Boundedness and blowup for nonlinear degenerate parabolic equations

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