Nonexistence of global solutions of abstract wave equations with high energies
- Jorge A Esquivel-Avila^{1}Email author
https://doi.org/10.1186/s13660-017-1546-1
© The Author(s) 2017
Received: 7 August 2017
Accepted: 17 October 2017
Published: 25 October 2017
Abstract
We consider an undamped second order in time evolution equation. For any positive value of the initial energy, we give sufficient conditions to conclude nonexistence of global solutions. The analysis is based on a differential inequality. The success of our result is based in a detailed analysis which is different from the ones commonly used to prove blow-up. Several examples are given improving known results in the literature.
Keywords
MSC
1 Introduction
2 Framework and previous results
We shall present some properties of a class of solutions of problem (1)-(2). In this functional framework, we assume that the following local existence and uniqueness theorem holds.
Theorem 2.1
Remark 2.2
Problem (1)-(2) is invariant if we reverse the time direction: \(t \mapsto- t\). The solution backwards \((u(t),v(t)), t < 0\), with the initial data \((u_{0},v_{0})\) corresponds to the solution forwards \((u(- t),- v(- t)), - t > 0\), with the initial data \((u_{0},- v_{0})\).
In particular \(u = 0\) is an equilibrium. The set of equilibria \(u \ne 0\) with minimal energy E is called ground state, and the corresponding minimal value of the energy is denoted by \(d > 0\), see [1]. The sign of \(I(u_{0})\) characterizes either blow-up in finite time or boundedness of solutions for small energies. Indeed, by means of the potential well method which works for energies \(E(u_{0},v_{0}) < d\), blow-up and boundedness properties are proved for nonlinear wave and Klein-Gordon equations in [2] if \(I(u_{0}) < 0\) and \(I(u_{0}) > 0\), respectively. After the work of Payne and Sattinger [2], several contributions have been published proving blow-up and globality of solutions of various types of equations by means of the potential well method. In particular, for generalized Boussinesq equations, we mention [3–6]. The qualitative behavior for high energies, \(E(u_{0},v_{0}) \geq d\), is unknown. Under sufficient conditions that involve upper bounds of the initial energy \(E(u_{0},v_{0})\), there are several recent works that prove blow-up of solutions of equations of the type (1) with high initial energies. Some of these results are particular cases of the one that we shall prove here. See the examples and references in the last section of this work. Our main result is obtained by means of the detailed analysis of a differential inequality. We do not apply any of the results known in the literature about differential inequalities commonly used to prove blow-up. See, for instance, [7–9] and the references therein for an account. However, we consider that these known results do not exploit the complete consequences of the differential inequalities involved. The purpose of this work is to get further in their analysis.
3 Main result
Here, we give sufficient conditions to get nonexistence of global solutions for any \(E(u_{0},v_{0})\) positive; however, these results are more relevant for \(E(u_{0},v_{0}) \geq d\). To that end, we investigate the consequences of a differential inequality.
Theorem 3.1
Corollary 3.2
Consider any solution of problem (1)-(2) in the sense of Theorem 2.1. Assume (H0) and (H1). For any positive constant Ẽ, we can always find initial data \(u_{0}, v_{0}\) satisfying (8) with the initial energy \(E(u_{0},v_{0}) = \tilde {E}\) such that the corresponding solution is not global.
Remark 3.3
For the proof of this theorem, some differential inequality is employed to prove that the solution only exists up to a finite time: \(T < \infty\). The estimate of the maximal time of existence by this means is not always optimal, that is, in general \(T > T_{\mathrm{MAX}}\). See [13–15] for more discussion. The technique described above belongs to the so-called functional method. That is, some functional in terms of a norm of the solution defined in the sense of Theorem 2.1 satisfies a differential inequality that necessarily implies that such norm blows up in finite time. Consequently, the solution cannot be global. This method has been used by many authors to show nonexistence of solutions of a wide class of equations. See, for instance, [9] for an early reference where a concavity argument is used. See also [7–9] and the references therein for an account of important contributions in the field, where several differential inequalities are studied. Here, we get further in the analysis of the differential inequality involved.
