The Lp–Lq estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media

We first obtain the Lp–Lq estimates of solutions to the Cauchy problem for one-dimensional damped wave equation Vtt−Vxx+Vt=0, (V,Vt)t=0=(V0,V1)(x), (x,t) R×R+,corresponding to that for the parabolic equation φt−φxx=0 φt=0=(V0+V1)(x).The estimates are shown by etc. for 1 q p ∞. To show (*), the explicit formula of the damped wave equation will be used. To apply the estimates to nonlinear problems is the second aim. We will treat the system of a compressible flow through porous media. The solution is expected to behave as the diffusion wave, which is the solution to the porous media equation due to the Darcy law. When the initial data has the same constant state at ±∞, a sharp Lp-convergence rate for p 2 has been recently obtained by Nishihara (Proc. Roy. Soc. Edinburgh, Sect. A, 133A (2003), 1–20) by choosing a suitably located diffusion wave. We will show the L1 convergence, applying (*).