Boundedness and blowup for nonlinear degenerate parabolic equations

The author deals with the quasilinear parabolic equation _ut=u^α+g(u)]Δu+bu^α+1+f(u,∇u)$u_t=u^{\alpha }+g\left(u\right)]\Delta u+b{u}^{\alpha +1}+f(u,\nabla u)$ with Dirichlet boundary conditions in a bounded domain Ω$\Omega$, where f$f$ and g$g$ are lower-order terms. He shows that, under suitable conditions on f$f$ and g$g$, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −Δ$-\Delta$ in Ω$\Omega$ with Dirichlet boundary condition. For some special cases, the result is sharp.