Singularly nonautonomous semilinear evolution equations with almost sectorial operators

Abstract

A great variety of phenomena that happens in nature relies on the theory of differential equations to be properly modeled and explained. In this work we focus on a specific class of partial differential equations named singularly nonautonomous semilinear evolution equations with almost sectorial operator, and we provide solution and study the asymptotic dynamics for this type of equation. The abstract model considered is given by u_t +A(t) u = F(u), t > s, where A(t), t a real number, is a family of closed linear operators defined on a fixed dense subspace D and F is a nonlinearity. The term singularly nonautonomous is used to express the fact that the linear operator A(t) is time-dependent and the term almost sectorial comes from a deficiency in the resolvent estimate for those operators. The motivation to consider this type of problem comes from the fact that, when a problem is being modeled, it is natural to expect variations with time in the linear operator. A situation in which the linear operator is time-dependent could be, for instance, the diffusion in a medium that is affected by the season of the year or time of the day. Moreover, almost sectorial operators naturally emerge in applications involving domains with a handle, which we explore in this work. For this semilinear problem in the abstract setting we study local well-posedness, regularity properties of the solution, global well-posedness and existence of pullback attractor. In order to obtain those results, we introduce the concepts of semigroups and linear proceses of growth 1- associated to almost sectorial operators A(t) and we present several of their properties. They play an essential role in the construction of the local solution as well as in the study of the regularity properties of the solution. As far as the long-time dynamics of the problem, the fact that A(t) is time-dependent prevented us to use the classical approaches to obtain global estimates for parabolic problems. To overcome this setback, we developed an iterative method that allows one to obtain estimates of the solution in the spaces Lˆ{2ˆk} for any natural number k. To illustrate the results and ideas developed, the abstract theory is intercalated with an application of it to a singularly nonautonomous reaction-diffusion equation in domain with a handle. This type of domain consists in two open subsets in R^{N} connected by a line segment R_0 and the family of linear operators associated to this type of problem belongs to the class of almost sectorial operators. In physical terms, this problem could model, for instance, two isolated tanks (the open subsets in R^{N}), at certain temperature distribution, to which we attach a cable (R_0) connecting them, and it might be of interest to determine how the heat flow occurs in this new domain, formed by the two open sets and the line segment. With the abstract theory developed in this work, semilinear evolutions equations with time-dependent almost sectorial operators can be analyzed in terms of existence of local/global well-posedness, regularity properties of those solutions and existence of compact attracting sets that captures the long-time dynamics of the problem. Moreover the iterative method developed to obtain global estimates of the solutions is quiet general and can be adapted to others second order parabolic equations in the divergent form.