Examples of nonsymmetric extinction and blow-up for quasilinear heat equations

Abstract

We present new asymptotic properties of solutions of some quasilinear heat equations with absorption or source including the equations $$ u_t = \nabla \cdot (u^{1/2} \nabla u) - u^{1/2}, \quad u_t = \nabla \cdot (u^{-1/2} \nabla u) - u^{1/2}, $$ $$ u_t = \nabla \cdot (u^{-1/2} \nabla u) + u^{3/2}, $$ $$ u_t = \nabla \cdot (u^{-4/(N+2)} \nabla u) + u^{(N+6)/(N+2)}, \quad x \in \Bbb R^N , \,\, N \ge 1. $$ We show that these equations admit explicit solutions which are nonsymmetric and nonmonotone in spatial variables near the extinction of blow-up time. The corresponding equations are shown to be reduced to finite dimensional dynamical systems on linear subspaces which are invariant under certain nonlinear reaction-diffusion operators.