Asymptotics of solutions to the periodic problem for the nonlinear damped wave equation

Abstract

We study large time asymptotic behavior of solutions to the periodic problem for the nonlinear damped wave equation $$ \left\{ \begin{array}{l} u_{tt}+2\alpha u_{t}-\beta u_{xx}=-\lambda \left| u\right| ^{\sigma}u,\text{ }x\in \Omega ,t >0 , u(0,x)=\phi \left( x\right) ,\text{}u_{t}(0,x)=\psi \left( x\right) ,\text{ }x\in \Omega , \end{array} \right. $$ where Ω = [ −π ,π ], α ,β ,λ ,σ > 0. We prove that if the initial data $${\phi \in {\bf H}^{1}}$$ and $${\psi \in {\bf L}^{2}}$$ , then there exists a unique solution $${u\left( t,x\right) \in {\bf C}\left( \left[ 0,\infty \right) ;{\bf H}^{1}\right)}$$ of the periodic problem which has the time decay estimates $$ \left\| u\left( t\right) \right\| _{{\bf L}^{\infty}} \leq C\left\langle t\right\rangle ^{-\frac{1}{\sigma}},\text{} \left\| \partial _{t}u\left( t\right) \right\| _{{\bf L}^{\infty}} \leq C\left\langle t\right\rangle ^{-\frac{1}{\sigma}-\frac{1}{2}} $$ for all t > 0. Moreover under some additional conditions we find the asymptotic formulas for the solutions.