Examples of nonsymmetric extinction and blow-up for quasilinear heat equations. (English) Zbl 0814.35047

Summary: We present new asymptotic properties of solutions of some quasilinear heat equations with absorption or source including the equations \[ u_ t = \nabla \cdot (u^{1/2} \nabla u) - u^{1/2},\;u_ t = \nabla \cdot (u^{-1/2} \nabla u) - u^{1/2}, \quad u_ t = \nabla \cdot (u^{- 1/2} \nabla u) + u^{3/2}, \]
\[ u_ t = \nabla \cdot (u^{-4/(N+2)} \nabla u) + u^{(N+6)/(N+2)}, \quad x \in \mathbb{R}^ N,\;N \geq 1. \] 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.


35K55 Nonlinear parabolic equations
35K65 Degenerate parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
35C05 Solutions to PDEs in closed form