4 Proofs
Proof of Theorem 3.1
Remark 4.1
If the potential well method is applicable, as in the examples in the next section, there exist conditions characterizing blow-up when \(E(u_{0},v_{0}) < d\) as we mentioned in the Introduction. In this situation, the blow-up problem when \(E(u_{0},v_{0}) \le\alpha _{Q(u_{0},v_{0})}\) is resolved as follows. (i) \(\alpha_{Q(u_{0},v_{0})} < d\), here the characterization for blow-up when \(E(u_{0},v_{0}) \le\alpha _{Q(u_{0},v_{0})} < d\) is given by the potential well method. (ii) \(\alpha _{Q(u_{0},v_{0})} \geq d\), here the characterization for blow-up is given by the potential well method only for \(E(u_{0},v_{0}) < d\), and for \(d \le E(u_{0},v_{0}) \le \alpha_{Q(u_{0},v_{0})}\) blow up can be proved like in [7, 11].
However, for any positive constant Ẽ, we can always find initial data \(u_{0}, v_{0}\) satisfying (8) with the initial energy \(E(u_{0},v_{0}) = \tilde{E}\) and with \(Q(u_{0},v_{0}) > 0\) sufficiently large so that \(\tilde{E} \in\mathcal{I}_{Q(u_{0},v_{0})}\); and consequently, the corresponding solution blows up in finite time, as is stated in Corollary 3.2.
Proof of Corollary 3.2
Remark 4.2
For small energies, the potential well method characterizes the qualitative behavior of any solution in terms of the sign of \(I(u_{0})\), see [2–6]. For high energies, previous results conclude qualitative properties based in part on the sign of \(I(u_{0})\), see [16–18]. By means of the invariance of some sets, along with the solution, it is proved in [5, 10, 12] that \(I(u_{0}) < 0\) holds under sufficient conditions on \((u_{0},v_{0})\) that imply blow-up. Here, we do not have invariance properties, and we need to analyze when Theorem 3.1 implies \(I(u_{0}) < 0\). Energy satisfies the inequality \(E(u_{0},v_{0}) < \beta_{Q(u_{0},v_{0})}\), where \(\beta _{Q(u_{0},v_{0})} = \frac{1}{2} (\Phi(u_{0},v_{0}) - \mathsf {g}(u_{0},v_{0}) )\), and \(\mathsf{g}(u_{0},v_{0}) > 0\) is a function that decreases as \(Q(u_{0},v_{0})\) increases. Consequently, such a condition on the initial energy is \(\Vert h_{0} \Vert _{\mathcal{P}}^{2} + \Vert u_{0} \Vert _{V}^{2} < c \Vert u_{0} \Vert _{\mathcal{P}}^{2} + 2 \mathcal{G}(u_{0}) - \mathsf{g}(u_{0},v_{0})\). Equivalently, \(I(u_{0}) < c \Vert u_{0} \Vert _{\mathcal{P}}^{2} - \Vert h_{0} \Vert _{\mathcal {P}}^{2} - \mathsf{g}(u_{0},v_{0}) + 2 \mathcal{G}(u_{0}) - (\mathcal {F}(u_{0}),u_{0})\). By (H1), \((\mathcal{F}(u_{0}),u_{0}) - 2 \mathcal{G}(u_{0}) \geq(r - 2) \mathcal{G}(u_{0}) > 0\). Hence, \(I(u_{0}) < - \Vert h_{0} \Vert _{\mathcal {P}}^{2} - \mathsf{g}(u_{0},v_{0}) + c \Vert u_{0} \Vert _{\mathcal{P}}^{2} - (r - 2) \mathcal{G}(u_{0})\). Then \(I(u_{0}) < 0\) if the displacement is such that the nonlinear source \((r - 2) \mathcal{G}(u_{0})\) dominates the term \(c \Vert u_{0} \Vert _{\mathcal{P}}^{2}\). Apparently, for high energies, the sign of \(I(u_{0})\) is not a sufficient condition to conclude nonexistence of global solutions, but it is a necessary one.
5 Some examples
5.1 Nonlinear Klein-Gordon equation
5.2 Nonlinear wave equation
5.3 Generalized Boussinesq equation
5.4 Sixth order generalized Boussinesq equation
6 Conclusions
Declarations
Acknowledgements
The final form of this work is due to the valuable suggestions of the referees whom I thank for their comments. This work was supported by the Universidad Autónoma Metropolitana, Unidad Azcapotzalco, through the project Evolution Equations, number: CB-02413.
Authors’ contributions
Only one author contributed to the manuscript and read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
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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