Blow up of solutions of two singular nonlinear viscoelastic equations with general source and localized frictional damping terms

Abstract

This work studies the blow-up result of the solution of a coupled nonlocal singular viscoelastic equation with general source and localized frictional damping terms under some suitable conditions. This work is a natural continuation of the previous recent articles by Boulaaras et al. (Appl. Anal., 2020, https://doi.org/10.1080/00036811.2020.1760250; Math. Methods Appl. Sci. 43:6140–6164, 2020; Topol. Methods Nonlinear Anal., 2020, https://doi.org/10.12775/TMNA.2020.014).

1 Introduction

This paper is devoted to a study of the blow-up of the following system of two singular nonlinear viscoelastic equations:

$$ \textstyle\begin{cases} u_{tt}-\frac{1}{x}(xu_{x})_{x}+ \int _{0}^{t}g_{1}(t-s)\frac{1}{x}(xu_{x}(x,s))_{x}\,ds+\mu ( x ) u_{t}=f_{1} ( u,v ) ,\quad \text{in }Q, \\ v_{tt}-\frac{1}{x}(xv_{x})_{x}+ \int _{0}^{t}g_{2}(t-s)\frac{1}{x}(xv_{x}(x,s))_{x}\,ds+\mu ( x ) v_{t}=f_{2} ( u,v ) ,\quad \text{in }Q, \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x),\quad x\in (0,L), \\ v(x,0)=v_{0}(x),\qquad v_{t}(x,0)=v_{1}(x),\quad x\in (0,L), \\ u(L,t)=v(L,t)=0, \qquad \int _{0}^{L}xu(x,t)\,dx= \int _{0}^{L}xv(x,t)\,dx=0,\end{cases} $$

(1)

where

$$ \textstyle\begin{cases} f_{1}(u,v)=a_{1} \vert u+v \vert ^{2(r+1)}(u+v)+b_{1} \vert u \vert ^{r}.u. \vert v \vert ^{r+2}, \\ f_{2}(u,v)=a_{1} \vert u+v \vert ^{2(r+1)}(u+v)+b_{1} \vert v \vert ^{r}.v. \vert u \vert ^{r+2}, \end{cases} $$

(2)

and $Q=(0,L)\times (0,T)$, $L<\infty$, $T<\infty$, $\mu \in C^{1} ( ( 0,L ) )$, $g_{1}(\cdot)$, $g_{2}(\cdot):\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ and $f_{1}(\cdot,\cdot)$, $f_{2}(\cdot,\cdot):\mathbb{R}^{2}\longrightarrow \mathbb{R}$ are functions given in (2).

The problems related with localized frictional damping have been extensively studied by many teams [5], where the authors obtained an exponential rate of decay for the solution of the viscoelastic nonlinear wave equation:

$$ u_{tt}-\Delta u+f ( x,t,u ) + \int _{0}^{t}g_{1}(t-s) \Delta u ( s ) \,ds+a ( x ) u_{t}=0\quad \text{in }(0,L) \times (0,T), $$

for a damping term $a ( x ) u_{t}$ that may be null for some part of the domain.

We used the techniques of [5], and we have proved in [3] the existence of a global solution using the potential well theory for the following viscoelastic system with nonlocal boundary condition and localized frictional damping:

$$ \textstyle\begin{cases} u_{tt}-\frac{1}{x}(xu_{x})_{x}+\int _{0}^{t}g_{1}(t-s)\frac{1}{x}(xu_{x}(x,s))_{x}\,ds+a(x)u_{t} \\ \quad = \vert v \vert ^{q+1} \vert u \vert ^{p-1}u,\quad \text{in }(0,L) \times (0,T), \\ v_{tt}-\frac{1}{x}(xv_{x})_{x}+\int _{0}^{t}g_{2}(t-s) \frac{1}{x}(xv_{x}(x,s))_{x}\,ds+a(x)v_{t} \\ \quad = \vert u \vert ^{p+1} \vert v \vert ^{q-1}v,\quad \text{in }(0,L) \times (0,T), \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x),\quad x\in (0,\alpha ), \\ v(x,0)=v_{0}(x),\qquad v_{t}(x,0)=v_{1}(x),\quad x\in (0,\alpha ), \\ u(\alpha ,t)=v(\alpha ,t)=0,\qquad \int _{0}^{\alpha }xu(x,t)\,dx=\int _{0}^{ \alpha }xv(x,t)\,dx=0.\end{cases}$$

