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General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms

Abstract

The paper studies the global existence and general decay of solutions using Lyapunov functional for a nonlinear wave equation, taking into account the fractional derivative boundary condition and memory term. In addition, we establish the blow-up of solutions with nonpositive initial energy.

Introduction

Extraordinary differential equations, also known as fractional differential equations, are a generalization of differential equations through fractional calculus. Much attention has been accorded to fractional partial differential equations during the past two decades due to the many chemical engineering, biological, ecological, and electromagnetism phenomena that are modeled by initial boundary value problems with fractional boundary conditions. See Tarasov [16], Magin [15].

In this work we consider the nonlinear wave equation

$$ \textstyle\begin{cases} u_{tt}-\Delta u+au_{t}+\int _{0}^{t}g ( t-s ) \Delta u ( s ) \,ds= \vert u \vert ^{p-2}u, & x\in \Omega ,t>0, \\ \frac{\partial u}{\partial \nu }=-b\partial _{t}^{\alpha ,\eta }u, & x \in \Gamma _{0},t>0, \\ u=0, & x\in \Gamma _{1},t>0, \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), & x\in \Omega ,\end{cases} $$
(1.1)

where Ω is a bounded domain in \(\mathbb{R} ^{n}\), \(n\geq 1\) with a smooth boundary Ω of class \(C^{2}\) and ν is the unit outward normal to \(\partial \Omega =\Gamma _{0}\cup \Gamma _{1}\), where \(\Gamma _{0}\) and \(\Gamma _{1}\) are closed subsets of Ω with \(\Gamma _{0}\cap \Gamma _{1}=\emptyset \).

\(a,b>0\), \(p>2\), and \(\partial _{t}^{\alpha ,\eta }\) with \(0<\alpha <1\) is the Caputo’s generalized fractional derivative (see [11] and [7]) defined by

$$ \partial _{t}^{\alpha ,\eta }u(t)=\frac{1}{\Gamma (1-\alpha )}\int _{0}^{t}(t-s)^{-\alpha }e^{-\eta (t-s)}u_{s}(s) \,ds, \quad \eta \geq 0, $$

where Γ is the usual Euler gamma function. It can also be expressed by

$$ \partial _{t}^{\alpha ,\eta }u(t)=I^{1-\alpha ,\eta }u^{\prime }(t), $$
(1.2)

where \(I^{\alpha ,\eta }\) is the exponential fractional integro-differential operator given by

$$ I^{\alpha ,\eta }u(t)=\frac{1}{\Gamma (\alpha )} \int _{0}^{t}(t-s)^{ \alpha -1}e^{-\eta (t-s)}u(s) \,ds, \quad \eta \geq 0. $$

In the context of boundary dissipations of fractional order problems, the main research focus is on asymptotic stability of solutions starting by writing the equations as an augmented system. Then, various techniques are used such as LaSalle’s invariance principle and the multiplier method mixed with frequency domain (see [116], and [18]).

Dai and Zhang [7] replaced \(\int _{0}^{t}K(x,t-s)u_{s}(x,s)\,ds\) with \(\partial _{t}^{\alpha }u(x,t)\) and \(h(x,t)\) with \(|u|^{m-1}u(x,t)\) and managed to prove exponential growth for the same problem.

Note that the nonlinear wave equation with boundary fractional damping case was first considered by authors in [4], where they used the augmented system to prove the exponential stability and blow-up of solutions in finite time.

Motivated by our recent work in [4] and based on the construction of a Lyapunov function, we prove in this paper under suitable conditions on the initial data the stability of a wave equation with fractional damping and memory term. This technique of proof was recently used by [9] and [4] to study the exponential decay of a system of nonlocal singular viscoelastic equations.

Here we also consider three different cases on the sign of the initial energy as recently examined by Zarai et al. [17], where they studied the blow-up of a system of nonlocal singular viscoelastic equations.

The organization of our paper is as follows. We start in Sect. 2 by giving some lemmas and notations in order to reformulate our problem (1.1) into an augmented system. In the following section, we use the potential well theory to prove the global existence result. Then, the general decay result is given in Sect. 4. In Sect. 5, following a direct approach, we prove blow-up of solutions.

Preliminaries

Let us introduce some notations, assumptions, and lemmas that are effective for proving our results.

Assume that the relaxation function g satisfies

\(( G_{1} ) \) \(g:\mathbb{R} _{+}\rightarrow \mathbb{R} _{+}\) is a nonincreasing differentiable function with

$$ g ( 0 ) >0, \quad 1- \int _{0}^{\infty }g ( s ) \,ds=l>0; $$
(2.1)

\(( G_{2} ) \) There exists a constant \(\xi >0\) such that

$$ g^{\prime } ( t ) \leq -\xi g ( t ) , \quad \forall t>0. $$
(2.2)

We denote

$$ ( g\circ u ) ( t ) = \int _{0}^{t}g ( t-s ) \bigl\Vert u ( t ) -u ( s ) \bigr\Vert ^{2}\,ds $$
(2.3)

and

$$\begin{aligned}& \aleph = \bigl\{ w\in H_{0}^{1}\vert I(w)>0 \bigr\} \cup \{0 \}, \\& H_{\Gamma _{1}} ^{1} (\Omega )= \bigl\{ u\in H ^{1} ( \Omega ),u \vert _{ \Gamma _{1}} =0 \bigr\} . \end{aligned}$$

Lemma 1

(Sobolev–Poincaré inequality)

If either \(1\leq q\leq \frac{N+2}{N-2}\) \(( N\geq 3 ) \) or \(1\leq q\leq +\infty \) \(( N=2 ) \), then there exists \(C_{\ast }>0\) such that

$$ \Vert u \Vert _{q+1}\leq C_{\ast } \Vert \nabla u \Vert _{2}, \quad \forall u\in H_{0}^{1}(\Omega ). $$

Lemma 2

(Trace–Sobolev embedding)

For all p such that

$$ 2< p\leq \frac{2(n-1)}{n-2}, $$
(2.4)

we have

$$ H_{\Gamma _{1}}^{1}(\Omega )\hookrightarrow L^{p}(\Gamma _{0}). $$

We denote by \(B_{q}\) the embedding constant, i.e.,

$$ \Vert u \Vert _{p,\Gamma _{0}}\leq B_{q} \Vert u \Vert _{2}. $$

Lemma 3

([17], p. 5, Lemma 2 or [3], p. 1406, Lemma 4.1)

Consider a nonnegative function \(B(t)\in C^{2}(0,\infty )\) satisfying

$$ B^{\prime \prime }(t)-4(\delta +1)B^{\prime }(t)+4(\delta +1)B(t) \geq 0, $$
(2.5)

where \(\delta >0\).

If

$$ B^{\prime }(0)>r_{2}B(0)+l_{0}, $$
(2.6)

then

$$ B^{\prime }(t)\geq l_{0}, \quad \forall t >0, $$
(2.7)

where \(l_{0} \in \mathbb{R}\), \(r_{2}\) represents the smallest root of the equation

$$ r^{2}-4(\delta +1)r+(\delta +1)=0, $$
(2.8)

i.e., \(r_{2}=2(\delta +1)-2\sqrt{(\delta +1)\delta }\).

Lemma 4

([17], p. 5, Lemma 3 or [3], p. 1406, Lemma 4.2)

Let \(J (t ) \) be a nonincreasing function on \([ t_{0},\infty ) \) verifying the differential inequality

$$ J^{\prime } ( t ) ^{2}\geq \alpha +bJ ( t ) ^{2+ \frac{1}{\delta }},\quad t\geq t_{0} \geq 0, $$
(2.9)

where \(\alpha >0\), \(b\in \mathbb{R} \), then there exists \(T^{\ast } >0\) such that

$$ \lim_{t\rightarrow T^{\ast -}}J ( t ) =0, $$
(2.10)

with the following upper bound cases for \(T^{\ast }\):

\(\mathbf{(i)}\) When \(b<0\) and \(J(t_{0})<\min \{ 1,\sqrt{\alpha /(-b)} \} \),

$$ T^{\ast }\leq t_{0}+\frac{1}{\sqrt{-b}}\ln \frac{\sqrt{\frac{\alpha }{-b}}}{\sqrt{\frac{\alpha }{-b}}-J(t_{0})}. $$
(2.11)

\(\mathbf{(ii)}\) When \(b=0\),

$$ T^{\ast }\leq t_{0}+\frac{J(t_{0})}{\sqrt{\alpha }}. $$
(2.12)

