Convergence to strong nonlinear diffusion waves for solutions of \(p\)-system with damping (Q5944087)

The author of this paper investigates a \(p\)-system with frictional damping, NEWLINE\[NEWLINEv_t -u_x =0,\;u_t +p(v)_x =-\alpha u, \quad \alpha >0,\;p'(v)<0,NEWLINE\]NEWLINE under the initial condition NEWLINE\[NEWLINE(v(t,x),u(t,x))|_{t=0}=(v_0(x),u_0(x)),NEWLINE\]NEWLINE which satisfies \((v_0(x),u_0(x))\to (v_{\pm },u_{\pm })\) as \(x\to \pm \infty \). NEWLINENEWLINENEWLINEIt is shown that for a certain class of given large initial data \((v_0(x),u_0(x))\), the above stated Cauchy problem admits a unique global smooth solution \((v(t,x),u(t,x))\) and such a solution tends time-asymptotically, at the optimal \(L^p(2\leq p\leq\infty)\) decay rates, to the corresponding nonlinear diffusion wave \((\bar v(t,x),\bar u(t,x))\) governed by the classical Darcy's law provided that the corresponding prescribed initial error function \((V_0(x),U_0(x))\) lies in \((H^3\times H^2)(\mathbb R)\cap (L^1\times L^1)(\mathbb R)\). It turns out that \((V_0(x),U_0(x))\) can be chosen arbitrarily large, and the nonlinear diffusion wave \((\bar v(t,x),\bar u(t,x))\) is not always weak. It is shown that these waves \((\bar v(t,x),\bar u(t,x))\) are nonlinear stable provided that the strength of the nonlinear diffusion waves is weak and that the initial disturbance \((V_0(x),U_0(x))\) satisfies the assumption that \(\|V_{0xx}(x)\|_{L^{\infty }}+\|U_{0x}(x)\|_{L^{\infty }}\) is sufficiently small. NEWLINENEWLINENEWLINEThe author also shows that the smallness assumption imposed on the strength of the diffusion waves is a necessary condition to guarantee the nonlinear stabiliy result which is compared with the corresponding results obtained by \textit{L. Hsiao} and \textit{T.-P. Liu} [Commun. Math. Phys. 143, No. 3, 599-605 (1992; Zbl 0763.35058)]. The smallness conditions imposed on the initial disturbance are much weaker.