(3)

Very recently, in [2] we have studied the following singular one-dimensional nonlinear equations that arise in generalized viscoelasticity with long-term memory:

$$ \textstyle\begin{cases} u_{tt}-\frac{1}{x}(xu_{x})_{x}+\int _{0}^{t}g_{1}(t-s) \frac{1}{x}(xu_{x}(x,s))_{x}\,ds=f_{1} ( u,v ) ,\quad \text{in }(0,L)\times (0,T), \\ v_{tt}-\frac{1}{x}(xv_{x})_{x}+\int _{0}^{t}g_{2}(t-s) \frac{1}{x}(xv_{x}(x,s))_{x}\,ds=f_{2} ( u,v ) ,\quad \text{in }(0,L)\times (0,T), \\ u(x,0)=u_{0}(x),\qquad u_{t}(x,0)=u_{1}(x), \quad x\in (0,L), \\ v(x,0)=v_{0}(x),\qquad v_{t}(x,0)=v_{1}(x),\quad x\in (0,L), \\ u(L,t)=v(L,t)=0,\qquad \int _{0}^{L}xu(x,t)\,dx=\int _{0}^{L}xv(x,t)\,dx=0.\end{cases}$$

(4)

Also in the field of blow-up, in [14], the authors studied the blow-up in finite time of solutions of an initial boundary value problem with nonlocal boundary conditions for a system of nonlinear singular viscoelastic equations.

In view of the articles mentioned above in [2, 3, 5] and a supplement to our recent study in [2], much less effort has been devoted to the blow-up of solutions of two singular nonlinear viscoelastic equations, where nonlocal boundary conditions, general source terms and localized frictional damping are considered.

The structure of the work is as follows: we start by giving the fundamental definitions and theorems on function spaces that we need, then we state the local existence theorem. Finally, we state and prove the main result, which under suitable conditions gives the blow-up in finite time of solutions for system 1.

2 Preliminaries

Let $L_{x}^{p}=L_{x}^{p}((0,L ))$ be the weighted Banach space equipped with the norm

$$ \Vert u \Vert _{L_{x}^{p}}= \biggl( \int _{0}^{L }x \vert u \vert ^{p}\,dx \biggr) ^{\frac{1}{p}}. $$

(5)

Let $H=L_{x}^{2}((0,L ))$ be the Hilbert space of square integral functions having the finite norm

$$ \Vert u \Vert _{H}= \biggl( \int _{0}^{L }xu^{2}\,dx \biggr) ^{ \frac{1}{2}}. $$

(6)

Let $V=V_{x}^{1}((0,L))$ be the Hilbert space equipped with the norm

$$ \Vert u \Vert _{V}= \bigl( \Vert u \Vert _{H}^{2}+ \Vert u_{x} \Vert _{H}^{2} \bigr) ^{\frac{1}{2}} $$

(7)

and

$$ V_{0}= \bigl\{ u\in V\text{ such that }u(L )=0 \bigr\} . $$

(8)

Lemma 1

(Poincaré-type inequality)

For any v in $V_{0}$ we have

$$ \int _{0}^{L }xv^{2}(x)\,dx\leq C_{p} \int _{0}^{L }x\bigl(v_{x}(x) \bigr)^{2}\,dx $$

(9)

and

$$ V_{0}= \bigl\{ v\in V\textit{ such that }v(L )=0 \bigr\} . $$

Remark 2

It is clear that $\Vert u \Vert _{V_{0}}= \Vert u_{x} \Vert _{H}$ defines an equivalent norm on $V_{0}$.