\(\mathbf{(iii)}\) When \(b>0\),

$$ T^{\ast }\leq \frac{J(t_{0})}{\sqrt{\alpha }} $$
(2.13)

or

$$ T^{\ast }\leq t_{0}+2^{\frac{3\delta +1}{2\delta }} \frac{\delta c}{\sqrt{\alpha }} \bigl( 1- \bigl[ 1+cJ(t_{0}) \bigr] ^{\frac{1}{2\delta }} \bigr) , $$
(2.14)

where

$$ c= \biggl( \frac{b}{\alpha } \biggr) ^{\delta / ( 2+\delta ) }. $$

Definition 1

We say that u is a blow-up solution of (1.1) at finite time \(T^{\ast }\) if

$$ \lim_{t\rightarrow T^{\ast -}} \frac{1}{ ( \Vert \nabla u \Vert _{2} ) }=0. $$
(2.15)

Theorem 1

([12], Theorem 1)

Consider the constant

$$ \varrho =(\pi )^{-1}\sin {(\alpha \pi )} $$

and the function μ given by

$$ \mu (\xi )= \vert \xi \vert ^{\frac{(2\alpha -1)}{2}},\quad 0< \alpha < 1, \xi \in \mathbb{R} . $$
(2.16)

Then we can obtain

$$ O=I^{1-\alpha ,\eta }U, $$
(2.17)

which is a relation between U the “input” of the system

$$ \partial _{t}\phi (\xi ,t)+ \bigl(\xi ^{2}+\eta \bigr)\phi ( \xi ,t)-U(L,t)\mu ( \xi )=0, \quad t>0,\eta \geq 0, \xi \in \mathbb{R} $$
(2.18)

and the “output” O given by

$$ O(t)=\varrho \int _{-\infty }^{+\infty }\phi (\xi ,t)\mu (\xi )\,d\xi , \quad \xi \in \mathbb{R} , t>0. $$
(2.19)

Now, using (1.2) and Theorem 1, the augmented system related to our system (1.1) may be given by

$$ \textstyle\begin{cases} u_{tt}-\Delta u+au_{t}+\int _{0}^{t}g ( t-s ) \Delta u ( s ) \,ds= \vert u \vert ^{p-2}u, & x\in \Omega ,t>0, \\ \partial _{t}\phi (\xi ,t)+(\xi ^{2}+\eta )\phi (\xi ,t)-u_{t}(x,t) \mu (\xi )=0, & x\in \Gamma _{0},\xi \in \mathbb{R} ,t>0, \\ \frac{\partial u}{\partial \nu }=-b_{1}\int _{-\infty }^{+\infty } \phi (\xi ,t)\mu (\xi )\,d\xi , & x\in \Gamma _{0},\xi \in \mathbb{R} ,t>0, \\ u=0, & x\in \Gamma _{1},t>0, \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), & x\in \Omega , \\ \phi (\xi ,0)=0, & \xi \in \mathbb{R} ,\end{cases} $$
(2.20)

where \(b_{1}=b\varrho \).

Lemma 5

([2], p. 3, Lemma 2.1)

For all \(\lambda \in D_{\eta }= \{ \lambda \in \mathbf{\mathbb{C}}:\Im m\lambda \neq 0 \} \cup \{ \lambda \in \mathbf{\mathbb{C}}:\Re e\lambda +\eta >0 \} \), we have

$$ A_{\lambda }= \int _{-\infty }^{+\infty } \frac{\mu ^{2}(\xi )}{\eta + \lambda +\xi ^{2}}\,d\xi = \frac{\pi }{\sin {(\alpha \pi )}}(\eta +\lambda )^{\alpha -1}. $$

Theorem 2

(Local existence and uniqueness)

Assume that (2.4) holds. Then, for all \((u_{0},u_{1},\phi _{0})\in H_{\Gamma _{0}}^{1}(\Omega )\times L^{2}( \Omega )\times L^{2}(-\infty ,+\infty )\), there exists some T small enough such that problem (2.20) admits a unique solution

$$ \textstyle\begin{cases} u\in C([0,T),H_{\Gamma _{0}}^{1}(\Omega )), & \\ u_{t}\in C([0,T),L^{2}(\Omega )), & \\ \phi \in C([0,T),L^{2}(-\infty ,+\infty ). & \end{cases} $$
(2.21)

Global existence

Before proving the global existence for problem (2.20), let us introduce the functionals

$$ I(t)= \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ ( g\circ \nabla u ) ( t ) - \Vert u \Vert _{p}^{p} $$

and

$$ J(t)=\frac{1}{2} \biggl[ \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ ( g\circ \nabla u ) ( t ) \biggr] -\frac{1}{p} \Vert u \Vert _{p}^{p}. $$

The energy functional E associated with system (2.20) is given as follows:

$$ \begin{aligned} E(t)&=\frac{1}{2} \Vert u_{t} \Vert _{2}^{2}+ \frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \frac{1}{2} ( g\circ \nabla u ) ( t )\\ &\quad {} -\frac{1}{p} \Vert u \Vert _{p}^{p}+\frac{b_{1}}{2} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho . \end{aligned} $$
(3.1)

Lemma 6

If \((u,\phi )\) is a regular solution to (2.20), then the energy functional given in (3.1) verifies

$$ \begin{aligned} \frac{d}{dt}E(t)&=-a \Vert u_{t} \Vert _{2}^{2}- \frac{1}{2}g ( t ) \Vert \nabla u \Vert _{2}^{2}+ \frac{1}{2} \bigl( g^{\prime } \circ \nabla u \bigr) ( t ) \\ &\quad {}-b_{1} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi ( \xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho \\ &\leq 0. \end{aligned} $$
(3.2)

Proof

Multiplying by \(u_{t}\) in the first equation from (2.20), using integration by parts over Ω, we get

$$\begin{aligned}& \frac{1}{2} \Vert u_{t} \Vert _{2}^{2}- \int _{\Omega } \Delta uu_{t}\,dx+a \Vert u_{t} \Vert _{2}^{2}+\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ \frac{1}{2} ( g\circ \nabla u ) ( t ) \\& \quad = \int _{\Omega } \vert u \vert ^{p-2}u u_{t} \,dx. \end{aligned}$$

Therefore

$$ \begin{aligned} &\frac{d}{dt} \biggl[ \frac{1}{2} \Vert u_{t} \Vert _{2}^{2} +\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2} ( g\circ \nabla u ) ( t ) -\frac{1}{p} \Vert u \Vert _{p}^{p} \biggr] \\ &\quad {}+a \Vert u_{t} \Vert _{2}^{2} +b_{1} \int _{\Gamma _{0}}u_{t}(x,t) \int _{-\infty }^{+\infty }\mu (\xi )\phi (\xi ,t)\,d\xi \,d\rho =0. \end{aligned} $$
(3.3)

Multiplying by \(b_{1}\phi \) in the second equation from (2.20) and integrating over \(\Gamma _{0}\times (-\infty ,+\infty )\), we get

$$ \begin{aligned} &\frac{b_{1}}{2}\frac{d}{dt} \int _{\Gamma _{0}} \int _{- \infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho +b_{1} \int _{ \Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d \xi \,d\rho \\ &\quad {} -b_{1} \int _{\Gamma _{0}}u_{t}(x,t) \int _{-\infty }^{+\infty }\mu ( \xi )\phi (\xi ,t)\,d\xi \,d\rho =0. \end{aligned} $$
(3.4)

From (3.1), (3.3), and (3.4) we obtain

$$ \begin{aligned} \frac{d}{dt}E(t)&=-a \Vert u_{t} \Vert _{2}^{2}- \frac{1}{2}g ( t ) \Vert \nabla u \Vert _{2}^{2}+ \frac{1}{2} \bigl( g^{\prime } \circ \nabla u \bigr) ( t ) \\ &\quad {}-b_{1} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi ( \xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho\\ & \leq 0. \end{aligned} $$

 □

Lemma 7

Assuming that (2.4) holds and that for all \((u_{0},u_{1},\phi _{0})\in H_{\Gamma _{0}}^{1}(\Omega )\times L^{2}( \Omega )\times L^{2}(-\infty ,+\infty )\) verifying

$$ \textstyle\begin{cases} \beta =C_{\ast }^{p} ( \frac{2p}{p-2}E(0) ) ^{ \frac{p-2}{2}}< 1, \\ I(u_{0})>0.\end{cases} $$
(3.5)

Then \(u(t)\in \aleph \), \(\forall t\in {[} 0,T]\).

