Large time behavior and Lp− Lq estimate of solutions of 2-dimensional nonlinear damped wave equations

Abstract

We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the L p − L q type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the L p − L q estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity | u| α u. Our result covers the whole super critical case α >1 , where the α=1 is well known as the Fujita exponent when n=2.