Open Access
July/August 2010 Global existence of solutions for a system of nonlinear damped wave equations
Takayoshi Ogawa, Hiroshi Takeda
Differential Integral Equations 23(7/8): 635-657 (July/August 2010). DOI: 10.57262/die/1356019188


We consider the Cauchy problem of the semilinear damped wave system: \begin{equation} \notag \begin{cases} \partial_{t}^2 u_{j} - \Delta u_{j} + \partial_{t} u_{j} = F_{j}(u), & t > 0, \quad x\in \mathbb R^{n},\\ u_{j}(0,x)=a_{j}(x),\quad \partial_{t} u_{j}(0,x) = b_{j}(x), & x\in \mathbb R^{n}, \end{cases} \end{equation} where $m \ge 2$ and $j = 1$, $\cdots$, $m$. We show the existence of a global-in-time solution for a small initial data under a sharp condition on the nonlinear exponents, which is a natural extension of the results for the single nonlinear damped wave equations ([28], [30]). The proof is based on $L^{p}$-$L^{q}$ type estimates of the fundamental solutions of the linear damped wave equations ([9]) and systematic choice of the function scale to adjust the nonlinear growth order.


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Takayoshi Ogawa. Hiroshi Takeda. "Global existence of solutions for a system of nonlinear damped wave equations." Differential Integral Equations 23 (7/8) 635 - 657, July/August 2010.


Published: July/August 2010
First available in Project Euclid: 20 December 2012

zbMATH: 1240.35339
MathSciNet: MR2654262
Digital Object Identifier: 10.57262/die/1356019188

Primary: 35L15 , 35L70

Rights: Copyright © 2010 Khayyam Publishing, Inc.

Vol.23 • No. 7/8 • July/August 2010
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