{\alpha_{1}-\alpha +L} \Vert \phi \Vert ^{r}_{C_{\gamma ,{L^{r}(\Omega )}}} + \frac{C e^{(L-\alpha )t}}{ \alpha_{1}-\alpha +L} \int^{t}_{-\infty } e^{(\alpha -L)s} \bigl\Vert g(s) \bigr\Vert _{L ^{r}(\Omega )}^{r}\,ds \\& \quad \overset{\vartriangle }{=} Q(\alpha_{1},\alpha ,L,\tau ,\phi ,g_{0},t). \end{aligned}$$

(92)

It is obvious that $Q(\alpha_{1},\alpha ,L,\tau ,\phi ,g_{0},t)$ is bounded, as $\tau \rightarrow -\infty$. From the well-posedness of (1), we know that the solution of (1) satisfies

$$\begin{aligned} u(t)=e^{A(t-\tau )}u(\tau )+ \int^{t}_{\tau }e^{A(t-s)}\bigl[-\lambda u+f(x,u _{s})+g(x,s)\bigr]\,ds. \end{aligned}$$

(93)

Therefore, using (90) and choosing $\alpha_{1}>0$, $\eta >0$, $q=\frac{r}{r-1}<2$, $r>2$ such that $0<\alpha <r(\delta -\eta )=\alpha_{1}<r\gamma$, for each $t\geq \tau$ we obtain

$$\begin{aligned} \bigl\Vert u(t) \bigr\Vert _{W^{1,r}{(\Omega )}}&= \biggl\Vert e^{A(t-\tau )}u(\tau )+ \int^{t}_{ \tau }e^{A(t-\tau )}\bigl[-\lambda u+f(x,u_{s})+g(x,s)\bigr]\,ds \biggr\Vert _{W^{1,r}{( \Omega )}} \\ & \leq \bigl\Vert e^{A(t -\tau )} u(\tau ) \bigr\Vert _{W^{1,r}{(\Omega )}}+ \lambda \int^{t }_{\tau } \bigl\Vert e^{A(t -s)} u \bigr\Vert _{W^{1,r}{(\Omega )}}\,ds \\ &\quad {} + \int^{t}_{\tau } \bigl\Vert e^{A(t-s)}f(x,u_{s}) \bigr\Vert _{W^{1,r}{(\Omega )}}\,ds + \int^{t}_{\tau } \bigl\Vert e^{A(t+z-s)}g(x,s) \bigr\Vert _{W^{1,r}{(\Omega )}}\,ds \\ & \leq M_{1}e^{-\delta (t-\tau )} \bigl\Vert u(\tau ) \bigr\Vert _{W^{1,r}{(\Omega )}}+ \lambda M_{2} \int^{t}_{\tau }(t-s)^{-\frac{1}{2}}e^{-\delta (t-s)} \Vert u \Vert _{L^{r}{(\Omega })}\,ds \\ &\quad {} +M_{3} \int^{t}_{\tau }(t-s)^{-\frac{1}{2}}e^{-\delta (t-s)} \bigl\Vert F\bigl(s,u\bigl(s- \rho (s)\bigr)\bigr) \bigr\Vert _{L^{r}{(\Omega })}\,ds \\ &\quad {} +M_{4} \int^{t}_{\tau }(t-s)^{-\frac{1}{2}}e^{-\delta (t-s)} \biggl\Vert \int ^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr)\,dz \biggr\Vert _{L^{r}{(\Omega })}\,ds \\ &\quad {} +M_{5} \int^{t }_{\tau }(t-s)^{-\frac{1}{2}}e^{-\delta (t-s)} \bigl\Vert g(x,s) \bigr\Vert _{L^{r}{(\Omega })}\,ds. \end{aligned}$$

(94)