Proof

As \(I(u_{0})>0\), there exists \(T^{\ast }\leq T\) such that

$$ I(u)\geq 0,\quad \forall t\in {[} 0,T^{\ast }). $$

This leads to

$$ \begin{aligned} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ ( g\circ \nabla u ) ( t ) & \leq \frac{2p}{p-2}J(t),\quad \forall t\in {[} 0,T^{\ast }) \\ & \leq \frac{2p}{p-2}E(0). \end{aligned} $$
(3.6)

Using the Poincare inequality, (3.1), (2.3), (3.5), and (3.6), we obtain

$$ \begin{aligned} \Vert u \Vert _{p}^{p}& \leq C_{\ast }^{p} \Vert \nabla u \Vert _{2}^{p} \\ & \leq C_{\ast }^{p} \biggl( \frac{2p}{p-2}E(0) \biggr) ^{ \frac{p-2}{2}} \Vert \nabla u \Vert _{2}^{2}. \end{aligned} $$
(3.7)

Thus

$$ \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+ ( g\circ \nabla u ) ( t ) - \Vert u \Vert _{p}^{p}>0,\quad \forall t\in {[} 0,T^{\ast }). $$

Consequently, \(u\in H\), \(\forall t\in {[} 0,T^{\ast })\).

Repeating the procedure, \(T^{\ast }\) can be extended to T, and that makes the proof of our global existence result within reach. □

Theorem 3

Assume that (2.4) holds. Then for all

$$ (u_{0},u_{1},\phi _{0})\in H_{\Gamma _{0}}^{1}( \Omega )\times L^{2}( \Omega )\times L^{2}(-\infty ,+\infty ) $$

verifying (3.5), the solution of system (2.20) is global and bounded.

Proof

From (3.2), we get

$$ \begin{aligned} E(0)&\geq E(t)\\ & =\frac{1}{2} \Vert u_{t} \Vert _{2}^{2}+ \frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2} ( g\circ \nabla u ) ( t ) - \frac{1}{p} \Vert u \Vert _{p}^{p} \\ &\quad {} +\frac{b_{1}}{2} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho \\ & \geq \frac{1}{2} \Vert u_{t} \Vert _{2}^{2}+ \frac{p-2}{2p} \Vert \nabla u \Vert _{2}^{2}+ \frac{1}{p}I(t)+\frac{b_{1}}{2} \int _{ \Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho . \end{aligned} $$
(3.8)

Or \(I(t)>0\), therefrom

$$ \Vert u_{t} \Vert _{2}^{2}+ \Vert \nabla u \Vert _{2}^{2}+b_{1} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi ( \xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho \leq C_{1}E(0), $$

where \(C_{1}=\max \{\frac{2}{b_{1}},\frac{2p}{p-2},2\}\). □

Decay of solutions

To proceed for the energy decay result, we construct an appropriate Lyapunov functional as follows:

$$ L(t)=\epsilon _{1}E(t)+\epsilon _{2}\psi _{1}(t)+ \frac{\epsilon _{2}b_{1}}{2}\psi _{2}(t), $$
(4.1)

where

$$ \begin{aligned} &\psi _{1}(t) = \int _{\Omega }u_{t}u\,dx, \\ &\psi _{2}(t) = \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+ \eta \bigr) \biggl( \int _{0}^{t}\phi (\xi ,s)\,ds \biggr) ^{2}\,d \xi \,d\rho , \end{aligned}$$

and \(\epsilon _{1}\), \(\epsilon _{2}\) are positive constants.

Lemma 8

If \((u,\phi )\) is a regular solution of problem (2.20), then the following equality holds:

$$ \begin{aligned} & \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+ \eta \bigr) \phi (\xi ,t) \int _{0}^{t}\phi (\xi ,s)\,ds \,d\xi \,d\rho \\ &\quad = \int _{\Gamma _{0}}u(x,t) \int _{-\infty }^{+\infty }\phi (\xi ,t) \mu (\xi )\,d\xi \,d\rho - \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho . \end{aligned}$$

Proof

From the second equation of (2.20), we have

$$ \bigl(\xi ^{2}+\eta \bigr)\phi (\xi ,t)=u_{t}(x,t)\mu (\xi )-\partial _{t}\phi ( \xi ,t),\quad \forall x\in \Gamma _{0}. $$
(4.2)

Integrating (4.2) over \([0, t ]\) and using equations 3 and 6 from system (2.20), we get

$$ \int _{0}^{t} \bigl(\xi ^{2}+\eta \bigr) \phi (\xi ,s)\,ds=u(x,t)\mu (\xi )-\phi ( \xi ,t),\quad \forall x\in \Gamma _{0}, $$
(4.3)

hence,

$$ \bigl(\xi ^{2}+\eta \bigr) \int _{0}^{t}\phi (\xi ,s)\,ds=u(x,t)\mu (\xi )-\phi ( \xi ,t),\quad \forall x\in \Gamma _{0}. $$
(4.4)

Multiplying by ϕ followed by integration over \(\Gamma _{0}\times (-\infty ,+\infty )\) leads to

$$ \begin{aligned} & \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+ \eta \bigr) \phi (\xi ,t) \int _{0}^{t}\phi (\xi ,s)\,ds \,d\xi \,d\rho \\ &\quad = \int _{\Gamma _{0}}u(x,t) \int _{-\infty }^{+\infty }\phi (\xi ,t) \mu (\xi )\,d\xi \,d\rho - \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho . \end{aligned}$$

 □

Lemma 9

For any \((u,\phi )\) solution of problem (2.20), we have

$$ \alpha _{1}E(t)\leq L(t)\leq \alpha _{2}E(t), $$
(4.5)

where \(\alpha _{1}\), \(\alpha _{2}\) are positive constants.

Proof

From (4.3), we get

$$ \int _{0}^{t}\phi (\xi ,s)\,ds=\frac{-\phi (\xi ,t)}{\xi ^{2}+\eta }+ \frac{u(x,t)\mu (\xi )}{\xi ^{2}+\eta },\quad \forall x\in \Gamma _{0}. $$
(4.6)

Thus

$$ \biggl( \int _{0}^{t}\phi (\xi ,s)\,ds \biggr) ^{2}= \frac{ \vert \phi (\xi ,t) \vert ^{2}}{(\xi ^{2}+\eta )^{2}}+ \frac{ \vert u(x,t) \vert ^{2}\mu ^{2}(\xi )}{(\xi ^{2}+\eta )^{2}}-2 \frac{\phi (\xi ,t)u(x,t)\mu (\xi )}{(\xi ^{2}+\eta )^{2}}. $$
(4.7)

Multiplying by \(\xi ^{2}+\eta \) in (4.7) followed by integration over \(\Gamma _{0}\times (-\infty ,+\infty )\) leads to

$$ \begin{aligned} \bigl\vert \psi _{2}(t) \bigr\vert & \leq \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty }\frac{ \vert \phi (\xi ,t) \vert ^{2}}{\xi ^{2}+\eta }\,d\xi \,d\rho + \int _{ \Gamma _{0}} \bigl\vert u(x,t) \bigr\vert ^{2} \int _{-\infty }^{+\infty } \frac{\mu ^{2}(\xi )}{\xi ^{2}+\eta }\,d\xi \,d\rho \\ &\quad {} +2 \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \frac{ \vert \phi (\xi ,t)u(x,t)\mu (\xi ) \vert }{\xi ^{2}+\eta }\,d\xi \,d\rho . \end{aligned} $$
(4.8)

Using Young’s inequality in order to have an estimation of the last term in (4.8), we get for any \(\delta >0\)

$$ \begin{aligned} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \frac{ \vert \phi (\xi ,t)u(x,t)\mu (\xi ) \vert }{\xi ^{2}+\eta }\,d\xi \,d\rho & = \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \frac{ \vert \phi (\xi ,t) \vert }{(\xi ^{2}+\eta )^{\frac{1}{2}}} \frac{ \vert u(x,t)\mu (\xi ) \vert }{(\xi ^{2}+\eta )^{\frac{1}{2}}} \,d\xi \,d\rho \\ & \leq \frac{1}{4\delta } \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty }\frac{ \vert \phi (\xi ,t) \vert ^{2}}{\xi ^{2}+\eta }\,d\xi \,d\rho \\ &\quad {} +\delta \int _{\Gamma _{0}} \bigl\vert u(x,t) \bigr\vert ^{2} \int _{-\infty }^{+\infty } \frac{\mu ^{2}(\xi )}{\xi ^{2}+\eta }\,d\xi \,d\rho . \end{aligned} $$
(4.9)

