Large time behavior of solutions to a degenerate parabolic equation not in divergence form

Abstract

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: u t = u p Δ u + a u q − b u r , subject to the null Dirichlet boundary condition. We at first discuss the existence and nonexistence of global solutions to the problem, and then study the large time behavior for the global solutions. When the positive source dominates the model, we prove that the global solutions uniformly tend to the positive steady state of the problem as t → ∞ . In particular, we establish the uniform asymptotic profiles for the decay solutions when the problem is governed by the nonlinear diffusion or absorption.