Asymptotic Analysis For A Nonlinear Diffusion Equation With A Weighted Localized Source

This paper deals with a zero-Dirichlet problem to a degenerate parabolic equation ut= ?um+ up+ λ(x)uq(0, t) with the local source and weighted localized source, where parameters p, q ≥ 0, max{p, q} > m > 1, weight function λ(x) ≥ 0 and λ(x) is radially decreasing. We study interactions among the local sources, localized sources and the weight function λ(x) and their influences to the blow-up behavior of solutions. First, we prove the blow-up rate estimate at the maximum point of the solution to the nonlinear diffusion function; Secondly, we study the influence of localized source and the weight function λ(x) to the uniform blow-up profile of the solution. Namely, we obtain the uniform blow-up profile of the solution when the localized source dominates(that is when p < q) the system, or when there is an exact balance between the local source and localized source(namely p = q); At last, we carefully discuss the property of the blow-up sets of the solution. In addition, we find that the total and single point blow-up of the solution are absolutely determined by the local source and localized source.