# Blow-up of solutions to a viscoelastic parabolic equation with positive initial energy

## Abstract

In this paper, a semilinear viscoelastic parabolic equation with nonlinear boundary flux is studied. Due to the comparison principle being invalid, potential well method and concavity argument are used to prove that the solutions blow up in finite time with positive initial energy. This result improves the one obtained by Han et al. (C. R. Math. Acad. Sci. Paris, Sér. I 353:825-830, 2015).

## Keywords

viscoelastic potential well method blow-up nonlinear boundary flux## 1 Introduction

*∂*Ω, \(\{\Gamma_{0},\Gamma_{1}\}\) is a partition of

*∂*Ω such that \(\partial\Omega=\Gamma_{0}\cup\Gamma_{1}\), \(\Gamma_{0}\cap\Gamma_{1}=\emptyset\) and \(\operatorname{meas}(\Gamma_{0})>0\),

*ν*is the unit outward normal on \(\Gamma_{1}\), and the relaxation function

*g*: \(\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) is a bounded \(C^{1}\) function satisfying some other conditions to be specified later.

*u*denotes the temperature, \(d>0\) is the diffusion coefficient and the integral term represents the capacity of the material to keep memory of their past trace. From a mathematical point of view, it is expected that the leading term can dominate the memory term and therefore the equation in (1) is of parabolic type. During the past few years, much work has been done on the study of equations with memory terms, and remarkable progress has been made on the local and global existence, uniqueness, finite time blow-up and regularities of weak or classical solutions. We only refer the interested readers to [5, 6] among a large number of literature sources.

In this paper, we confine ourselves to the finite time blow-up property of solutions to Problem (1), an important property possessed by many nonlinear evolution equations. There have been many methods to choose from when determining whether the solutions to the given evolution problem blow up in finite time or not, for instance, the (first) eigenvalue method, the concavity argument, the comparison method based on maximum principle and other methods based on delicate integration. Interested reader may refer to [7] for the outline of each method and their applications to typical examples. Mainly by using the methods mentioned above, blow-up profiles including blow-up time, blow-up rate, blow-up set and boundary layers of solutions to semilinear equations like (1) have been widely investigated when \(g(t)\equiv0\). We only refer to the survey papers [8, 9] here.

Compared the blow-up results obtained in [11, 12] with the ones in [1, 10], it is expected that the solutions to Problem (1) will also blow up in finite time with positive but small initial energy, which is the main purpose of this paper. Since the source is on the boundary, we cannot establish the connection between \(\frac{d}{dt} \Vert u \Vert _{2}\) and \(\Vert u \Vert _{2}^{r}\), as was done in [10]. By using the famous potential well method proposed by Sattinger and Payne [13, 14], we will show that when the initial data falls outside of the potential well, the \(L^{2}(\Omega)\) norm of the corresponding solution has a positive lower bound, which then can be applied to control the positive initial energy and to derive the finite time blow-up of the corresponding solutions. Similar procedures were used by Marin et al. in dealing with thermoelasticity of micropolar bodies, see [15, 16]. There are some other interesting works that we have some ideas from, of which we only mention [17, 18, 19, 20].

The rest of this paper is organized as follows. In Section 2, as preliminaries, we define some sets and functionals and prove their basic properties. The main result will be stated and proved in Section 3.

## 2 Preliminaries

*H*with the norm \(\Vert u \Vert _{H}= \Vert \nabla u \Vert _{2}\) that is equivalent to the standard one (see [21]).

*g*and the parameter

*p*are supposed to satisfy

Before going further, we present the definition of strong solutions to Problem (1), which was given in [1, 22]. Local existence of such a solution was proved in [23] (the first three steps in the proof of Theorem 6) for a little more general problems which contain Problem (1) as a special case, by applying Galerkin’s method and the contraction mapping principle. The lengthy proof will not be repeated here.

