Best asymptotic profile for linear damped \(p\)-system with boundary effect (Q982521)

The authors consider quasi-linear two-dimensional \(p\)-system with linear damping in the quadrant \(\mathbb R^+\times\mathbb R^+\): \[ v_t = u_x,\quad u_t + p(v)_x = -\alpha u. \] Such models describe compressible flows in porous media. Boundary condition is \(u(0,t)=0\) or \(u(0,t)=v_-\), initial values of \(u\) and \(v\) are given and have positive limits at infinity. The authors show by heuristic analysis that the best asymptotic profile is the parabolic solution of the corresponding porous media equation with special initial data. Convergence rates are estimated.