Global existence and blowup solutions for quasilinear parabolic equations

Abstract

The authors discuss the quasilinear parabolic equation u t = ∇ ⋅ ( g ( u ) ∇ u ) + h ( u , ∇ u ) + f ( u ) with u | ∂ Ω = 0 , u ( x , 0 ) = ϕ ( x ) . If f, g and h are polynomials with proper degrees and proper coefficients, they show that the blowup property only depends on the first eigenvalue of −Δ in Ω with Dirichlet boundary condition. For a special case, they obtain a sharp result.