Theorem 3

(See [1])

For anyvin$V_{0}$and$2< p<4$, we have

$$ \int _{0}^{L }x \vert v \vert ^{p}\,dx\leq C_{\ast } \Vert v_{x} \Vert _{H=L_{x}^{2}(0,L )}^{p}, $$

(10)

where$C_{\ast }$is a constant depending onLandponly.

We prove the blow-up result under the following suitable assumptions.

• (A1) $g_{1},g_{2}: \mathbb{R}_{+}\rightarrow \mathbb{R}_{+}$ are differentiable and decreasing functions such that

  $$ \begin{aligned} &g_{1}(t)\geq 0 ,\qquad 1- \int _{0}^{\infty }g_{1} ( s ) \,ds=l_{1}>0, \\ &g_{2}(t)\geq 0 ,\qquad 1- \int _{0}^{\infty }g_{2} ( s ) \,ds=l_{2}>0. \end{aligned} $$

  (11)

• (A2) There exist constants $\xi _{1},\xi _{2}>0$ such that

  $$ \begin{aligned} &g_{1}^{\prime } ( t ) \leq -\xi _{1} g_{1} ( t ) ,\quad t\geq 0, \\ &g_{2}^{\prime } ( t ) \leq -\xi _{2} g_{2} ( t ) , \quad t\geq 0. \end{aligned} $$

  (12)

• (A3) $\mu :[0,L]\rightarrow \mathbb{R}_{+}$ is a $C^{1}$ function so that

  $$ \mu \geq 0, \qquad \mu >0 \quad \mbox{in } (L_{0},L]. $$

  (13)

Theorem 4

Assume (11), (12), and (13) hold. Let

$$ \textstyle\begin{cases} -1< r< \frac{4-n}{n-2},\quad n\geq 3; \\ r\geq -1,\quad n=1,2.\end{cases} $$

(14)

Then, for any$(u_{0},v_{0})\in V_{0}^{2}$and$(v_{1},v_{2})\in H^{2}$, problem (1) has a unique local solution

$$ u\in C\bigl(\bigl(0,T^{*}\bigr);V_{0}\bigr)\cap C^{1}\bigl(\bigl(0,T^{*}\bigr);H\bigr), $$

for$T^{*}>0$small enough.

Lemma 5

There exists a function $F(u, v)$ such that

$$\begin{aligned} F(u, v) =&\frac{1}{2(r+2)} \bigl[u f_{1}(u, v)+v f_{2}(u, v) \bigr] \\ =&\frac{1}{2(r+2)} \bigl[a_{1} \vert u+v \vert ^{2(r+2)}+2 b_{1} \vert u v \vert ^{r+2} \bigr] \geq 0, \end{aligned}$$

where

$$ \frac{\partial F}{\partial u}=f_{1}(u, v), \qquad \frac{\partial F}{\partial v}=f_{2}(u, v). $$

We take $a_{1}=b_{1} = 1$ for convenience.

Lemma 6

([9])

There exist two positive constants $c_{0}$ and $c_{1}$ such that

$$ \frac{c_{0}}{2(r+2)} \bigl( \vert u \vert ^{2(r+2)}+ \vert v \vert ^{2(r+2)} \bigr) \leq F(u, v) \leq \frac{c_{1}}{2(r+2)} \bigl( \vert u \vert ^{2(r+2)}+ \vert v \vert ^{2(r+2)} \bigr) .$$

(15)

We now define the energy functional.

