Singularly nonautonomous semilinear evolution equations with almost sectorial operators ; Ecuaciones de evolución semilineales singularmente no autónomas con operadores casi sectoriales

In this work we consider the singularly nonautonomous semilinear parabolic problem ut + A(t)u = F(u); t > ; u( ) = u0; in a Banach space X, where A(t); t 2 R, is a family of uniformly almost sectorial operators. The term singularly nonautonomous express the fact that the linear part of the equation, A(t) : D X ! X, is time-dependent and the almost sectoriality of the family A(t) comes from a deficiency in its resolvent estimate. For this semilinear problem in the abstract setting we study local well-posedness, regularity of the solution and the asymptotic dynamics of the problem. To illustrate the ideas developed for the abstract initial value problem, we consider a singularly nonautonomous reaction-diffusion equation in a domain with a handle. This type of domain consists in a subset of RN, 0 = [ R0, where is an open set of RN and R0 is diffeomorphic to a subset (0; 1) R. The “handle” refers to this line segment R0 attached to . In 0 we consider the following reaction-diffusion equation [equation]. This equation generates a singularly nonautonomous evolution equation with almost sectorial operator and local well-posedness, existence of strong solution and existence of pullback attractor are studied, in the lights of the abstract theory developed. In particular, in order to obtain existence of attractors, the system above will be decouple, originating two evolution equations: one with Neumann homogeneous boundary condition in and another with nonhomogeneous and time-dependent Dirichlet boundary conditions in R0. The properties of those two decoupled equations are thoughtfully studied and from them, estimates on the pullback attractor are obtained. ; En este trabajo consideramos el problema parabólico semilineal singularmente no autónomo ut + A(t)u = F(u); t > ; u( ) = u0; en un espacio de Banach X, donde A(t); t 2 R, es una familia de operadores uniformemente casi sectoriales. El término singularmente no autónomo expresa el hecho de que la parte lineal de la ecuación, A(t) : DX ! X, es dependiente del ...