A singular parabolic equation: Existence, stabilization

作者：Badra Mehdi; Bal Kaushik; Giacomoni Jacques^*

来源：Journal of Differential Equations, 2012, 252(9): 5042-5075.

DOI：10.1016/j.jde.2012.01.035

• Quasilinear parabolic equation
• Singular nonlinearity
• Existence of weak solution
• Weak comparison principle
• Sub and supersolutions
• Cone condition
• Time-semi-discretization
• Semigroup theory for nonlinear operators

We investigate the following quasilinear parabolic and singular equation, %26lt;br%26gt;[GRAPHICS] %26lt;br%26gt;where Omega is an open bounded domain with smooth boundary in R-N, 1 %26lt; p %26lt; infinity, 0 %26lt; delta and T %26gt; 0. We assume that (x,s) is an element of Omega x R+ -%26gt; f(x, s) is a bounded below Caratheodory function, locally Lipschitz with respect to s uniformly in x is an element of Omega and asymptotically sub-homogeneous, i.e. %26lt;br%26gt;[GRAPHICS] %26lt;br%26gt;(where lambda(1) (Omega) is the first eigenvalue of -Delta(p) in Omega with homogeneous Dirichlet boundary conditions) and u(o) is an element of L-infinity (Omega) boolean AND W-o(1,p) (Omega), satisfying a cone condition defined below. Then, for any delta is an element of (0, 2 + 1/p-1), we prove the existence and the uniqueness of a weak solution is u is an element of V(Q(T)) to (P-t). Furthermore, is u is an element of C([0, T], W-0(1,p)(Omega)) and the restriction delta %26lt; 2 + 1/p-1 is sharp. The proof relies on a semidiscretization 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 C defined below and by the weak comparison principle that 1/u(delta) is an element of L-infinity (0, T; W--1,W-p (Omega)) and u(1-delta) is an element of L-infinity(0, T; L-1(Omega)). When t -%26gt; f(x,t)/t(p-1) is nonincreasing for a.e. x is an element of Omega, we show that u(t) -%26gt; u(infinity) in L-infinity(Omega) as t -%26gt; infinity, where u(infinity) is the unique solution to the stationary problem. This stabilization property is proved by using the accretivity of a suitable operator in L-infinity (Omega). %26lt;br%26gt;Finally, in the last section we analyze the case p = 2. Using the interpolation spaces theory and the semigroup theory, we prove the existence and the uniqueness of weak solutions to (P-t) for any delta %26gt; 0 in C([0, T], L-2(Omega))boolean AND L-infinity (Q(T)) and under suitable assumptions on the initial data we give additional regularity results. Finally, we describe their asymptotic behaviour in L-infinity(Omega) boolean AND H-0(1)(Omega) when delta %26lt; 3.

• 出版日期2012-5-1