Large time behaviour of solutions of scalar viscous and nonviscous conservation laws (Q1808531)

The authors study the large time behavior of entropy solutions to the Cauchy problem for a possibly degenerate nonlinear diffusion equation with a nonlinear convection term \[ (1)\quad u_t+f(u)_x=\varphi(u)_{xx}, \qquad (2)\quad u(x,0)=u_0(x), \] in a half-plane \({\mathbb R}\times{\mathbb R}^+\). The function \(\varphi(u)\) is nondecreasing and continuous in \({\mathbb R}\), \(f(u)\) is locally Lipschitz continuous in \({\mathbb R}\). The initial function \(u_0(x)\) is assumed to have bounded total variation. The authors establish that the entropy solution of the problem (1), (2) converges, as \(t\to +\infty\), to the entropy solution \(u(x/t)\) of the corresponding Riemann problem for a first-order conservation law \(u_t+f(u)_x=0\).