Lemma 7

Assume (11), (12), (13), and (14) hold, let$(u,v)$be a solution of (1), then$E(t)$is non-increasing, that is,

$$\begin{aligned} E(t) =&\frac{1}{2} \Vert u_{t} \Vert _{H}^{2}+\frac{1}{2} \Vert v_{t} \Vert _{H}^{2}+\frac{1}{2}l_{1} \Vert u_{x} \Vert _{H}^{2}+ \frac{1}{2}l_{2} \Vert v_{x} \Vert _{H}^{2} \\ &{}+\frac{1}{2}(g_{1}o u_{x})+ \frac{1}{2}(g_{2}o v_{x})- \int _{0}^{L}xF(u,v)\,dx \end{aligned}$$

(16)

satisfies

$$\begin{aligned} E^{\prime }(t) =& - \int _{0}^{L}x\mu (x) u_{t}^{2} \,dx- \int _{0}^{L}x \mu (x) v_{t}^{2} \,dx+\frac{1}{2}g^{\prime }_{1}\circ u_{x}+\frac{1}{2}g^{ \prime }_{2}\circ v_{x} \\ &{}- \int _{0}^{t}g_{1}(s)\,ds \int _{0}^{L}x u_{x}^{2} \,dx- \int _{0}^{t}g_{2}(s)\,ds \int _{0}^{L}x v_{x}^{2} \,dx \\ \leq & 0, \end{aligned}$$

(17)

where

$$ \int _{0}^{L}xF(u,v)\,dx=\frac{1}{2(r+2)} \bigl( \Vert u+v \Vert ^{2(r+2)}_{L^{2(r+2)}_{x}}+2 \Vert uv \Vert ^{(r+2)}_{L^{(r+2)}_{x}} \bigr) $$

(18)

and

$$ (g\circ u_{x}) (t)= \int _{0}^{L } \int _{0}^{t}xg(t-s) \bigl\vert u_{x}(x,t)-u_{x}(x,s) \bigr\vert ^{2}\,ds\,dx. $$

(19)

Proof

By multiplying (1)₁, (1)₂ by $xu_{t}$, $xv_{t}$, respectively, and integrating over $(0,L)$, we get

$$\begin{aligned}& \frac{d}{dt} \biggl\{ \frac{1}{2} \Vert u_{t} \Vert _{H}^{2}+ \frac{1}{2} \Vert v_{t} \Vert _{H}^{2}+\frac{1}{2}l_{1} \Vert u_{x} \Vert _{H}^{2}+ \frac{1}{2}l_{2} \Vert v_{x} \Vert _{H}^{2} +\frac{1}{2}(g_{1} \circ u_{x}) \\& \qquad {} +\frac{1}{2}(g_{2}\circ u_{x})- \int _{0}^{L}xF(u,v)\,dx \biggr\} \\& \quad = - \int _{0}^{L}x\mu (x) u_{t}^{2} \,dx- \int _{0}^{L}x\mu (x) v_{t}^{2} \,dx+ \frac{1}{2}g^{\prime }_{1}\circ u_{x}+\frac{1}{2}g^{\prime }_{2}\circ v_{x} \\& \qquad {} -\biggl( \int _{0}^{t}g_{1}(s)\,ds\biggr) \Vert u_{x} \Vert _{H}^{2}-\biggl( \int _{0}^{t}g_{2}(s)\,ds\biggr) \Vert v_{x} \Vert _{H}^{2} . \end{aligned}$$

(20)

And by using (11), (12) and (13), we obtain (17). □

3 Blow-up

In this section, we prove the blow-up result of solution of problem (1).

Now we define the functional

$$\begin{aligned} \mathbb{H}(t) =&-E(t) \\ =&-\frac{1}{2} \Vert u_{t} \Vert _{H}^{2}- \frac{1}{2} \Vert v_{t} \Vert _{H}^{2}-\frac{1}{2}l_{1} \Vert u_{x} \Vert _{H}^{2}- \frac{1}{2}l_{2} \Vert v_{x} \Vert _{H}^{2} \\ &{}-\frac{1}{2}(g_{1}ou_{x})- \frac{1}{2}(g_{2}ov_{x}) \\ &{}+\frac{1}{2(r+2)} \bigl[ \Vert u+v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+2 \Vert uv \Vert _{L_{x}^{r+2}}^{r+2} \bigr]. \end{aligned}$$

(21)

Theorem 8

Assume (11)–(13), and (14) hold. Assume further that$E(0)<0$, then the solution of problem (1) blows up in finite time.

