Lp-Lq estimates for damped wave equations and their applications to semi-linear problem

In this paper we study the Cauchy problem to the linear damped wave equation u(n) - Deltau + 2au(t) = 0 in (0, infinity) x R-n (n greater than or equal to 2). It has been asserted that the above equation has the diffusive structure as t --> infinity. 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) - Deltau + 2au(t) = \u\(sigma)u in (0, infinity) x R-n (2 less than or equal to n less than or equal to 5). If the power sigma is larger than the critical exponent 2/n (Fujita critical exponent) and it satisfies sigma less than or equal to 2/(n-2) when n greater than or equal to 3, then the time global existence of small solution is proved, and the decay estimates of several norms of the solution are derived.