Asymptotic behaviour of solutions of quasi-linear parabolic equations with nonlinear flux
The authors study the large time behaviour of solutions of the problem \(u_t=\text{div} a(x,Du)\) in \((0,\infty)\times\Omega\), \(-\partial u/\partial\eta_a\in\beta(u)\) on \((0,\infty)\times\partial\Omega\), \(u(x,0)=u_0(x)\) in \(\Omega\), where \(\Omega\) is smoothly bounded domain in \(\mathbb R^N\), \(\beta\) is a maximal monotone graph in \(\mathbb R\times\mathbb R\), \(0\in\beta(0)\), \(u_0\in L^1(\Omega)\) and, typically, \(a(x,\xi)=| \xi| ^{p-2}\xi\), \(p>1\). Using the Lyapunov method for semigroups of nonlinear contractions, they show that the solution converges in \(L^1(\Omega)\) to a constant function. In Section 4 of the paper, a decay rate in \(L^p(\Omega)\) is obtained in the case of homogeneous Dirichlet or Neumann boundary conditions.
Lyapunov method for semigroups of nonlinear contractions
scientific article; zbMATH DE number 1229870