Proof

From (17), we have

$$ E(t)\leq E(0)\leq 0. $$

(22)

Therefore

$$ \mathbb{H}^{\prime }(t)=-E^{\prime }(t)\geq 0. $$

By (18) and (15), we have

$$\begin{aligned} 0 \leq &\mathbb{H}(0)\leq \mathbb{H}(t) \leq \frac{1}{2(r+2)} \bigl[ \Vert u+v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+2 \Vert uv \Vert _{L_{x}^{(r+2)}}^{r+2} \bigr] \\ \leq &\frac{c_{1}}{2(r+2)} \bigl[ \Vert u \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+ \Vert v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)} \bigr]. \end{aligned}$$

(23)

We set

$$ \mathcal{K}(t)=\mathbb{H}^{1-\alpha }+\varepsilon \int _{0}^{L}x(uu_{t}+vv_{t}) \,dx+ \frac{\varepsilon }{2} \int _{0}^{L}x\mu (x) \bigl(u^{2}+v^{2} \bigr)\,dx, $$

(24)

where

$$ 0< \alpha < \frac{2r+2}{4(r+2)}< 1. $$

(25)

By multiplying (1)₁, (1)₂ by xu, xv and taking the derivative of (24), we get

$$\begin{aligned} \mathcal{K}^{\prime }(t) =&(1-\alpha )\mathbb{H}^{-\alpha } \mathbb{H}^{\prime }(t)+\varepsilon \bigl( \Vert u_{t} \Vert _{H}^{2}+ \Vert v_{t} \Vert _{H}^{2}\bigr)-\varepsilon \bigl( \Vert u_{x} \Vert _{H}^{2}+ \Vert v_{x} \Vert _{H}^{2}\bigr) \\ &{}+\varepsilon \int _{0}^{L}u_{x} \int _{0}^{t}g_{1}(t-s)xu_{x}(s)\,ds\,dx+ \varepsilon \int _{0}^{L}v_{x} \int _{0}^{t}g_{2}(t-s)xv_{x}(s)\,ds\,dx \\ &{}+\varepsilon \bigl[ \Vert u+v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+2 \Vert uv \Vert _{L_{x}^{(r+2)}}^{r+2} \bigr], \end{aligned}$$

(26)

we have

$$\begin{aligned}& \varepsilon \int _{0}^{t}g_{1}(t-s)\,ds \int _{0}^{L}u_{x}.xu_{x}(s) \,dx\,ds \\& \quad = \varepsilon \int _{0}^{t}g_{1}(t-s)\,ds \int _{0}^{L}u_{x}. \bigl(xu_{x}(s)-xu_{x}(t)\bigr)\,dx\,ds +\varepsilon \biggl( \int _{0}^{t}g_{1}(s)\,ds\biggr) \Vert u_{x} \Vert _{H}^{2} \\& \quad \geq \varepsilon \biggl(\frac{1}{2} \int _{0}^{t}g_{1}(s)\,ds\biggr) \Vert u_{x} \Vert _{H}^{2}- \frac{\varepsilon }{2}(g_{1}\circ u_{x}), \end{aligned}$$

(27)

$$\begin{aligned}& \varepsilon \int _{0}^{t}g_{2}(t-s)\,ds \int _{0}^{L}v_{x}.xv_{x}(s) \,dx\,ds \\& \quad = \varepsilon \int _{0}^{t}g_{2}(t-s)\,ds \int _{0}^{L}v_{x}. \bigl(xv_{x}(s)-xv_{x}(t)\bigr)\,dx\,ds +\varepsilon \biggl( \int _{0}^{t}g_{2}(s)\,ds\biggr) \Vert v_{x} \Vert _{H}^{2} \\& \quad \geq \varepsilon \biggl(\frac{1}{2} \int _{0}^{t}g_{2}(s)\,ds\biggr) \Vert v_{x} \Vert _{H}^{2}- \frac{\varepsilon }{2}(g_{2}\circ u_{x}). \end{aligned}$$

