## Taiwanese Journal of Mathematics

- Taiwanese J. Math.
- Volume 23, Number 6 (2019), 1461-1477.

### Bounds for the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Level

Bin Guo and Xiaolei Li

#### 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].

#### Article information

**Source**

Taiwanese J. Math., Volume 23, Number 6 (2019), 1461-1477.

**Dates**

Received: 10 October 2018

Revised: 20 December 2018

Accepted: 10 January 2019

First available in Project Euclid: 17 January 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.twjm/1547715693

**Digital Object Identifier**

doi:10.11650/tjm/190103

**Mathematical Reviews number (MathSciNet)**

MR4033554

**Zentralblatt MATH identifier**

07142982

**Subjects**

Primary: 35L05: Wave equation 35L20: Initial-boundary value problems for second-order hyperbolic equations 35L71: Semilinear second-order hyperbolic equations

**Keywords**

concavity argument energy estimate mountain pass level

#### Citation

Guo, Bin; Li, Xiaolei. Bounds for the Lifespan of Solutions to Fourth-order Hyperbolic Equations with Initial Data at Arbitrary Energy Level. Taiwanese J. Math. 23 (2019), no. 6, 1461--1477. doi:10.11650/tjm/190103. https://projecteuclid.org/euclid.twjm/1547715693