Lp decay problem for the dissipative wave equation in odd dimensions

Consider the Cauchy problem in odd dimensions for the dissipative wave equation: ( +∂t )u = 0 in R2n+1 ×(0,∞) with (u, ∂t u)|t=0 = (u0,u1). Because the L2 estimates and the L∞ estimates of the solution u(t) are well known, in this paper we pay attention to the Lp estimates with 1 p <2 (in particular, p = 1) of the solution u(t) for t 0. In order to derive Lp estimates we first give the representation formulas of the solution u(t) = ∂t S(t)u0 + S(t)(u0 + u1) and then we directly estimate the exact solution S(t)g and its derivative ∂t S(t)g of the dissipative wave equation with the initial data (u0,u1) = (0,g). In particular, when p = 1 and n 1, we get the L1 estimate: u(t) L1 Ce−t/4( u0 Wn,1 + u1 Wn−1,1 )+ C( u0 L1 + u1 L1 ) for t 0.  2004 Elsevier Inc. All rights reserved.