Blow-up of solutions to a class of Kirchhoff equations with strong damping and nonlinear dissipation
- Qingying Hu^{1},
- Jian Dang^{1} and
- Hongwei Zhang^{1}Email author
Received: 6 March 2017
Accepted: 20 July 2017
Published: 4 August 2017
Abstract
The initial boundary value problem of a class of Kirchhoff equations with strong damping and nonlinear dissipation is considered. By modifying Vitillaro’s argument, we prove a blow-up result for solutions with positive and negative initial energy respectively.
Keywords
Kirchhoff equation blow-up strong damping nonlinear dissipation1 Introduction
When M is not a constant function, equation (1.1) without the damping and source terms is often called a Kirchhoff-type wave equation; it has first been introduced by Kirchhoff [7] in order to describe the nonlinear vibrations of an elastic string. When \(\omega =0\) or \(h(u_{t})=0\), the nonexistence of the global solutions of Kirchhoff equations was investigated by many authors (see [8–24] and the references therein). The work of Ono [10, 11] dealt with equation (1.1) with \(\omega =0\) and \(f(u)=\vert u\vert ^{p-2}u\). When \(h(u_{t} ) =- \Delta u_{t}\) or \(u_{t}\), Ono showed that the local solutions blow up in finite time with \(E(0)\le 0\) by applying the concavity method. Ono also combined the so-called potential well method and concavity method to show blow-up properties with \(E(0)>0\). When \(h(u_{t} )=\vert u_{t} \vert ^{m-2}u_{t} \), \(m>2\), Ono proved that the local solution is not global when \(p>\max\{2r+ 2,m\}\) and \(E(0)<0\). Wu [13] extended the result of [11, 12] in the case of \(h(u_{t} ) =- \Delta u _{t}\) or \(u_{t}\) by the energy method and gave some estimates for the life span of solutions. Wu also extended the result of [10] to general \(M(s)\) and to the condition that \(E(0)\ge 0\) for nonlinear dissipative term \(h(u_{t} )=\vert u_{t} \vert ^{m-2}u _{t}\) by Vitillaro’s argument [2]. For more blow-up results of problem (1.1)-(1.3) with \(\omega =0\), \(h(u_{t} )=\vert u _{t} \vert ^{m-2}u_{t}\) and \(f(u)=\vert u\vert ^{p-2}u\) see [14–21].
However, a natural question is whether nonlinear sources can cause finite time blow-up for solutions to problem (1.1)-(1.3) when introducing both the presence of the nonlinear weak damping term \(h(u_{t})=\vert u_{t} \vert ^{m-2}u_{t}\) and the linear strong damping term \(\Delta u_{t}\) (i.e. \(\omega \neq 0\)). This question has been addressed for the wave equation (1.4) by Gazzola and Squassina [3] and Yu [4] (see also Graber and Said-Houari [25] for a strongly damped wave equation with dynamic boundary conditions). From the physics point of view, the strong damping term \(\Delta u_{t}\) and the nonlinear dissipative damping term \(h(u_{t})\) play a dissipative or inhibitive part in the energy accumulation in the configurations, which dissipates energy and drives the system toward stability, while the nonlinear source term \(f(u)\) models an external force that amplifies the energy and drives the system to possible solutions that blow up in finite time. It is well known that if \(\omega =0\), \(h(u_{t})=\vert u_{t}\vert ^{m-2}u_{t}\), \(f(u)=\vert u\vert ^{p-2}u\), the solutions of (1.4) with any initial data continue to exist globally ‘in time’ if \(m \ge p\) and blow up in finite time if \(p > m\) and the initial energy is sufficiently negative or certain positive initial energy (see [1–6] and the references therein). However, introducing both a nonlinear weak damping term \(h(u_{t})\) and a linear strong damping term \(\Delta u _{t}\) makes the problem very interesting but difficult as well. Indeed, a strong action of dissipative terms could make the existence of global solutions easier, since they play the role of stabilizing terms and their smoothing effect makes the blow-up more difficult [21, 26]. Introducing a strong damping term \(\Delta u_{t}\) makes the problem different from the one mentioned in [1]. The most frequently used technique in the proof of blow-up named ‘concavity argument’ is no longer applied, and the techniques in the papers mentioned above also cannot be used directly due to the term \(\Delta u_{t}\). Thereby, at present, less results are at present time known for the wave equation with a strong damping term, and there still exist many other unsolved problems; see Gazzola and Squassina [3] for the case \(m = 2\) (see also [4–6, 25] and the references therein).
Motivated by these papers, the purpose of this paper is to investigate the nonexistence result of global solutions of the problem (1.1)-(1.3) with both terms \(\Delta u_{t}\) and \(h(u_{t})\). More precisely, we shall show global nonexistence results of the problem (1.1)-(1.3) by adopting and modifying the method of [2, 17, 26] and combining with potential well theory. We will construct a function \(L(t)\) (see Section 3) which is different from that in [2, 5, 17, 26]. The method can also be extended to equation (1.1) with the general function \(M(s)\), \(h(s)\) and \(g(s)\) as in [26], and it can also be extended to equation (1.5) as in [27]. The plan of this article is as follows. In Section 2, some notations, assumptions and preliminaries are introduced and the main results of this article are shown in Section 3.
2 Preliminaries
In this section, we give some assumptions and preliminary results in order to state the main results of this article. Throughout this article, the following notations are used for precise statements: \(L^{p}(\Omega)\) \((1< p<\infty)\) denotes the usual space of all \(L^{p}\)-functions on Ω with norm \(\Vert u\Vert _{L^{p}(\Omega)}= \Vert u\Vert _{p}\) and the inner product \((u,v)=\int_{\Omega }uv\,dx\). For simplicity, we denote \(\Vert u\Vert _{L^{2}(\Omega)}= \Vert u \Vert \). The constants C used in this paper are positive generic constants, which may be different in various occurrences. For simplicity, we take \(\omega =a=b=1\).
- (A)
\(p>\max\{2(r+1),m\}\) and \(1< m< p \le \frac{2(n-1)}{n-2}\) if \(n\ge 3\), \(1< m< p \le \infty \) if \(n=1,2\).
Next, we present the following local existence theorem, which can be founded in [27].
Theorem 2.1
[27]
Lemma 2.2
[27]
A similar argument from [2, 13, 15, 27] gives the following result.
Lemma 2.3
Proof
3 Main results
Now, we give our main results.
Theorem 3.1
Assuming that (A) holds and \(u_{0},u_{1}\in H^{2} \cap H^{1}_{0}\), then any solution u of the problem (1.1)-(1.3) with initial data satisfying \(E(0)< E_{1}\) and \(\Vert \nabla u_{0} \Vert ^{2}+\Vert \nabla u_{0}\Vert ^{2(r+1)}\ge \lambda_{1}^{2(r+1)}\) will blow up in finite time.
Proof
Theorem 3.2
Assuming that \(u_{0}\in H^{2}\cap H^{1}_{0}\), \(u_{1} \in H^{1}_{0}\), and \(p >\max\{2(r+1), m\}\), \(E(0) < 0\), then the local solution of the problem (1.1)-(1.3) blows up in finite time.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China (Nos. 11526077, 11601122).
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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