(28)

We obtain, from (26),

$$\begin{aligned} \mathcal{K}^{\prime }(t) \geq &(1-\alpha )\mathbb{H}^{-\alpha } \mathbb{H}^{\prime }(t)+\varepsilon \bigl( \Vert u_{t} \Vert _{H}^{2}+ \Vert v_{t} \Vert _{H}^{2}\bigr) \\ &{}-\varepsilon \biggl(\biggl(1-\frac{1}{2} \int _{0}^{t}g_{1}(s)\,ds\biggr) \Vert u_{x} \Vert _{H}^{2}+\biggl(1- \frac{1}{2} \int _{0}^{t}g_{2}(s)\,ds\biggr) \Vert v_{x} \Vert _{H}^{2} \biggr) \\ &{}-\frac{\varepsilon }{2}(g_{1}\circ u_{x})- \frac{\varepsilon }{2}(g_{2}\circ v_{x})+ \varepsilon \bigl[ \Vert u+v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+2 \Vert uv \Vert _{L_{x}^{(r+2)}}^{r+2} \bigr]. \end{aligned}$$

(29)

For $0< a<1$, from (21)

$$\begin{aligned} \varepsilon \bigl[ \Vert u+v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+2 \Vert uv \Vert _{L_{x}^{(r+2)}}^{r+2} \bigr] =&\varepsilon a \bigl[ \Vert u+v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+2 \Vert uv \Vert _{L_{x}^{(r+2)}}^{r+2} \bigr] \\ &{}+2\varepsilon (r+2) (1-a)\mathbb{H}(t) \\ &{}+\varepsilon (r+2) (1-a) \bigl( \Vert u_{t} \Vert _{H}^{2}+ \Vert v_{t} \Vert _{H}^{2}\bigr) \\ &{}+\varepsilon (r+2) (1-a) \biggl(1- \int _{0}^{t}g_{1}(s)\,ds\biggr) \Vert u_{x} \Vert _{H}^{2} \\ &{}+\varepsilon (p+2) (1-a) \biggl(1- \int _{0}^{t}g_{2}(s)\,ds\biggr) \Vert v_{x} \Vert _{H}^{2} \\ &{}+\varepsilon (r+2) (1-a) (g_{1}\circ u_{x}) \\ &{}+\varepsilon (r+2) (1-a) (g_{2}\circ v_{x}). \end{aligned}$$

(30)

Substituting in (29), we get

$$\begin{aligned} \mathcal{K}^{\prime }(t) \geq &(1-\alpha )\mathbb{H}^{-\alpha } \mathbb{H}^{\prime }(t)+\varepsilon {}\bigl[ (r+2) (1-a)+1\bigr] \bigl( \Vert u_{t} \Vert _{H}^{2}+ \Vert v_{t} \Vert _{H}^{2}\bigr) \\ &{}+\varepsilon \biggl[(r+2) (1-a) \biggl(1- \int _{0}^{t}g_{1}(s)\,ds\biggr)- \biggl(1- \frac{1}{2}\int _{0}^{t}g_{2}(s)\,ds\biggr) \biggr] \Vert u_{x} \Vert _{H}^{2} \\ &{}+\varepsilon \biggl[(r+2) (1-a) \biggl(1- \int _{0}^{t}g_{2}(s)\,ds\biggr)- \biggl(1- \frac{1}{2}\int _{0}^{t}g_{2}(s)\,ds\biggr) \biggr] \Vert v_{x} \Vert _{H}^{2} \\ &{}+\varepsilon \biggl[(r+2) (1-a)-\frac{1}{2} \biggr](g_{1}ou_{x}+g_{2}ov_{x}) \\ &{}+\varepsilon a \bigl[ \Vert u+v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+2 \Vert uv \Vert _{L_{x}^{(r+2)}}^{r+2} \bigr]+2\varepsilon (r+2) (1-a) \mathbb{H}(t). \end{aligned}$$

