Semigroup Approach to Nonlinear Diffusion Equations

The following sections are included:

• Preface
• Contents

The modern theory of partial differential equations is based on the functional analysis and theory of Sobolev spaces. The aim of this chapter is to provide for later use some basic results of functional analysis pertaining to geometric properties of infinite-dimensional normed spaces, convex functions, spaces of distributions, and the variational theory of linear elliptic boundary value problems. Most of these results, which can be easily found in standard textbooks or monographs, are given without proof or with a sketch of proof and suitable references.

In this chapter, we present the basic results of the theory of maximal monotone and m-accretive operators in Banach spaces. Moreover, the fundamentals of the existence theory for the Cauchy problem associated with nonlinear m-accretive operators in Banach spaces is briefly treated. The presentation is confined to the essential results necessary for the treatment of nonlinear partial differential equations, and so most of them are given here without proof and refer the reader to the book [15] for the complete treatment.

We shall apply here the general theory presented in Chapter 2 pertaining to the existence theory for nonlinear infinite dimensional equations with maximal monotone and m-accretive operators to some classes of nonlinear boundary value problems on open subsets Ω ⊂ ℝ^N. In particular, the theory covers semilinear elliptic problems, nonlinear diffusion equations, nonlinear stationary Fokker–Planck equations and the conservation law equation. Most of these problems are not treated in their most general framework because, as pointed out earlier, the emphasis is not put on the generality but on the method. As a matter of fact, more general existence results can be obtained by combining the above maximal monotonicity and m-accretiveness theory with specific compactness arguments based on the Schauder theorem or the degree theory in Banach spaces, but we shall not develop this approach here.

In this chapter we present several applications of general theory to nonlinear dynamics governed by partial differential equations of dissipative type illustrating the ideas and general existence theory developed in the previous section. Most of the significant well posed dynamics described by partial differential equations can be written in the abstract form (2.96) with an appropriate quasi-m-accretive operator A in a Banach space X. The boundary value conditions are incorporated in the domain of A. Usually, X is a space of functions on an open subset Ω of the Euclidean space ℝ^N and A is a nonlinear differential operator in space variables. In this approach, the time variable t has a privileged role. The whole existence strategy for a given dynamics is to find the appropriate operator A and to prove that it is quasi-m-accretive. The basic result to be frequently used here is the existence theorem for the Cauchy problem and its consequences which provides existence and uniqueness of a mild (or strong) solution as limit of a convergent finite difference scheme. The main emphasis here is put on parabolic-like boundary value problems although the area of problems covered by the general theory is much larger and includes also nonlinear hyperbolic problems. In most cases, the operator A is single-valued but there are, however, notable situations where A is multivalued and this happens, for instance, in the case of problems with free boundary or with discontinuous coefficients.

The following sections are included:

• Bibliography
• Index