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CPAA

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

PROC

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.

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