Differential and Integral Equations

Global existence of solutions for a system of nonlinear damped wave equations

Takayoshi Ogawa and Hiroshi Takeda

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Abstract

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.

Article information

Source
Differential Integral Equations, Volume 23, Number 7/8 (2010), 635-657.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019188

Mathematical Reviews number (MathSciNet)
MR2654262

Zentralblatt MATH identifier
1240.35339

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations 35L15: Initial value problems for second-order hyperbolic equations

Citation

Ogawa, Takayoshi; Takeda, Hiroshi. Global existence of solutions for a system of nonlinear damped wave equations. Differential Integral Equations 23 (2010), no. 7/8, 635--657. https://projecteuclid.org/euclid.die/1356019188


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