Combining (4.8) and (4.9), we obtain

$$ \begin{aligned} \bigl\vert \psi _{2}(t) \bigr\vert & \leq \biggl(\frac{2\delta +1}{2\delta } \biggr) \int _{ \Gamma _{0}} \int _{-\infty }^{+\infty } \frac{ \vert \phi (\xi ,t) \vert ^{2}}{\xi ^{2}+\eta }\,d\xi \,d \rho \\ &\quad {} +(2\delta +1) \int _{\Gamma _{0}} \bigl\vert u(x,t) \bigr\vert ^{2} \int _{-\infty }^{+ \infty }\frac{\mu ^{2}(\xi )}{\xi ^{2}+\eta }\,d\xi \,d\rho . \end{aligned} $$
(4.10)

Since \(\frac{1}{\xi ^{2}+\eta }\leq \frac{1}{\eta }\), then

$$ \begin{aligned} \bigl\vert \psi _{2}(t)& \bigr\vert \leq \biggl(\frac{2\delta +1}{2\delta \eta } \biggr) \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d \xi \,d\rho \\ &\quad {} +(2\delta +1) \int _{\Gamma _{0}} \bigl\vert u(x,t) \bigr\vert ^{2} \int _{-\infty }^{+ \infty }\frac{\mu ^{2}(\xi )}{\xi ^{2}+\eta }\,d\xi \,d\rho . \end{aligned} $$
(4.11)

Applying Lammas 2 and 5, we get

$$ \bigl\vert \psi _{2}(t) \bigr\vert \leq \biggl(\frac{2\delta +1}{2\delta \eta } \biggr) \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho +A_{0}B_{q}(2 \delta +1) \Vert \nabla u \Vert _{2}^{2}. $$
(4.12)

By Poincare-type inequality and Young’s inequality, we obtain

$$ \bigl\vert \psi _{1}(t) \bigr\vert \leq \frac{1}{2} \Vert u_{t} \Vert _{2}^{2}+ \frac{C_{\ast }}{2} \Vert \nabla u \Vert _{2}^{2}. $$
(4.13)

Adding (4.13) to (4.12), we get

$$ \begin{aligned} \biggl\vert \psi _{1}(t)+ \frac{b_{1}}{2}\psi _{2}(t) \biggr\vert & \leq \bigl\vert \psi _{1}(t) \bigr\vert + \frac{b_{1}}{2} \bigl\vert \psi _{2}(t) \bigr\vert \\ & \leq \frac{1}{2} \Vert u_{t} \Vert _{2}^{2}+ \frac{1}{2} \bigl[ A_{0}B_{q}b_{1}(2 \delta +1)+C_{\ast } \bigr] \Vert \nabla u \Vert _{2}^{2} \\ &\quad {} +\frac{b_{1}}{2} \biggl[ \frac{2\delta +1}{2\delta \eta } \biggr] \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d \xi \,d\rho . \end{aligned} $$
(4.14)

Therefore, by the energy definition given in (3.1), for all \(N>0\), we have

$$ \begin{aligned} \biggl\vert \psi _{1}(t)+ \frac{b_{1}}{2}\psi _{2}(t) \biggr\vert & \leq NE(t)+ \frac{1-N}{2} \Vert u_{t} \Vert _{2}^{2}+ \frac{N}{p} \Vert u_{t} \Vert _{p}^{p} \\ &\quad{}+\frac{1}{2} \bigl[ A_{0}B_{q}b_{1}(2 \delta +1)+C_{\ast }-N \bigr] \Vert \nabla u \Vert _{2}^{2} \\ &\quad{}+\frac{b_{1}}{2} \biggl[ \frac{2\delta +1}{2\delta \eta }-N \biggr] \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d \xi \,d\rho . \end{aligned} $$
(4.15)

From (3.7) and (4.15), we finally get

$$ \begin{aligned} \biggl\vert \psi _{1}(t)+ \frac{b_{1}}{2}\psi _{2}(t) \biggr\vert & \leq NE(t)+ \frac{1-N}{2} \Vert u_{t} \Vert _{2}^{2} \\ &\quad{}+\frac{1}{2} \biggl[ A_{0}B_{q}b_{1}(2 \delta +1)+C_{\ast }- \frac{p-2}{2p}N \biggr] \Vert \nabla u \Vert _{2}^{2} \\ &\quad{}+\frac{b_{1}}{2} \biggl[ \frac{2\delta +1}{2\delta \eta }-N \biggr] \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d \xi \,d\rho , \end{aligned} $$
(4.16)

where N and \(\epsilon _{1}\) are chosen as follows:

$$\begin{aligned}& N>\max \biggl\{ \frac{2\delta +1}{2\delta \eta }, \frac{2p(A_{0}B_{q}b_{1}(2\delta +1)+C_{\ast })}{p-2}, 1 \biggr\} , \\& \epsilon _{1}\geq N\epsilon _{2}. \end{aligned}$$

Then we conclude from (4.16)

$$ \alpha _{1}E(t)\leq L(t)\leq \alpha _{2}E(t), $$

where

$$ \alpha _{1}=\epsilon _{1}-N\epsilon _{2} $$

and

$$ \alpha _{2}=\epsilon _{1}+N\epsilon _{2}. $$

 □

Now, we prove the exponential decay of global solution.

Theorem 4

If (2.4) and (3.5) hold, then there exist k and K, positive constants such that the global solution of (2.20) verifies

$$ E(t)\leq Ke^{-kt}. $$
(4.17)

Proof

By differentiation in (4.1), we get

$$ \begin{aligned} L^{\prime }(t)& =\epsilon _{1}E^{\prime }(t)+ \epsilon _{2} \Vert u_{t} \Vert _{2}^{2}+\epsilon _{2} \int _{\Omega }u_{tt}u\,dx \\ &\quad{}+\epsilon _{2}b_{1} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \phi (\xi ,t) \int _{0}^{t}\phi (\xi ,s)\,ds \,d\xi \,d\rho . \end{aligned} $$
(4.18)

Combining with (2.20) to obtain

$$ \begin{aligned} L^{\prime }(t)& =\epsilon _{1}E^{\prime }(t)+ \epsilon _{2} \biggl[ \Vert u_{t} \Vert _{2}^{2}- \Vert \nabla u \Vert _{2}^{2}+ \Vert u \Vert _{p}^{p}-a \int _{\Omega }uu_{t}\,dx \biggr] \\ &\quad{}-b_{1}\epsilon _{2} \int _{\Gamma _{0}}u(x,t) \int _{-\infty }^{+ \infty }\mu (\xi )\phi (\xi ,t)\,d\xi \,d\rho \\ &\quad{}+b_{1}\epsilon _{2} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \phi (\xi ,t) \int _{0}^{t}\phi (\xi ,s)\,ds \,d\xi \,d\rho . \end{aligned} $$
(4.19)

An application of Lemma 8 leads to

$$ \begin{aligned} L^{\prime }(t)& =\epsilon _{1}E^{\prime }(t)+ \epsilon _{2} \Vert u_{t} \Vert _{2}^{2}-\epsilon _{2} \Vert \nabla u \Vert _{2}^{2}+\epsilon _{2} \Vert u \Vert _{p}^{p} \\ &\quad{}-b_{1}\epsilon _{2} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho -a\epsilon _{2} \int _{\Omega }uu_{t}\,dx. \end{aligned} $$
(4.20)

Using Poincare-type inequality and Young’s inequality on the last term of (4.20), we get, for all \(\delta ^{\prime }>0\),

$$ \int _{\Omega }uu_{t}\,dx\leq \frac{1}{4\delta ^{\prime }} \Vert u_{t} \Vert _{2}^{2}+C_{\ast } \delta ^{\prime } \Vert \nabla u \Vert _{2}^{2}. $$
(4.21)

From (4.20), (4.21), and (3.2), we obtain

$$ \begin{aligned} L^{\prime }(t)& \leq \biggl[ -a\epsilon _{1}+\epsilon _{2} \biggl(1+ \frac{a}{4\delta ^{\prime }} \biggr) \biggr] \Vert u_{t} \Vert _{2}^{2}+ \epsilon _{2} \bigl[ -1+\delta ^{\prime }C_{\ast }a \bigr] \Vert \nabla u \Vert _{2}^{2} \\ &\quad{}+\epsilon _{2} \Vert u \Vert _{p}^{p}-b_{1} \epsilon _{2} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d \xi \,d\rho . \end{aligned} $$
(4.22)

We use (3.7) to get

$$ \begin{aligned} L^{\prime }(t)& \leq \biggl[ -a\epsilon _{1}+\epsilon _{2} \biggl(1+ \frac{a}{4\delta ^{\prime }} \biggr) \biggr] \Vert u_{t} \Vert _{2}^{2}+ \epsilon _{2} \biggl[ -1+\delta ^{\prime }C_{\ast }a+C_{\ast }^{p} \biggl( \frac{2p}{p-2} \biggr)^{\frac{p-2}{2}} \biggr] \Vert \nabla u \Vert _{2}^{2} \\ &\quad{}-b_{1}\epsilon _{2} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho . \end{aligned} $$
(4.23)