Then, by (46), (92), Hold’s inequality and Young’s inequality, we have

$$\begin{aligned}& \lambda M_{2} \int^{t}_{\tau }(t -s)^{-\frac{1}{2}}e^{-\delta (t -s)} \Vert u \Vert _{L^{r}{(\Omega })}\,ds \\& \quad \leq \lambda M_{2} \biggl( \int^{t}_{\tau }(t -s)^{-\frac{1}{2}q}e ^{-q\eta (t -s)}\,ds \biggr)^{\frac{1}{q}} \times \biggl( \int^{t}_{\tau }e ^{-r(\delta -\eta )(t -s)} \Vert u \Vert ^{r}_{L^{r}{(\Omega })} \biggr)^{ \frac{1}{r}} \\& \quad \leq \frac{\lambda M_{2}}{q} \biggl( \int^{t }_{\tau }(t -s)^{- \frac{1}{2}q}e^{-q\eta (t -s)}\,ds \biggr)+\frac{\lambda M_{2}}{r} \biggl( \int^{t}_{\tau }e^{-r(\delta -\eta )(t-s)} \Vert u \Vert ^{r}_{L^{r}{(\Omega })}\,ds \biggr) \\& \quad \leq \frac{\lambda M_{2}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}}+\frac{\lambda M_{2}}{r} \biggl( \int^{t}_{\tau }e ^{-r(\delta -\eta )(t-s)} \Vert u \Vert ^{r}_{L^{r}{(\Omega })}\,ds \biggr) \\& \quad \leq \frac{\lambda M_{2}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}}+\frac{\lambda M_{2}}{r}Q\bigl(r(\delta -\eta ), \tau ,\phi ,g_{0},t\bigr) \\& \quad \overset{\vartriangle }{=} \frac{\lambda M_{2}\Gamma (1-\frac{q}{2})}{q ^{2-\frac{1}{2}q}\eta^{1-\frac{1}{2}q}} +R_{2,W^{1,r}(\Omega )}\bigl(r( \delta -\eta ),\tau ,\phi ,g_{0},t\bigr). \end{aligned}$$

(95)

Similarly, combining (2), (3), and (6), we have

$$\begin{aligned}& M_{3} \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e^{-\delta (t -s)} \bigl\Vert F\bigl(x,u\bigl(s- \rho (s)\bigr)\bigr) \bigr\Vert _{L^{r}{(\Omega })}\,ds \\& \quad \leq M_{3} \biggl( \int^{t }_{\tau }(t -s)^{-\frac{1}{2}q}e^{-q \eta (t -s)}\,ds \biggr)^{\frac{1}{q}} \times \biggl( \int^{t}_{\tau }e^{-r( \delta -\eta )(t -s)} \Vert F \Vert ^{r}_{L^{r}{(\Omega })} \biggr)^{\frac{1}{r}} \\& \quad \leq \frac{ M_{3}}{q} \biggl( \int^{t }_{\tau }(t -s)^{-\frac{1}{2}q}e ^{-q\eta (t -s)}\,ds \biggr)+\frac{ M_{3}}{r} \biggl( \int^{t }_{\tau }e^{-r( \delta -\eta )(t -s)} \Vert F \Vert ^{r}_{L^{r}{(\Omega })}\,ds \biggr) \\& \quad \leq \frac{M_{3}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}} +\frac{ M_{3}}{r} \int^{t}_{\tau }e^{-r(\delta -\eta )(t-s)} \bigl(k_{1}^{r} \Vert \Omega \Vert ^{r}_{L^{r}{(\Omega })}+k_{2} ^{r} e^{-r\gamma \rho (t)} \bigl\Vert u\bigl(s-\rho (s)\bigr) \bigr\Vert ^{r}_{L^{r}{(\Omega })} \bigr)\,ds \\& \quad \leq \frac{M_{3}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}} +\frac{M_{3} k_{1}^{r} \vert \Omega \vert ^{r}_{L^{r}( \Omega )}}{r^{2}(\delta -\eta )} +\frac{k_{2}^{r}M_{3}}{r}Q\bigl(r(\delta -\eta ),\tau ,\phi ,g_{0},h,t\bigr) \\& \quad \overset{\vartriangle }{=} \frac{ M_{3}\Gamma (1-\frac{r}{2})}{q^{2- \frac{1}{2}q}\eta^{1-\frac{1}{2}q}} +\frac{M_{3} k_{1}^{r} \vert \Omega \vert ^{r} _{L^{r}(\Omega )}}{r^{2}(\delta -\eta )} +R_{3,W^{1,r}(\Omega )}\bigl(r( \delta -\eta ),\tau ,\phi ,g_{0},t\bigr), \end{aligned}$$

(96)

$$\begin{aligned}& M_{4} \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e^{-\delta (t -s)} \biggl\Vert \int^{0}_{-\infty }G\bigl(s,z,u(s+z)\bigr)\,dz \biggr\Vert _{L^{r}{(\Omega })}\,ds \\& \quad \leq M_{4} \biggl( \int^{t }_{\tau }(t-s)^{-\frac{1}{2}q}e^{-q\eta (t -s)}\,ds \biggr)^{\frac{1}{q}} \\& \quad\quad {} \times \biggl( \int^{t }_{\tau }e^{-r(\delta -\eta )(t -s)} \biggl\Vert \int ^{0}_{-\infty }\bigl(m_{0}(z)+{m}_{1}(z) \bigl\vert u(s+z_{0}) \bigr\vert \bigr)\,dz \biggr\Vert _{L^{r}{(\Omega })} ^{r} \biggr)^{\frac{1}{r}} \\& \quad \leq \frac{M_{4}\Gamma (1-\frac{q}{2})}{q^{2-\frac{1}{2}q} \eta^{1-\frac{1}{2}q}}+\frac{ M_{4}}{r} \biggl(m_{0}^{r} \vert \Omega \vert ^{r} \int^{t}_{\tau }e^{-r(\delta -\eta )(t-s)} \,ds+{m}_{1}^{r} \int^{t}_{ \tau }e^{-r(\delta -\eta )(t-s)} \Vert u_{s} \Vert ^{r}_{C_{\gamma ,L^{r}( \Omega )}}\,ds \biggr) \\& \quad \overset{\vartriangle }{=} \frac{M_{4}\Gamma (1-\frac{q}{2})}{q^{2- \frac{1}{2}q}\eta^{1-\frac{1}{2}q}}+\frac{2^{r-1} M_{4} m_{0}^{r} \vert \Omega \vert ^{r}}{r^{2}(\delta -\eta )}+R_{4,W^{1,r}(\Omega )} \bigl(r(\delta - \eta ), \tau ,\phi ,g_{0},t\bigr), \end{aligned}$$

