Asymptotic behavior of solutions to a degenerate parabolic equation with a gradient source term

In this paper, we investigate positive solutions of the degenerate parabolic equation not in divergence form: u_t=u^pΔu+u^q|∇u|²−u^r, subject to null Dirichlet boundary condition. We study the existence of global solutions and the large time behavior for them. The main effort is paid to obtain uniform asymptotic profiles for decay solutions, under various dominations of the nonlinear diffusion or absorption. It is shown that the large time property of the solution u behaves just like (1+(r−1)t)^(−1/r−1) if the decay is governed by the nonlinear absorption with 1<r<p+1. Otherwise, the asymptotic profiles would possess the form of (1+pt)^−1/pW, with W solving various homogeneous Dirichlet elliptic equations: (i) −ΔW=W^1−p−W if r=p+1<q+2; (ii) −ΔW=W^1−p−W+W⁻¹|∇W|² if r>p+1=q+2; and (iii) −ΔW=W^1−p if r,q+2>p+1.