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 Ufimsk. Mat. Zh., 2017, Volume 9, Issue 4, Pages 45–54 (Mi ufa402)

On global instability of solutions to hyperbolic equations with non-Lipschitz nonlinearity

Y. Sh. Il'yasov, E. E. Kholodnov

Institute of Mathematics, Ufa Scientific Center, RAS, Chernyshevsky str. 112, 450077, Ufa, Russia

Abstract: In a bounded domain $\Omega \subset \mathbb{R}^n$, we consider the following hyperbolic equation
\begin{equation*} \begin{cases} v_{tt} = \Delta_p v+\lambda |v|^{p-2}v-|v|^{\alpha-2}v,& x\in \Omega,
v\bigr{|}_{\partial \Omega}=0. \end{cases} \end{equation*}
We assume that $1<\alpha<p<+\infty$; this implies that the nonlinearity in the right hand side of the equation is of a non-Lipschitz type. As a rule, this type of nonlinearity prevent us from applying standard methods from the theory of nonlinear differential equations. An additional difficulty arises due to the presence of the $p$-Laplacian $\Delta_p (\cdot):=div(|\nabla(\cdot)|^{p-2}\nabla(\cdot))$ in the equation. In the first result, the theorem on the existence of the so-called stationary ground state of the equation is proved. The proof of this result is based on the Nehari manifold method. In the main result of the paper we state that each stationary ground state is unstable globally in time. The proof is based on the development of an approach by Payne and Sattinger introduced for studying the stability of solutions to hyperbolic equations.

Keywords: stability of solutions, nonlinear hyperbolic equations, Nehari manifold method, $p$-Laplacian.

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English version:
Ufa Mathematical Journal, 2017, 9:4, 44–53 (PDF, 372 kB); https://doi.org/10.13108/2017-9-4-44

Bibliographic databases:

UDC: 517.957
MSC: 35J61, 35J92, 35J50

Citation: Y. Sh. Il'yasov, E. E. Kholodnov, “On global instability of solutions to hyperbolic equations with non-Lipschitz nonlinearity”, Ufimsk. Mat. Zh., 9:4 (2017), 45–54; Ufa Math. J., 9:4 (2017), 44–53

Citation in format AMSBIB
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\paper On global instability of solutions to hyperbolic equations with non-Lipschitz nonlinearity
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\pages 45--54
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\jour Ufa Math. J.
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This publication is cited in the following articles:
1. E. E. Kholodnov, “Ob osnovnykh sostoyaniyakh i resheniyakh s kompaktnymi nositelyami ellipticheskikh uravnenii s nelipshitsevymi nelineinostyami”, Differentsialnye uravneniya, Itogi nauki i tekhn. Ser. Sovrem. mat. i ee pril. Temat. obz., 163, VINITI RAN, M., 2019, 108–112
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