(97)

and

$$\begin{aligned}& \int^{t }_{\tau } \bigl\Vert e^{A(t -s)} g(x,s) \bigr\Vert _{W^{1,r}{(\Omega )}}\,ds \\& \quad \leq M_{5} \int^{t}_{\tau }(t -s)^{-\frac{1}{2}}e^{-\delta (t -s)} \Vert g \Vert _{L^{r} (\Omega )}\,ds \\& \quad \leq M_{5} \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e^{-(\delta - \eta )(t -s)}e^{-\delta (t -s)} \Vert g \Vert _{L^{r}(\Omega )}\,ds \\& \quad \leq M_{5} \biggl( \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e^{-q\delta (t -s)}\,ds \biggr)^{\frac{1}{q}}\times \biggl( \int^{t}_{\tau }e^{-r( \delta -\eta )(t -s)} \Vert g \Vert ^{r}_{L^{r}(\Omega )}\,ds \biggr)^{\frac{1}{r}} \\& \quad \leq \frac{M_{5}}{q} \biggl( \int^{t }_{\tau }(t -s)^{-\frac{1}{2}}e ^{-q\delta (t -s)}\,ds \biggr)+ \frac{M_{5}}{r} \biggl( \int^{t}_{-\infty }e ^{-r(\delta -\eta )(t -s)} \Vert g \Vert ^{r}_{L^{r}(\Omega )}\,ds \biggr) \\& \quad \overset{\vartriangle }{=} \frac{ M_{5}\Gamma (1-\frac{q}{2})}{q^{2- \frac{1}{2}q}\eta^{1-\frac{1}{2}q}} + R_{5,W^{1,r}(\Omega )}\bigl(r( \delta -\eta ),\tau ,q,g,t\bigr). \end{aligned}$$

(98)

Similar to the arguments in Lemma 4.1, for each $t\in \mathbb{R}$, we can conclude that by (91)

$$\begin{aligned}& \sup_{z\in [-\infty ,0]}e^{-r\gamma z} \bigl\Vert u(t+z) \bigr\Vert _{W^{1,r}{(\Omega )}} \\& \quad \leq M_{1}e^{-\delta (t-\tau )} \bigl\Vert u(\tau ) \bigr\Vert _{W^{1,r}{(\Omega )}}+\frac{( \lambda M_{2}+M_{3}+M_{4}+M_{5})\Gamma (1-\frac{r}{2})}{r^{2- \frac{1}{2}r} \eta^{1-\frac{1}{2}r}} \\& \quad\quad {} +R_{2,W^{1,r}{(\Omega )}}\bigl(r(\delta -\eta ),\tau ,\phi ,g_{0} ,t\bigr) +\frac{M _{3} k_{1}^{r} \vert \Omega \vert ^{r}}{r^{2}(\delta -\eta )} \\& \quad\quad {} +R_{3,W^{1,r}}\bigl(r(\delta -\eta ),\phi ,\tau ,g_{0} ,t \bigr)+\frac{2^{r-1} M_{4} m_{0}^{r} \vert \Omega \vert ^{r}}{r^{2}(\delta -\eta )} \\& \quad\quad {} +R_{4,W^{1,r}}\bigl(r(\delta -\eta ),\tau ,\phi ,g_{0} ,t \bigr)+ R_{5,W ^{1,r}(\Omega )}\bigl(r(\delta -\eta ),\tau ,q,g,t\bigr) \\& \quad \overset{\vartriangle }{=} R_{6,W^{1,r}(\Omega )}\bigl(r(\delta -\eta ), \tau ,r, \phi ,g_{0}, t\bigr),\quad \text{for each } t\in \mathbb{R}. \end{aligned}$$

(99)

