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December, 2019 Bounds for the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Level
Bin Guo, Xiaolei Li
Taiwanese J. Math. 23(6): 1461-1477 (December, 2019). DOI: 10.11650/tjm/190103

Abstract

This paper deals with lower and upper bounds for the lifespan of solutions to a fourth-order nonlinear hyperbolic equation with strong damping: \[ u_{tt} + \Delta^{2} u - \Delta u - \omega \Delta u_t + \alpha(t) u_t = |u|^{p-2} u. \] First of all, the authors construct a new control function and apply the Sobolev embedding inequality to establish some qualitative relationships between initial energy value and the norm of the gradient of the solution for supercritical case ($2(N-2)/(N-4) \lt p \lt 2N/(N-4)$, $N \geq 5$). And then, the concavity argument is used to prove that the solution blows up in finite time for initial data at low energy level, at the same time, an estimate of the upper bound of blow-up time is also obtained.

Subsequently, for initial data at high energy level, the authors prove the monotonicity of the $L^{2}$ norm of the solution under suitable assumption of initial data, furthermore, we utilize the concavity argument and energy methods to prove that the solution also blows up in finite time for initial data at high energy level.

At last, for the supercritical case, a new control functional with a small dissipative term and an inverse Hölder inequality with correction constants are employed to overcome the difficulties caused by the failure of the embedding inequality ($H^{2}(\Omega) \cap H^{1}_{0}(\Omega) \hookrightarrow L^{2p-2}$ for $2(N-2)/(N-4) \lt p \lt 2N/(N-4)$) and then an explicit lower bound for blow-up time is obtained. Such results extend and improve those of [S. T. Wu, J. Dyn. Control Syst. 24 (2018), no. 2, 287--295].

Citation

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Bin Guo. Xiaolei Li. "Bounds for the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Level." Taiwanese J. Math. 23 (6) 1461 - 1477, December, 2019. https://doi.org/10.11650/tjm/190103

Information

Received: 10 October 2018; Revised: 20 December 2018; Accepted: 10 January 2019; Published: December, 2019
First available in Project Euclid: 17 January 2019

zbMATH: 07142982
MathSciNet: MR4033554
Digital Object Identifier: 10.11650/tjm/190103

Subjects:
Primary: 35L05 , 35L20 , 35L71

Keywords: concavity argument , energy estimate , mountain pass level

Rights: Copyright © 2019 The Mathematical Society of the Republic of China

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Vol.23 • No. 6 • December, 2019
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