On the other hand, from (3.5)

$$ -1+C_{\ast }^{p} \biggl(\frac{2p}{p-2} \biggr)^{\frac{p-2}{2}}< 0. $$

For a small enough \(\delta ^{\prime }\), we may have

$$ -1+\delta ^{\prime }C_{\ast }a+C_{\ast }^{p} \biggl(\frac{2p}{p-2} \biggr)^{ \frac{p-2}{2}}< 0. $$

Then choose \(d>0\) depending only on \(\delta ^{\prime }\) such that

$$ \begin{aligned} L^{\prime }(t )&\leq \biggl[ -a\epsilon _{1}+\epsilon _{2} \biggl(1+ \frac{a}{4\delta ^{\prime }} \biggr) \biggr] \Vert u_{t} \Vert _{2}^{2}- \epsilon _{2}d \Vert \nabla u \Vert _{2}^{2} \\ &\quad{}-b_{1}\epsilon _{2} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho . \end{aligned} $$
(4.24)

Equivalently, for all positive constant M, we have

$$ \begin{aligned} L^{\prime }(t)& \leq \biggl[ -a\epsilon _{1}+\epsilon _{2} \biggl(1+ \frac{a}{4\delta ^{\prime }}+ \frac{M}{2} \biggr) \biggr] \Vert u_{t} \Vert _{2}^{2}+\epsilon _{2} \biggl[ \frac{M}{2}-d \biggr] \Vert \nabla u \Vert _{2}^{2} \\ &\quad{}+b_{1}\epsilon _{2} \biggl[ \frac{M}{2}-1 \biggr] \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho -\epsilon _{2}ME(t). \end{aligned} $$
(4.25)

For \(\epsilon _{1}\) and \(M<\min \{2,2\,d\}\) chosen such that

$$ \epsilon _{1}> \frac{\epsilon _{2}(1+\frac{a}{4\delta ^{\prime }}+\frac{M}{2})}{a}. $$

We obtain from (4.25)

$$ L^{\prime }(t)\leq -M\epsilon _{2}E(t)\leq \frac{-\epsilon _{2}M}{\alpha _{2}}L(t), $$
(4.26)

as a result of (4.5). Now, a simple integration of (4.26) yields

$$ L(t)\leq L(0)e^{-kt}, $$

where \(k=\frac{\epsilon _{2}M}{\alpha _{2}}\). Another use of (4.5) provides (4.17). □

Blow-up

In the current section, we follow the same approach given in [11] to prove the blow-up of solution of problem (2.20).

Remark 1

By integration of (3.2) over \((0,t)\), we have

$$ \begin{aligned} E(t)& =E(0)-a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds \\ &\quad{}+\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}+\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ &\quad{}-b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds. \end{aligned} $$
(5.1)

Now, let us define \(F(t)\):

$$\begin{aligned} F(t) =& \Vert u \Vert _{2}^{2}+a \int _{0}^{t} \Vert u \Vert _{2}^{2} \,ds \\ &{}-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) +b_{1}H(t), \end{aligned}$$
(5.2)

where

$$ H(t)= \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+ \eta \bigr) \biggl( \int _{0}^{s}\phi (\xi ,z)\,dz \biggr) ^{2}\,d\xi \,d\rho \,ds. $$
(5.3)

Lemma 10

Assume that \(\| \nabla u\| _{2}^{2}\) is bounded on \([0,T)\), then

$$ H(t)\leq C< +\infty . $$
(5.4)

More precisely

$$ H(t)\leq \frac{1}{2}C_{1}B_{q}e^{-\eta C_{2}} \bigl[ C_{2}^{2\alpha -1} \alpha +C_{2}^{3-2\alpha }\eta \bigr] \Gamma (\alpha )T^{4}, $$

where

$$ C_{1}=\sup_{t\in {[} 0,T)} \bigl\{ \Vert \nabla u \Vert _{2}^{2},1 \bigr\} . $$

Proof

Using (2.18), we obtain

$$ \phi (\xi ,t)= \int _{0}^{t}\mu (\xi )e^{-(\xi ^{2}+\eta )(t-s)}u(x,s)\,ds, \quad \forall x\in \Gamma _{0}. $$
(5.5)

Hölder’s inequality yields

$$ \phi (\xi ,t)\leq \biggl( \int _{0}^{t}\mu ^{2}(\xi )e^{-2(\xi ^{2}+ \eta )(t-s)}\,ds \biggr) ^{\frac{1}{2}} \biggl( \int _{0}^{t} \bigl\vert u(x,s) \bigr\vert ^{2}\,ds \biggr) ^{\frac{1}{2}},\quad \forall x\in \Gamma _{0}. $$
(5.6)

On the other hand,

$$ \biggl( \int _{0}^{t}\phi (\xi ,s)\,ds \biggr) ^{2} \leq T \int _{0}^{t} \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,ds. $$
(5.7)

From (5.6) in (5.7), we obtain

$$ \biggl( \int _{0}^{t}\phi (\xi ,s)\,ds \biggr) ^{2} \leq T \int _{0}^{t} \biggl[ \int _{0}^{s}\mu ^{2}(\xi )e^{-2(\xi ^{2}+\eta )(s-z)}\,dz \int _{0}^{s} \bigl\vert u(x,z) \bigr\vert ^{2}\,dz \biggr] \,ds. $$
(5.8)

Applying Lemma 2 leads to

$$ \int _{\Gamma _{0}} \biggl( \int _{0}^{t}\phi (\xi ,s)\,ds \biggr) ^{2}\,d \rho \leq B_{q}C_{1}T \int _{0}^{t} \biggl[ \int _{0}^{s}\mu ^{2}(\xi )e^{-2( \xi ^{2}+\eta )(s-z)}\,dz \biggr] \,ds. $$
(5.9)

Since \(z\in (0,s)\), we choose \(\exists C_{2}\geq 0\) such that \(s-z\geq \frac{C_{2}}{2}\) to term (5.9) into

$$ \int _{\Gamma _{0}} \biggl( \int _{0}^{t}\phi (\xi ,s)\,ds \biggr) ^{2}\,d \rho \leq \frac{1}{2}B_{q}C_{1}T^{3} \mu ^{2}(\xi )e^{-C_{2}(\xi ^{2}+ \eta )}. $$
(5.10)

Multiplication by \(\xi ^{2}+\eta \) followed by integration over \((0,t)\times (-\infty ,+\infty )\) yields

$$ \begin{aligned} H(t)& \leq C_{1}B_{q}e^{-\eta C_{2}}T^{3} \int _{0}^{t} \biggl[ \int _{0}^{+\infty }\xi ^{2\alpha +1}e^{-C_{2}\xi ^{2}}\,d \xi \biggr] \,ds \\ &\quad{}+C_{1}B_{q}e^{-\eta C_{2}}\eta T^{3} \int _{0}^{t} \biggl[ \int _{0}^{+ \infty }\xi ^{2\alpha -1}e^{-C_{2}\xi ^{2}}\,d \xi \biggr] \,ds. \end{aligned} $$
(5.11)

Then

$$ \begin{aligned} H(t)& \leq \frac{1}{2}C_{1}B_{q}e^{-\eta C_{2}}C_{2}^{2 \alpha -1}T^{3} \int _{0}^{t} \biggl[ \int _{0}^{+\infty }y^{\alpha }e^{-y}\,dy \biggr] \,ds \\ &\quad{}+\frac{1}{2}C_{1}B_{q}e^{-\eta C_{2}}C_{2}^{3-2\alpha } \eta T^{3} \int _{0}^{t} \biggl[ \int _{0}^{+\infty }y^{\alpha -1}e^{-y}\,dy \biggr] \,ds. \end{aligned} $$
(5.12)

Applying a special integral (Euler gamma function), we obtain

$$ H(t)\leq \frac{1}{2}C_{1}B_{q}e^{-\eta C_{2}} \bigl[ C_{2}^{2\alpha -1} \alpha +C_{2}^{3-2\alpha }\eta \bigr] \Gamma (\alpha )T^{4}. $$
(5.13)

 □

Lemma 11

Suppose \(p>2\), then

$$ \begin{aligned} F^{\prime \prime }(t)&\geq (p+2) \Vert u_{t} \Vert _{2}^{2} \\ &\quad{}+2p \biggl\{ -E(0)+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds- \frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ &\quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \biggr\} . \end{aligned} $$
(5.14)