(31)

In this point, we take $a>0$ small enough so that

$$ \alpha _{1}=(r+2) (1-a)-1>0 $$

and we assume

$$ \max \biggl\{ \int _{0}^{\infty }g_{1}(s)\,ds, \int _{0}^{\infty }g_{2}(s)\,ds \biggr\} < \frac{(r+2)(1-a)-1}{ ((r+2)(1-a)-\frac{1}{2} )}= \frac{2\alpha _{1}}{2\alpha _{1}+1} ;$$

(32)

then we have

$$\begin{aligned}& \alpha _{2} = \biggl\{ (r+2) (1-a)-1)- \int _{0}^{t}g_{1}(s)\,ds \biggl((r+2) (1-a)- \frac{1}{2}\biggr) \biggr\} >0, \\& \alpha _{3} = \biggl\{ (r+2) (1-a)-1)- \int _{0}^{t}g_{2}(s)\,ds \biggl((r+2) (1-a)- \frac{1}{2}\biggr) \biggr\} >0, \end{aligned}$$

we pick ε small enough such that

$$ \mathbb{H}(0)+\varepsilon \int _{0}^{L}x(u_{0}u_{1}+v_{0}v_{1}) \,dx>0. $$

Thus, for some $\beta >0$, estimate (31) becomes

$$\begin{aligned} \mathcal{K}^{\prime }(t) \geq &\beta \bigl\{ \mathbb{H}(t)+ \Vert u_{t} \Vert _{H}^{2}+ \Vert v_{t} \Vert _{H}^{2}+ \Vert u_{x} \Vert _{H}^{2}+ \Vert v_{x} \Vert _{H}^{2} \\ &{}+(g_{1}ou_{x})+(g_{2}ov_{x})+ \bigl[ \Vert u+v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+2 \Vert uv \Vert _{L_{x}^{(r+2)}}^{r+2} \bigr] \bigr\} . \end{aligned}$$

(33)

By (15), for some $\beta _{1}>0$, we obtain

$$\begin{aligned} \mathcal{K}^{\prime }(t) \geq &\beta _{1} \bigl\{ \mathbb{H}(t)+ \Vert u_{t} \Vert _{H}^{2}+ \Vert v_{t} \Vert _{H}^{2}+ \Vert u_{x} \Vert _{H}^{2}+ \Vert v_{x} \Vert _{H}^{2} \\ &{}+(g_{1}ou_{x})+(g_{2}ov_{x})+ \bigl[ \Vert u \Vert _{L_{x}^{2(r+2)}}^{2(p+2)}+ \Vert u \Vert _{L_{x}^{2(r+2)}}^{2(r+2)} \bigr] \bigr\} \end{aligned}$$

(34)

and

$$ \mathcal{K}(t)\geq \mathcal{K}(0)>0,\quad t>0. $$

(35)

Next, using Hölder’s and Young’s inequalities, we have

$$\begin{aligned} \biggl\vert \int _{0}^{L}x(uu_{t}+vv_{t}) \,dx \biggr\vert ^{\frac{1}{1-\alpha }} \leq &C\bigl[ \Vert u \Vert _{L_{x}^{2(r+2)}}^{\frac{\theta }{1-\alpha }}+ \Vert u_{t} \Vert _{H}^{\frac{\mu }{1-\alpha }} \\ &{} + \Vert v \Vert _{L_{x}^{2(r+2)}}^{\frac{\theta }{1-\alpha }}+ \Vert v_{t} \Vert _{H}^{\frac{\mu }{1-\alpha }}\bigr], \end{aligned}$$

(36)

where $\frac{1}{\mu }+\frac{1}{\theta }=1$.

