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

摘要

In this paper we study the Cauchy problem to the linear damped wave equation u_(tt) - Δu + 2au_t = 0 in (0, ∞) x R~n (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 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) - Δu + 2au_t = |u|~σ u in (0, ∞) x R~n (2 ≤ n ≤ 5). If the power a is larger than the critical exponent 2 (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.