Proof

We differentiate with respect to t in (5.2), then we get

$$ \begin{aligned} F^{\prime }(t)& =2 \int _{\Omega }uu_{t}\,dx+a \Vert u \Vert _{2}^{2} \\ &\quad{}+\frac{1}{2}g ( t ) \Vert \nabla u \Vert _{2}^{2}- \frac{1}{2} \bigl( g^{\prime }\circ \nabla u \bigr) ( t ) \\ &\quad{}+2b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \phi (\xi ,s) \int _{0}^{s}\phi (\xi ,z)\,dz \,d\xi \,d\rho \,ds. \end{aligned} $$
(5.15)

Using divergence theorem and (2.20), we obtain

$$ \begin{aligned} F^{\prime \prime }(t)& =2 \Vert u_{t} \Vert _{2}^{2}-2 \int _{\Omega }\nabla u \int _{0}^{t}g ( t-s ) \nabla u ( s ) \,ds \,dx \\ &\quad{}+2 \Vert u \Vert _{p}^{p}+2b_{1} \int _{\Gamma _{0}}u(x,t) \int _{-\infty }^{+\infty }\mu (\xi )\phi (\xi ,t)\,d\xi \,d\rho \\ &\quad{}+2b_{1} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+ \eta \bigr) \phi (\xi ,t) \int _{0}^{t}\phi (\xi ,s)\,ds \,d\xi \,d\rho . \end{aligned} $$
(5.16)

By definition of energy functional in (3.1) and relation (5.1), we give the following evaluation of the third term of (5.16):

$$ \begin{aligned} 2 \Vert u \Vert _{p}^{p}& =p \Vert u_{t} \Vert _{2}^{2}+p \Vert \nabla u \Vert _{2}^{2}+pb_{1} \int _{\Gamma _{0}} \int _{- \infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho -2pE(0) \\ &\quad{}+2p \biggl[ a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds- \frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ & \quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \biggr] . \end{aligned} $$
(5.17)

We can also estimate the last term of (5.16) using Lemma 8:

$$ \begin{aligned} & \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+ \eta \bigr) \phi (\xi ,t) \int _{0}^{t}\phi (\xi ,s)\,ds \,d\xi \,d\rho \\ & \quad = \int _{\Gamma _{0}}u(x,t) \int _{-\infty }^{+\infty }\phi (\xi ,t) \mu (\xi )\,d\xi \,d\rho - \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho . \end{aligned} $$
(5.18)

From (5.17), (5.18), and (5.16), we get

$$ \begin{aligned} F^{\prime \prime }(t)&\geq (p+2) \Vert u_{t} \Vert _{2}^{2}+(p-2) \Vert \nabla u \Vert _{2}^{2}+b_{1}(p-2) \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl\vert \phi (\xi ,t) \bigr\vert ^{2}\,d\xi \,d\rho \\ &\quad{}+2p \biggl[ -E(0)+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds- \frac{1}{2} \biggl(1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}- \frac{1}{2} ( g\circ \nabla u ) ( t ) \\ & \quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \biggr] . \end{aligned} $$
(5.19)

Taking \(p>2\), we obtain the needed estimation

$$ \begin{aligned} F^{\prime \prime }(t)&\geq (p+2) \Vert u_{t} \Vert _{2}^{2} \\ &\quad{}+2p \biggl\{ -E(0)+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds- \frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ &\quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \biggr\} . \end{aligned}$$

 □

Lemma 12

Suppose that \(p>2\) and that either one of the next assumptions is verified:

(i) \(E(0)<0\);

(ii) \(E(0)=0\), and

$$ F^{\prime }(0)>a \Vert u_{0} \Vert _{2}^{2}; $$
(5.20)

(iii) \(E(0)>0\), and

$$ F^{\prime }(0)> \bigl[ F(0)+l_{0} \bigr] +a \Vert u_{0} \Vert _{2}^{2}, $$
(5.21)

where

$$ r=p-2\sqrt{p^{2}-p} $$

and

$$ l_{0}=a \Vert u_{0} \Vert _{2}^{2}-2E(0). $$
(5.22)

Then \(F^{\prime }(t)>a\| u_{0}\| _{2}^{2}\) for \(t>t_{0}\), where

$$ t^{\ast }>\max \biggl\{ 0, \frac{F^{\prime }(0)-a \Vert u_{0} \Vert _{2}^{2}]}{2pE(0)} \biggr\} , $$
(5.23)

where \(t_{0}=t^{\ast }\) in case (i), and \(t_{0}=0\) in cases (ii) and (iii).

Proof

(i) Case of \(E(0)<0\).

From (5.14), we have

$$ F^{{\prime \prime }}(t)\geq -2pE(0), $$

which clearly leads to

$$ F^{{\prime }}(t)\geq F^{{\prime }}(0)-2pE(0)t. $$

Then

$$ F^{{\prime }}(t)>a \Vert u_{0} \Vert _{2}^{2},\quad \forall t \geq t^{\ast }, $$

where \(t^{\ast }\) as given in (5.23).

(ii) Case \(E(0)=0\).

Using (5.14) we get

$$ F^{\prime \prime }(t)\geq 0,\quad \forall t\geq 0. $$

Thus

$$ F^{\prime }(t)\geq F^{\prime }(0),\quad \forall t\geq 0. $$

Then, by (5.20),

$$ F^{{\prime }}(t)>a \Vert u_{0} \Vert _{2}^{2},\quad \forall t \geq 0. $$

(iii) Case \(E(0)>0\).

The proof of this case consists of getting to a differential inequality: \(B^{\prime \prime }(t)-pB^{\prime }(t)+pB(t)\geq 0\) pursued by a use of Lemma 3. Indeed, from (5.15) we have

$$ \begin{aligned} F^{\prime }(t)&=2 \int _{\Omega }uu_{t}\,dx+a \Vert u \Vert _{2}^{2} \\ &\quad{}+\frac{1}{2}g ( t ) \Vert \nabla u \Vert _{2}^{2}- \frac{1}{2} \bigl( g^{\prime }\circ \nabla u \bigr) ( t ) \\ &\quad{}+2b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \phi (\xi ,s) \int _{0}^{s}\phi (\xi ,z)\,dz \,d\xi \,d\rho \,ds. \end{aligned} $$
(5.24)

Or, the last term in (5.24) can be estimated using Young’s inequality

$$ \begin{aligned} & \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \phi (\xi ,s) \int _{0}^{s}\phi (\xi ,z)\,dz\,d \xi \,d\rho \,ds \\ &\quad \leq \frac{1}{2} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \\ &\qquad {} +\frac{1}{2} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \biggl( \int _{0}^{s}\phi (\xi ,z)\,dz \biggr) ^{2}\,d \xi \,d\rho \,ds. \end{aligned} $$
(5.25)

On the other hand,

$$ 2 \int _{0}^{t} \int _{\Omega }u_{s}u\,dx \,ds= \int _{0}^{t}\frac{d}{ds} \Vert u_{s} \Vert _{2}^{2}\,ds= \Vert u \Vert _{2}^{2}- \Vert u_{0} \Vert _{2}^{2}. $$
(5.26)

By Young’s inequality, we get

$$ \Vert u \Vert _{2}^{2}\leq \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds+ \int _{0}^{t} \Vert u \Vert _{2}^{2} \,ds+ \Vert u_{0} \Vert _{2}^{2}. $$
(5.27)

Now, we remake (5.24) using (5.25) and (5.27):

$$ \begin{aligned} F^{\prime }(t)& \leq \Vert u \Vert _{2}^{2}+ \Vert u_{t} \Vert _{2}^{2}+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds+a \int _{0}^{t} \Vert u \Vert _{2}^{2} \,ds+a \Vert u_{0} \Vert _{2}^{2} \\ &\quad{}-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ &\quad {}+b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+ \eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \\ &\quad{}+b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \biggl( \int _{0}^{s}\phi (\xi ,z)\,dz \biggr) ^{2}\,d\xi \,d \rho \,ds. \end{aligned} $$
(5.28)

From the definition of F in (5.2), inequality (5.28) also becomes

$$ \begin{aligned} F^{\prime }(t)&\leq F(t)+ \Vert u_{t} \Vert _{2}^{2}+b_{1}\int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl(\xi ^{2}+ \eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \\ &\quad{}+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds+a \Vert u_{0} \Vert _{2}^{2}. \end{aligned} $$
(5.29)

