Lp-Lq 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 uu – Δu + 2aut = 0 in (0, ∞) × Rn (n ≥ 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 Lp-Lq estimates. We apply the above Lp-Lq estimates to the Cauchy problem for the semilinear damped wave equation uu – Δu + 2aut = |u|σu in (0, ∞) × Rn (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 ≥ 3, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.© 2004, The Mathematical Society of Japan. All rights reserved.