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,∂tu)|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)=∂tS(t)u0+S(t)(u0+u1) and then we directly estimate the exact solution S(t)g and its derivative ∂tS(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.