Thus, by (5.14), we get

$$ \begin{aligned} F^{\prime \prime }(t)-p \bigl\{ F^{\prime }(t)-F(t) \bigr\} & \geq 2 \Vert u_{t} \Vert _{2}^{2}+ap \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds-pa \Vert u_{0} \Vert _{2}^{2}-2pE(0) \\ &\quad{}+pb_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds. \end{aligned} $$
(5.30)

Hence

$$ F^{\prime \prime }(t)-pF^{\prime }(t)+pF(t)+pl_{0}\geq 0, $$
(5.31)

where

$$ l_{0}=a \Vert u_{0} \Vert _{2}^{2}-2E(0). $$

Posing

$$ B(t)=F(t)+l_{0} $$

leads to

$$ B^{\prime \prime }(t)-pB^{\prime }(t)+pB(t)\geq 0. $$
(5.32)

By Lemma 3 and for \(p=\delta +1\), we conclude that if

$$ B^{\prime }(t)> \bigl(p-2\sqrt{p^{2}-p} \bigr)B(0)+a \Vert u_{0} \Vert _{2}^{2}, $$
(5.33)

then

$$ F^{\prime }(t)=B^{\prime }(t)>a \Vert u_{0} \Vert _{2}^{2} \quad \forall t\geq 0. $$

 □

Theorem 5

Suppose that \(p>2\) and that either one of the next assumptions is verified:

(i) \(E(0)<0\);

(ii) \(E(0)=0\) and (5.20) holds;

(iii) \(0< E(0)< \frac{(2p-4) ( F^{\prime }(t_{0})-a\| u_{0}\| _{2}^{2} ) ^{2}J(t_{0})^{\frac{1}{\gamma _{1}}}}{16p}\) and (5.21) holds.

Then, in the sense of Definition 1, the solution \((u,\phi )\) blows up at finite time \(T^{\ast }\).

For case (i):

$$ T^{\ast }\leq t_{0}-\frac{J(t_{0})}{J^{\prime }(t_{0})}. $$
(5.34)

Moreover, if \(J(t_{0})<\min \{ 1,\sqrt{\frac{\sigma }{-b}} \} \), we get

$$ T^{\ast }\leq t_{0}+\frac{1}{\sqrt{-b}}\ln \frac{\sqrt{\frac{\sigma }{-b}}}{\sqrt{\frac{\sigma }{-b}}-J(t_{0})}. $$
(5.35)

For case (ii), we get either

$$ T^{\ast }\leq t_{0}-\frac{J(t_{0})}{J^{\prime }(t_{0})} $$
(5.36)

or

$$ T^{\ast }\leq t_{0}+\frac{J(t_{0})}{J^{\prime }(t_{0})}. $$
(5.37)

For case (iii):

$$ T^{\ast }\leq \frac{J(t_{0})}{\sqrt{\sigma }}, $$
(5.38)

or else

$$ T^{\ast }\leq t_{0}+2^{\frac{3\gamma _{1}+1}{2\gamma _{1}}} \frac{\gamma _{1}c}{\sqrt{\sigma }} \bigl\{ 1- \bigl[1-cJ(t_{0}) \bigr]^{\frac{1}{2\gamma _{1}}} \bigr\} , $$
(5.39)

where \(\gamma _{1}=\frac{p-4}{4}\), \(c=(\frac{b}{\sigma })^{\frac{\gamma _{1}}{2+\gamma _{1}}}\), \(J(t)\), b and σ are as in (5.40) and (5.54) respectively.

Note that \(t_{0} =0\) in cases (ii) and (iii). For case (i), we have as in (5.23): \(t_{0}=t^{*}\).

Proof

Consider

$$ J(t)= \bigl[ F(t)+a(T-t) \Vert u_{0} \Vert _{2}^{2} \bigr] ^{- \gamma _{1}},\quad t\in {[} t_{0},T]. $$
(5.40)

We differentiate on \(J(t)\) to get

$$ J^{{\prime }}(t)=-\gamma _{1}J(t)^{1+\frac{1}{\gamma _{1}}} \bigl[ F^{ \prime }(t)-a \Vert u_{0} \Vert _{2}^{2} \bigr] $$
(5.41)

and again

$$ J^{{\prime \prime }}(t)=-\gamma _{1}J(t)^{1+\frac{2}{\gamma _{1}}}G(t), $$
(5.42)

where

$$ G(t)=F^{{\prime \prime }}(t) \bigl[ F(t)+a(T-t) \Vert u_{0} \Vert _{2}^{2} \bigr] -(1+\gamma _{1}) \bigl\{ F^{{\prime }}(t)-a \Vert u_{0} \Vert _{2}^{2} \bigr\} ^{2}. $$
(5.43)

Using (5.14), we obtain

$$ \begin{aligned} F^{\prime \prime }(t)&\geq (p+2) \Vert u_{t} \Vert _{2}^{2} \\ &\quad{}+2p \biggl\{ -E(0)+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds- \frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ &\quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \biggr\} . \end{aligned}$$

Consequently,

$$ \begin{aligned} F^{\prime \prime }(t)&\geq -2pE(0) \\ &\quad {}\times p \biggl\{ \Vert u_{t} \Vert _{2}^{2}+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ &\quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \biggr\} . \end{aligned} $$
(5.44)

Or, from (5.15) and the fact that \(\| u\| _{2}^{2}-\| u_{0}\| _{2}^{2}=2\int _{0}^{t}\int _{\Omega }u_{s}u\,dx \,ds\), we attain

$$ \begin{aligned} F^{\prime }(t)-a \Vert u_{0} \Vert _{2}^{2}& =2 \int _{\Omega }uu_{t}\,dx+2a \int _{0}^{t} \int _{\Omega }u_{s}u\,dx \,ds \\ &\quad{}+2b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \phi (\xi ,s) \int _{0}^{s}\phi (\xi ,z)\,dz \,d\xi \,d\rho \,ds. \end{aligned} $$
(5.45)

Going back to (5.43) with (5.44) and (5.45) in hand, we get

$$ \begin{aligned} G(t) &\geq -2pE(0)J(t)^{\frac{-1}{\gamma _{1}}} \\ &\quad{}+p \biggl\{ \Vert u_{t} \Vert _{2}^{2}+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ & \quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds \biggr\} \\ &\quad{}\times \biggl[ \Vert u \Vert _{2}^{2}+a \int _{0}^{t} \Vert u \Vert _{2}^{2}\,ds-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ & \quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \biggl( \int _{0}^{s}\phi (\xi ,z)\,dz \biggr) ^{2}\,d \xi \,d\rho \,ds \biggr] \\ &\quad{}-4(1+\gamma _{1}) \biggl\{ \int _{\Omega }uu_{t}\,dx+a \int _{0}^{t} \int _{\Omega }u_{s}u\,dx \,ds+\frac{1}{2}g ( t ) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} \bigl( g^{\prime }\circ \nabla u \bigr) ( t ) \\ & \quad {} +b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+ \infty } \bigl(\xi ^{2}+\eta \bigr) \phi (\xi ,s) \int _{0}^{s}\phi (\xi ,z)\,dz\,d \xi \,d\rho \,ds \biggr\} ^{2}. \end{aligned} $$
(5.46)

For the sake of simplicity, we introduce the following notations:

$$\begin{aligned}& \begin{aligned} \mathbf{A}& = \Vert u \Vert _{2}^{2}+a \int _{0}^{t} \Vert u \Vert _{2}^{2}\,ds-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ &\quad{}+b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \biggl( \int _{0}^{s}\phi (\xi ,z)\,dz \biggr) ^{2}\,d\xi \,d \rho \,ds, \end{aligned} \\& \begin{aligned} \mathbf{B}& = \int _{\Omega }uu_{t}\,dx+a \int _{0}^{t} \int _{\Omega }u_{s}u\,dx \,ds+\frac{1}{2}g ( t ) \Vert \nabla u \Vert _{2}^{2}- \frac{1}{2} \bigl( g^{\prime }\circ \nabla u \bigr) ( t ) \\ &\quad{}+b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \phi (\xi ,s) \int _{0}^{s}\phi (\xi ,z)\,dz \,d\xi \,d\rho \,ds, \end{aligned} \\& \begin{aligned} \mathbf{C}& = \Vert u_{t} \Vert _{2}^{2}+a \int _{0}^{t} \Vert u_{s} \Vert _{2}^{2}\,ds-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \\ &\quad{}+b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \bigl\vert \phi (\xi ,s) \bigr\vert ^{2}\,d\xi \,d\rho \,ds. \end{aligned} \end{aligned}$$