Hence, we can see that $\sup_{z\in [-\infty ,0]}e^{-r\gamma z} \Vert u(t+z) \Vert _{W^{1,r}{(\Omega )}}$ is bounded, for each $t\in \mathbb{R}$, $z\in (-\infty ,0]$, as $\tau \rightarrow -\infty$, which implies the process $\{U(t,\tau )\}$ has pullback absorbing sets in $C_{\gamma ,W^{1,r}(\Omega )}$. □

7 Existence of the pullback attractors in $C_{\gamma,W^{1,r}(\Omega)}$

Theorem 7.1

Suppose in additional to the hypotheses in Lemma 6.1 and $g(s)\in C( \mathbb{R}, W^{1,r}(\Omega ))$, $F\in C^{1}(\mathbb{R}\times \mathbb{R}; \mathbb{R})$, $G\in C^{1}(\mathbb{R} \times \mathbb{R}\times \mathbb{R};\mathbb{R})$, $\frac{\partial F}{\partial x}$, $\frac{\partial G}{\partial x}$ are both bounded. Then the processes $\{U(t,\tau )\}$ on $C_{\gamma ,W^{1,r}(\Omega )}$ generated by the solution of Eq. (1) with $\phi \in C_{\gamma ,W^{1,r}(\Omega )}$ has the pullback attractors $\mathcal{A}_{C_{\gamma ,W^{1,r}(\Omega )}}$.

Proof

We divide the proof into three steps.

Step 1. Taking gradient operator ∇ to act on (1), we can obtain

$$\begin{aligned} \frac{\partial \nabla u}{\partial t}-\Delta \nabla u+\lambda \nabla u =& \frac{\partial F}{\partial x}+\frac{\partial F}{\partial u}\nabla u\bigl(t-\rho (t),x\bigr) + \int^{0}_{-\infty } \frac{\partial G}{\partial x}\,dz \\ & {} + \int^{0}_{-\infty } \frac{\partial G}{\partial u}\nabla u(t+z,x)\,dz+ \nabla g(t,x). \end{aligned}$$

(100)

Multiplying (100) by $\vert \nabla u \vert ^{r-2}\nabla u$ and integrating by parts, we get

$$\begin{aligned}& \frac{1}{r}\frac{d}{dt} \bigl\Vert \nabla u(t) \bigr\Vert ^{r}_{L^{r}(\Omega )}+\frac{4(r-1)}{r ^{2}} \int_{\Omega } \bigl\vert \nabla \bigl( \bigl\vert \nabla u(t) \bigr\vert ^{\frac{r}{2}}\bigr) \bigr\vert ^{2}\,dx+ \int_{\Omega } \lambda \bigl\vert \nabla u(t) \bigr\vert ^{r}\,dx \\& \quad = \int_{\Omega } \frac{\partial F}{\partial x} \bigl\vert \nabla u(t) \bigr\vert ^{r-2} \nabla u(t)\,dx+ \int_{\Omega } \frac{\partial F}{\partial u}\nabla u\bigl(t- \rho (t),x\bigr) \vert \nabla u \vert ^{r-2}\nabla u\,dx \\& \quad\quad {} + \int_{\Omega } \int^{0}_{-\infty }\frac{\partial G}{\partial x} \bigl\vert \nabla u(t) \bigr\vert ^{r-2}\nabla u(t)\,dz\,dx+ \int_{\Omega } \int^{0}_{-\infty }\frac{ \partial G}{\partial u}\nabla u(t+z,x) \vert \nabla u \vert ^{r-2}\nabla u\,dz\,dx \\& \quad\quad {} + \int_{\Omega }\nabla g(t,x) \bigl\vert \nabla u(t) \bigr\vert ^{r-2}\nabla u(t)\,dx. \end{aligned}$$

(101)

By the same arguments as Lemma 4.1, we also obtain the process $\{U(t,\tau )\}$ generating by (100) has pullback absorbing sets in $C_{\gamma ,W^{1,r}(\Omega )}$.

Step 2. According to Theorem 15 [10], Eq. (1) has a pullback attractor $\mathcal{A}_{C_{\gamma ,H^{1}(\Omega )}}$. Hence, by the same arguments as Theorem 5.6, we also obtain the process $\{U(t,\tau )\}$ generating by Eq. (100) on $C_{\gamma ,L^{2}( \Omega )}$ is ω-limit compact.

Step 3. Combining step 1, step 2, and Lemma 6.1, as the proof of Theorem 5.6, we find that the process $\{U(t,\tau )\}$ generated by Eq. (100) on $C_{\gamma ,W^{1,r}(\Omega )}$ has pullback absorbing sets and is $\mathcal{D}$ pullback ω-limit compact. Thus, we know from Theorem 5.6 the process $\{U(t,\tau )\}$ generating by Eq. (1) has the pullback attractors $\mathcal{A}_{C_{\gamma ,W ^{1,r}(\Omega )}}$. □