A singular parabolic equation: Existence, stabilization

We investigate the following quasilinear parabolic and singular equation,(Pt){ut−Δpu=1uδ+f(x,u)in (0,T)×Ω,u=0on (0,T)×∂Ω,u>0in (0,T)×Ω,u(0,x)=u0(x)in Ω, where Ω   is an open bounded domain with smooth boundary in RNRN, 10T>0. We assume that (x,s)∈Ω×R+→f(x,s)(x,s)∈Ω×R+→f(x,s) is a bounded below Caratheodory function, locally Lipschitz with respect to s   uniformly in x∈Ωx∈Ω and asymptotically sub-homogeneous, i.e.equation(0.1)0⩽limt→+∞f(x,t)tp−1=αf<λ1(Ω) (where λ1(Ω)λ1(Ω) is the first eigenvalue of −Δp−Δp in Ω   with homogeneous Dirichlet boundary conditions) and u0∈L∞(Ω)∩W01,p(Ω), satisfying a cone condition defined below. Then, for any δ∈(0,2+1p−1), we prove the existence and the uniqueness of a weak solution u∈V(QT)u∈V(QT) to (Pt)(Pt). Furthermore, u∈C([0,T],W01,p(Ω)) and the restriction δ<2+1p−1 is sharp. The proof relies on a semi-discretization in time with implicit Euler method and on the study of the stationary problem. The key points in the proof is to show that u   belongs to the cone CC defined below and by the weak comparison principle that 1uδ∈L∞(0,T;W−1,p′(Ω)) and u1−δ∈L∞(0,T;L1(Ω))u1−δ∈L∞(0,T;L1(Ω)). When t→f(x,t)tp−1 is nonincreasing for a.e. x∈Ωx∈Ω, we show that u(t)→u∞u(t)→u∞ in L∞(Ω)L∞(Ω) as t→∞t→∞, where u∞u∞ is the unique solution to the stationary problem. This stabilization property is proved by using the accretivity of a suitable operator in L∞(Ω)L∞(Ω).Finally, in the last section we analyze the case p=2p=2. Using the interpolation spaces theory and the semigroup theory, we prove the existence and the uniqueness of weak solutions to (Pt)(Pt) for any δ>0δ>0 in C([0,T],L2(Ω))∩L∞(QT)C([0,T],L2(Ω))∩L∞(QT) and under suitable assumptions on the initial data we give additional regularity results. Finally, we describe their asymptotic behaviour in L∞(Ω)∩H01(Ω) when δ<3δ<3.