Therefore

$$ Q(t)\geq -2pE(0)J(t)^{\frac{-1}{\gamma _{1}}}+p \bigl\{ \mathbf{A} \mathbf{C}-\mathbf{B}^{2} \bigr\} . $$
(5.47)

Note that, \(\forall w\in R\) and \(\forall t>0\),

$$ \begin{aligned} \mathbf{A}w^{2}+2\mathbf{B}w+\mathbf{C}& = \biggl[ w^{2} \Vert u \Vert _{2}^{2}+2w \int _{\Omega }uu_{t}\,dx+ \Vert u_{t} \Vert _{2}^{2} \biggr] \\ &\quad{}+a \int _{0}^{t} \biggl[ w^{2} \Vert u \Vert _{2}^{2}+2w \int _{ \Omega }uu_{s}\,dx+ \Vert u_{s} \Vert _{2}^{2} \biggr] \,ds \\ &\quad{}+ \bigl( w^{2}+1 \bigr) \biggl( -\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \biggr) \\ &\quad{}+w \biggl( \frac{1}{2}g ( t ) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} \bigl( g^{\prime }\circ \nabla u \bigr) ( t ) \biggr) \\ &\quad{}+b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \biggl[ w^{2} \biggl( \int _{0}^{s}\phi (\xi ,z)\,dz \biggr) ^{2} \\ & \quad {} +2w\phi (\xi ,s) \int _{0}^{s}\phi (\xi ,z)\,dz+ \bigl\vert \phi ( \xi ,s) \bigr\vert ^{2} \biggr] \,d\xi \,d\rho \,ds. \end{aligned} $$
(5.48)

Hence

$$ \begin{aligned} &\mathbf{A}w^{2} +2\mathbf{B}w+\mathbf{C}\\ &\quad = \bigl[ w \Vert u \Vert _{2}+ \Vert u_{t} \Vert _{2} \bigr] ^{2}+a \int _{0}^{t} \bigl[ w \Vert u \Vert _{2}+ \Vert u_{s} \Vert _{2} \bigr] ^{2}\,ds \\ &\qquad{}+ \bigl( w^{2}+1 \bigr) \biggl( -\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} ( g\circ \nabla u ) ( t ) \biggr) \\ &\qquad {}+w \biggl( \frac{1}{2}g ( t ) \Vert \nabla u \Vert _{2}^{2}-\frac{1}{2} \bigl( g^{\prime } \circ \nabla u \bigr) ( t ) \biggr) \\ &\qquad{}+b_{1} \int _{0}^{t} \int _{\Gamma _{0}} \int _{-\infty }^{+\infty } \bigl( \xi ^{2}+\eta \bigr) \biggl[ w \int _{0}^{s}\phi (\xi ,z)\,dz+ \bigl\vert \phi ( \xi ,s) \bigr\vert \biggr] ^{2}\,d\xi \,d\rho \,ds. \end{aligned} $$
(5.49)

It is clear that

$$ \mathbf{A}w^{2}+2\mathbf{B}+\mathbf{C}\geq 0 $$

and

$$ \mathbf{B}^{2}-\mathbf{A}\mathbf{C}\leq 0. $$
(5.50)

Then, from (5.47) and (5.50), we obtain

$$ G(t)\geq -2pE(0)J(t)^{\frac{-1}{\gamma _{1}}},\quad t\geq t_{0}. $$
(5.51)

Hence, by (5.42) and (5.51),

$$ J^{\prime \prime }(t)\leq \frac{p^{2}-4p}{2}E(0)J(t)^{1+ \frac{1}{\gamma _{1}}},\quad t\geq t_{0}. $$
(5.52)

Or, by Lemma [6], \(J^{{\prime }}(t)<0\), where \(t\geq t_{0}\).

Multiplication by \(J^{{\prime }}(t)\) in (5.52), followed by integration from \(t_{0}\) to t, leads to

$$ J^{{\prime }}(t)^{2}\geq \sigma +bJ(t)^{2+\frac{1}{\gamma _{1}}}, $$
(5.53)

where

$$ \textstyle\begin{cases} \sigma = [ \frac{(p-4)^{2}}{16} ( F^{{ \prime }}(t_{0})- \Vert u_{0} \Vert _{2}^{2} ) ^{2}-\frac{p(p-4)^{2}}{2p-4}E(0)J(t_{0})^{\frac{-1}{\gamma _{1}}} ] J(t_{0})^{2+\frac{2}{\gamma _{1}}}, \\ b =\frac{p(p-4)^{2}}{2p-4}E(0). \end{cases} $$
(5.54)

Note that \(\sigma >0\) is equivalent to \(E(0)< \frac{(2p-4) ( F^{\prime }(t_{0})-a\| u_{0}\| _{2}^{2} ) ^{2}J(t_{0})^{\frac{1}{\gamma _{1}}}}{16p}\), which by Lemma 4 ensures the existence of a finite time \(T^{\ast }>0\) such that

$$ \lim_{t\rightarrow T^{\ast -}}J ( t ) =0. $$

That involves

$$ \begin{aligned} &\lim_{t\rightarrow T^{\ast -}} \biggl[ \Vert u \Vert _{2}^{2}+a \int _{0}^{t} \Vert u \Vert _{2}^{2} \,ds-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}\\ &\quad {}- \frac{1}{2} ( g\circ \nabla u ) ( t ) +b_{1}H(t) \biggr] ^{-1}=0, \end{aligned} $$
(5.55)

i.e.,

$$ \begin{aligned} &\lim_{t\rightarrow T^{\ast -}} \biggl[ \Vert u \Vert _{2}^{2}+a \int _{0}^{t} \Vert u \Vert _{2}^{2} \,ds-\frac{1}{2} \biggl( 1- \int _{0}^{t}g(s)\,ds \biggr) \Vert \nabla u \Vert _{2}^{2}\\ &\quad {}- \frac{1}{2} ( g\circ \nabla u ) ( t ) +b_{1}H(t) \biggr] =+ \infty . \end{aligned} $$
(5.56)

So, there exists T such that \(t_{0}< T\leq T^{*}\) and \(\| \nabla u\| _{2}^{2}\longrightarrow +\infty \) as \(t\longrightarrow T^{-}\).

Indeed, if it is not the case, then \(\| \nabla u\| _{2}^{2}\) remained bounded on \([t_{0},T^{\ast })\), which by Lemma 10 leads to

$$ \lim_{t\rightarrow T^{\ast -}} \bigl[ \Vert u \Vert _{2}^{2}+b_{1}H(t) \bigr] =C< +\infty , $$

contradicting (5.56). □

Conclusion

Much attention has been accorded to fractional partial differential equations during the past two decades due to the many chemical engineering, biological, ecological, and electromagnetism phenomena that are modeled by initial boundary value problems with fractional boundary conditions. In the context of boundary dissipations of fractional order problems, the main research focus is on asymptotic stability of solutions starting by writing the equations as an augmented system. Then, various techniques are used such as LaSalle’s invariance principle and the multiplier method mixed with frequency domain. We prove in this paper under suitable conditions on the initial data the stability of a wave equation with fractional damping and memory term. This technique of proof was recently used by [4] to study the exponential decay of a system of nonlocal singular viscoelastic equations. Here we also considered three different cases on the sign of the initial energy as recently examined by Zarai et al. [17], where they studied the blow-up of a system of nonlocal singular viscoelastic equations.

In the next work, we will try to extend the same study of this paper to a general source term case.

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Acknowledgements

The authors are grateful to the anonymous referees for the careful reading and their important observations/suggestions for the sake of improving this paper. The first author (Pr. Salah Boulaaras) would like to thank all the professors of the mathematics department at the University of Annaba in Algeria, especially his Professors/Scientists Pr. Mohamed Haiour, Pr. Ahmed-Salah Chibi, and Pr. Azzedine Benchettah for the important content of masters and PhD courses in pure and applied mathematics which he received during his studies. Moreover, he thanks them for the additional help they provided to him during office hours in their office about the few concepts/difficulties he had encountered, and he appreciates their talent and dedication for their postgraduate students currently and previously.

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Correspondence to Rafik Guefaifia.

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On the occasion of the 44th birthday of the first author’s brother, Professor Djemai Mahmoud Mouha Boulaaras.

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Boulaaras, S., Kamache, F., Bouizem, Y. et al. General decay and blow-up of solutions for a nonlinear wave equation with memory and fractional boundary damping terms. Bound Value Probl 2020, 172 (2020). https://doi.org/10.1186/s13661-020-01470-w

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MSC

  • General decay
  • Global existence
  • Fractional boundary dissipation
  • Blow-up
  • Memory term

Keywords

  • 35L35
  • 35L20
\