We take $\theta =2(1-\alpha )$, to get

$$ \frac{\mu }{1-\alpha }=\frac{2}{1-2\alpha }\leq 2(r+2). $$

Subsequently, for $s=\frac{2}{(1-2\alpha )}$ and by using (21), we obtain

$$\begin{aligned}& \Vert u \Vert _{L_{x}^{2(r+2)}}^{\frac{2}{1-2\alpha }} \leq d\bigl( \Vert u \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+\mathbb{H}(t)\bigr), \\& \Vert v \Vert _{L_{x}^{2(r+2)}}^{\frac{2}{1-2\alpha }} \leq d\bigl( \Vert v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+\mathbb{H}(t)\bigr), \quad \forall t\geq 0. \end{aligned}$$

Therefore,

$$ \biggl\vert \int _{0}^{L}x(uu_{t}+vv_{t}) \,dx \biggr\vert ^{\frac{1}{1-\alpha }}\leq c_{3}\bigl[ \Vert u \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+ \Vert v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+ \Vert u_{t} \Vert _{H}^{2}+ \Vert v_{t} \Vert _{H}^{2}+\mathbb{H}(t)\bigr]. $$

Subsequently,

$$\begin{aligned} \mathcal{K}^{\frac{1}{1-\alpha }}(t) =& \biggl(\mathbb{H}^{1-\alpha }+ \varepsilon \int _{0}^{L}x(uu_{t}+vv_{t}) \,dx+\frac{\varepsilon }{2}\int _{0}x\mu (x) \bigl(u^{2}+v^{2} \bigr)\,dx \biggr)^{\frac{1}{1-\alpha }} \\ \leq &c \biggl\{ \mathbb{H}(t)+ \biggl\vert \int _{0}^{L}x(uu_{t}+vv_{t}) \,dx \biggr\vert ^{ \frac{1}{1-\alpha }}+ \Vert u \Vert _{H}^{\frac{2}{1-\alpha }}+ \Vert v \Vert _{H}^{ \frac{2}{1-\alpha }} \biggr\} \\ \leq &c \bigl[\mathbb{H}(t)+ \Vert u_{t} \Vert _{H}^{2}+ \Vert v_{t} \Vert _{H}^{2}+ \Vert u_{x} \Vert _{H}^{2}+ \Vert v_{x} \Vert _{H}^{2}+(g_{1}ou_{x}) \\ &{}+(g_{2}ov_{x})+ \Vert u \Vert _{L_{x}^{2(r+2)}}^{2(r+2)}+ \Vert v \Vert _{L_{x}^{2(r+2)}}^{2(r+2)} \bigr]. \end{aligned}$$

(37)

From (33) and (37), we have

$$ \mathcal{K}^{\prime }(t)\geq \lambda \mathcal{K}^{\frac{1}{1-\alpha }}(t), $$

(38)

where $\lambda >0$, this depends only on $\beta _{1}$ and c.

By integration of (38), we obtain

$$ \mathcal{K}^{\frac{\alpha }{1-\alpha }}(t)\geq \frac{1}{\mathcal{K}^{\frac{-\alpha }{1-\alpha }}(0)-\lambda \frac{\alpha }{(1-\alpha )}t}. $$

Hence, $\mathcal{K}(t)$ blows up in time

$$ T\leq T^{\ast }= \frac{1-\alpha }{\lambda \alpha \mathcal{K}^{\alpha /(1-\alpha )}(0)}. $$

Then the proof is completed. □

4 Conclusion

Mixed non-local problems for hyperbolic and parabolic PDEs have been studied intensively in recent decades. Such equations or systems with constraints modelize many time-dependant physical phenomena. These constraints can be data measured directly on the boundary or giving integral boundary conditions (see for example [1, 4–7, 10–13]). In view of the articles mentioned above in [2, 3, 5] and a supplement to our recent study in [2, 8], we have proved in this work the blow-up of solutions of two singular nonlinear viscoelastic equations, where nonlocal boundary conditions, general source terms and localized frictional damping are considered.