Large time behavior of solutions for a nonlinear damped wave equation
Hiroshi Takeda
Communications on Pure & Applied Analysis 2016, 15(1): 41-55 doi: 10.3934/cpaa.2016.15.41
We study the large time behavior of small solutions to the Cauchy problem for a nonlinear damped wave equation. We proved that the solution is approximated by the Gauss kernel with suitable choice of the coefficients and powers of $t$ for $N+1$ th order for all $N \in \mathbb{N}$. Our analysis is based on the approximation theorem of the linear solution by the solution of the heat equation [37]. In particular, as pointed out by Galley-Raugel [4], we explicitly observe that from third order expansion, the asymptotic behavior of the solutions of a nonlinear damped wave equation is different from that of a nonlinear heat equation.
keywords: hyperbolic equations asymptotic profile Damped wave equation perturbation. Cauchy problem
Global existence of solutions for higher order nonlinear damped wave equations
Hiroshi Takeda
Conference Publications 2011, 2011(Special): 1358-1367 doi: 10.3934/proc.2011.2011.1358
We consider a Cauchy problem for a polyharmonic nonlinear damped wave equation. We obtain a critical condition of the nonlinear term to ensure the global existence of solutions for small data. Moreover, we show the op-timal decay property of solutions under the sharp condition on the nonlinear exponents, which is a natural extension of the results for the nonlinear damped wave equations. The proof is based on $L^p-L^q$ type estimates of the fundamental solutions of the linear polyharmonic damped wave equations.
keywords: Global solutions optimal decay Critical exponents

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