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 Lp2Lq estimates of solutions to the Cauchy problem for onedimensional damped wave equation Vtt Vxx þ Vt ¼ 0; ðV; VtÞjt¼0 ¼ ðV0; V1ÞðxÞ; ðx;tÞAR Rþ; corresponding to that for the parabolic equation ft fxx ¼ 0 fj t¼0 ¼ ðV0 þ V1ÞðxÞ: The estimates are shown by ðV fÞð ;tÞ et=2 V0ð þ tÞ þ V0ð tÞ 2 Lp pCt1 2 1 q1 p 1 jjV0; V1jjLq ; tX1; ðÞ etc. for 1pqpppN: To show ðÞ; 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 7N; a sharp Lp-convergence rate for pX2 has been recently obtainedby Nishihara (Proc. Roy. Soc. Edinburgh, Sect. A, 133A (2003), 1–20) by choosing a suitably locateddiffusion wave. We will show the L1 convergence, applying