Three types of self-similar blow-up for the fourth-order p-Laplacian equation with source

摘要

Self-similar blow-up behaviour for the fourth-order quasilinear p-Laplacian equaion with source, u(t) = -(vertical bar u(xx)vertical bar(u)u(xx))(xx) + vertical bar u vertical bar(p-1)u in R x R+, where n> 0. p>1. is studied. Using variational setting for p=n+1 and branching techniques for p not equal n+1, finite and countabel families of blow-up patterns of the self-similar form u(S)(x,t)=(T-1)(-1/p-1)f(y). where y = x/(T-1)(beta).beta=-p-(n+1)/2(n+2)(p-1). are described by an analytic-numerical approach. Three parameter ranges: p = n+1 (regional). p > n+1 (single point). and 1 < p < n +1 (global blow-up) are studied. This blow-up model is motivated by the second-order reaction-diffusion counterpart u(t) = (vertical bar u(x)vertical bar(n)u(x))(x) + u(p) (u >= 0) that was studied in the middle of the 1980s, while first results on blow-up of solutions were estabilished by Tsutsumi in 1972. (C) 2008 Elsevier B.V. All rights reserved.