Blow-up phenomenon for a nonlinear wave equation with anisotropy and a source term

ABSTRACT

This paper deals with the blow-up phenomenon for a nonlinear wave equation with anisotropy and a source term: $\begin{array}{r}\begin{array}{rl}& u_{tt}-\sum _{i=1}^{i=n}\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left({\left|\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right|}^{p_i-2}\frac{\mathrm{\partial }u}{\mathrm{\partial }{x}_i}\right)-\mathrm{\Delta }{u}_t=u|u{|}^{\sigma -2},x\in \mathrm{\Omega },t>0,\\ & u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),x\in \mathrm{\Omega },\\ & u(x,t)=0,x\in \mathrm{\partial \Omega },t\ge 0,\end{array}\end{array}$ where $\mathrm{\Omega }\subseteq {\mathbb{R}}^n,n\ge 1,$ is a bounded domain with smooth boundary. Here, $u_0$ and $u_1$ are initial functions and $\sigma >2$ as well as $\sigma \ge p_i\ge 2$ for $i=1,\dots ,n.$

We present a new theorem for studying the blow-up phenomena and apply this theorem to the above mentioned problem. For this problem, we prove that the solutions blow up in finite time with negative initial energy without any restrictions on initial data. We also prove the solutions blow up in finite time with positive initial energy under some suitable conditions on initial data. Besides, we present some key remarks based on the conception of limit the energy function in the case of non-negative initial energy. These results extend the recent results obtained by Lu, Li and Hao [Existence and blow up for a nonlinear hyperbolic equation with anisotropy. Appl Math Lett. 2012; 25:1320–1326] which assert the solutions blow up in finite time with non-positive initial energy provided that $(\sigma -2){\int }_{\mathrm{\Omega }}{u}_0\left(x\right)u_1\left(x\right)\mathrm{d}x>\parallel \mathrm{\nabla }{u}_0{\parallel }_2^2.$

COMMUNICATED BY:

• Scott Hansen

KEYWORDS:

• Nonlinear wave equation
• blow-up
• concavity argument

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Lxx
• 35L05
• 35B44

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The second author was supported by Institute for Research in Fundamental Sciences (IPM) and Iran National Science Foundation (INSF) [grant number 94027514].