Blow-up of solutions to a viscoelastic parabolic equation with positive initial energy
- Haixia Li^{1} and
- Yuzhu Han^{2}Email author
Received: 18 March 2017
Accepted: 30 May 2017
Published: 8 June 2017
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
1 Introduction
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 [17–20].
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
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
Proof
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., \(\Vert u \Vert _{\Gamma_{1},p}\leq S \Vert \nabla u \Vert _{2}\), \(\forall u\in H\).
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
Proof
Declarations
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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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