Large time behavior of solutions to the Neumann problem for a quasilinear second order degenerate parabolic equation in domains with noncompact boundary (Q5945079)

The authors study the optimal rate of stabilization at large time of the solution to the Neumann problem NEWLINE\[NEWLINEu_t = \sum_{i = 1}^N\frac{\partial}{\partial x_i}(a_i(t,x,\nabla u)) -b(x,t,u)\quad \text{in } \Omega\times (0,T),\;T > 0,NEWLINE\]NEWLINE NEWLINE\[NEWLINE\sum_{i = 1}^Na_i(t,x,\nabla u)n_i = 0 \quad \text{on } \partial\Omega \times (0,T),NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,0) = u_0(x)\quad \text{in }\Omega,\;x \in \Omega,\;u_0(x) \geq 0,NEWLINE\]NEWLINE where \(\Omega\subset\mathbb{R}^N\), \(N\geq 2\), is an unbounded domain with sufficiently smooth noncompact boundary \(\partial\Omega\) satisfying a certain so-called isoperemetrical inequality. The main aim of the present article is to find the optimal bound of \(\|u(\cdot,t)\|_{L^\infty(\Omega)}\) for \(t\) large, where \(u\) is a nonnegative solution of the above problem with initial datum belonging to \(L^1(\Omega)\) or slowly decaying at infinity. Moreover, when \(b = 0\), \(m > 1\) and \(u_0\) is compactly supported, it is established a sharp bound of the interface.