Boundedness and blowup for nonlinear degenerate parabolic equations

Abstract

The author deals with the quasilinear parabolic equation u t = [ u α + g ( u ) ] Δ u + b u α + 1 + f ( u , ∇ u ) with Dirichlet boundary conditions in a bounded domain Ω , where f and g are lower-order terms. He shows that, under suitable conditions on f and g , whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of − Δ in Ω with Dirichlet boundary condition. For some special cases, the result is sharp.