Parabolic equations with nonlinear singularities

We show the existence of positive solutions u∈L2(0,T;H01(Ω)) for nonlinear parabolic problems with singular lower order terms of the asymptote-type. More precisely, we shall consider both semilinear problems whose model is {ut−Δu+u1−u=f(x,t)inΩ×(0,T),u(x,0)=u0(x)inΩ,u(x,t)=0on∂Ω×(0,T), and quasilinear problems having natural growth with respect to the gradient, whose model is {ut−Δu+∣∇u∣2uγ=f(x,t)inΩ×(0,T),u(x,0)=u0(x)inΩ,u(x,t)=0on∂Ω×(0,T), with γ>0γ>0. Moreover, we prove a comparison principle and, as an application, we study the asymptotic behavior of the solution as tt goes to infinity.