Asymptotic behavior of solution to nonlinear damped \(p\)-system with boundary effect (Q2905670)

This paper deals with the initial-boundary value problem (IBVP) for \(2\times 2\) damped \(p\)-system with nonlinear source \(-\alpha u-\beta|u|^{q-1} u\), \(q> 2\), where \(\beta> 0\) or \(\beta< 0\) but \(|\beta|<{\alpha\over|u_+|^{q-1}}\). The initial data \((v_0,u_0)\to (v_+,u_+)\), \(u_+\neq 0\) as \(x\to\infty\). It is proved that there exists a global solution \((v,u)(x,t)\) of the IBVP under consideration which converges to the solution \((\overline v,\overline u)\) of the corresponding porous media equations. The latter have specially selected initial data \(\overline v_0(x)\). Supposing that the initial perturbation belongs to \(L_1\cap H^3\) the authors find the optimal convergence rates of \(\|\partial^k_x(v-\overline v,u-\overline u)\|_{L^2}\), \(k= 0,1\) for \(t\to\infty\). Under the assumption that the initial perturbation is in the weighted Lebesgue space \(L^{1,\gamma}\cap H^3\) with weight \((1+x)^\gamma\), \(\gamma> 0\), new and much better decay rates for \(\|\partial^k_x(v-\overline v)\|_{L^2}\) are found. Contrary, when \(\beta<0\) and \(|\beta|> {\alpha\over|u_+|^{q-1}}\) the solution of the \(p\)-system blows up for finite time.NEWLINENEWLINE At the end of the paper numerical simulations confirming the theoretical results are carried out.