L^p- L^q estimates for damped wave equations and their applications to semi-linear problem

Abstract

In this paper we study the Cauchy problem to the linear damped wave equation u_tt-Δ u+2au_t=0 in (0, ∞)× \bm{R}ⁿ (n≥q 2). It has been asserted that the above equation has the diffusive structure as t→∞. We give the precise interpolation of the diffusive structure, which is shown by L^{p_-}L^q estimates. We apply the above L^{p_-}L^q estimates to the Cauchy problem for the semilinear damped wave equation u_tt-Δ u+ 2au_t=|u|^σu in (0, ∞)× \bm{R}ⁿ (2≤ n≤ 5). If the power σ is larger than the critical exponent 2/n (Fujita critical exponent) and it satisfies σ≤ 2/(n-2) when n≥q 3, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.