### Definition 2.1

Assumption (7) is necessary to ensure that the equation in (1) is of parabolic type, and assumption (8) implies \(\vert u \vert ^{p-2}u\in L^{2}(\Gamma_{1})\) by the Sobolev trace embedding theorem (Theorem 5.36 in [24]) and hence \(\int_{\Gamma_{1}} \vert u \vert ^{p-2}u\phi\,\mathrm{d}\sigma\) makes sense.

### Lemma 2.1

*defined in*(9)

*is nonincreasing in*

*t*

*and satisfies*

### Proof

*t*. The proof is complete. □

*l*be the constant given in (7). For any \(u\in H\), set

*d*of the potential well

*W*is characterized as

*d*is given in the next lemma.

### Lemma 2.2

*The depth* *d* *of the potential well* *W* *is positive*.

### Proof

Since *p* satisfies (8), *H* can be embedded into \(L^{p}(\Gamma _{1})\) continuously. Let \(S>0\) be the best embedding constant, i.e., \(\Vert u \Vert _{\Gamma_{1},p}\leq S \Vert \nabla u \Vert _{2}\), \(\forall u\in H\).

*d*, it is seen that \(d>0\). The proof is complete. □

The next lemma describes the invariance of *V* with respect to the semiflow of (1) under some additional conditions.

### Lemma 2.3

*Let* (7) *and* (8) *hold and* \(u(x,t)\) *be a local solution to Problem *(1). *If there exists* \(t_{0}\in[0,T)\) *such that* \(u(\cdot, t_{0})\in V\) *and* \(E(u(t_{0}))< d\), *then* \(u(x,t)\) *remains inside* *V* *for any* \(t\in[t_{0},T)\), *where* *T* *is the maximal existence time of* \(u(x,t)\).

### Proof

Suppose on the contrary that there exists \(t_{1}\in[0,T)\) such that \(u(x,t)\in V\) for \(t\in[t_{0},t_{1})\) and \(u(x,t_{1})\notin V\). By the definition of *V* and the continuity of \(J(u(x,t))\) and \(I(u(x,t))\) with respect to *t*, we have either (i) \(J(u(x,t_{1}))=d\) or (ii) \(I(u(x,t_{1}))=0\).

Since \(u(x,t)\in V\) for \(t\in[t_{0},t_{1})\), we have \(\Vert \nabla u(\cdot,t) \Vert _{2}\geq c^{*}\) for all \(t\in[t_{0},t_{1})\). By continuity, it also holds that \(\Vert \nabla u(\cdot,t_{1}) \Vert _{2}\geq c^{*}\). This together with (ii) implies that \(u(x,t_{1})\in\mathcal{N}\). By the definition of *d*, we have \(J(u(x,t_{1}))\geq d\), a contradiction. The proof is complete. □

## 3 Main result

The main result of this paper is the following.

### Theorem 3.1

*Suppose that*(7), (8)

*hold and that*\(u_{0}\in V\)

*satisfies*\(E(u_{0})<\frac{1-l}{2} (\frac{l}{S^{p}} )^{\frac{2}{p-2}}\),

*where*

*l*

*and*

*S*

*are the positive constants defined in*(7)

*and Lemma*2.2,

*respectively*.

*If*

*then any strong solution*\(u(x,t)\)

*to Problem*(1)

*blows up in finite time*

*T*

*in the sense that*\(\lim_{t\rightarrow T} \Vert u(\cdot,t) \Vert _{2}=+\infty\).

### Proof

*u*is global, then

*ε*and

*α*small enough such that

## Notes

### Acknowledgements

The authors would like to thank the referees for their valuable comments and suggestions regarding the original manuscript and for pointing out several references that are quite helpful to us. They would also like to express their sincere gratitude to Professor Wenjie Gao for his enthusiastic guidance and constant encouragement. The first author is supported by NSFC (11626044) and by the Natural Science Foundation of Changchun Normal University. The second author is supported by NSFC (11401252) and by Science and Technology Development Project of Jilin Province (20160520103JH).

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