The \(L^{p}\)--\(L^{q}\) estimates of solutions to one-dimensional damped wave equations and their application to the compressible flow through porous media
The paper refers to \(L^p\)-\(L^q\) estimates for the solution of the Cauchy problem for the damped wave equations of the form \(V_{tt}- V_{xx}+ V_t= 0\), \(x\in\mathbb{R}^1\), \(t> 0\), \((V,V_t)|_{t=0}= (V_0, V_1)(x)\), corresponding to the solution for the parabolic equation \(\varphi_t- \varphi_{xx}= 0\), \(\varphi|_{t=0}= (V_0+ V_1)(x)\). The following types of estimates are obtained: \[ \begin{multlined}\Biggl\|(V- \varphi)(.,t)- e^{-t/2}{V_0(.+t)+ V_0(.-t)\over 2}\Biggr\|_{L^p}\leq\\ Ct^{-{1\over 2}({1\over q}-{1\over p})-1}\|V_0, V_1\|_{L^q},\quad t\geq 1,\quad 1\leq q\leq p\leq\infty.\end{multlined} \] As an application, the authors investigate the Cauchy problem for the one-dimensional compressible flow through a porous media.
scientific article; zbMATH DE number 1957270