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Parabolic evolution equations and nonlinear boundary conditions. (English) Zbl 0658.34011
The author considers the initial boundary value problem \[ (1)\quad \dot u(t)={\mathcal A}(t)u=f(t,u),\quad {\mathcal B}(t)u=g(t,u),\quad 0<t\leq T,\quad u(0)=u_ 0. \] \({\mathcal A}(t)\), \({\mathcal B}(t)\) are linear operators. Nonlinearities are in the right-hand side, and are present both in the main equation and in the boundary values. The goal of the author was to present an abstract theory for this problem, which allows further study of qualitative properties of the solutions, in the spirit of the geometric theory of semilinear parabolic equations by D. Henry [Lect. Notes Math. 840 (1981; Zbl 0456.35001)]. The additional fact here is the presence of nonlinear boundary conditions. An important tool for the geometric treatment is the variation of constants formula. It is precisely the purpose of this paper to derive such a formula in this “doubly” non-linear context in a rigorous way, notably regarding its domain of validity (the functional analytic framework in which it has a meaning) and its interpretation in terms of the equation (what kinds of solutions verify this formula). The paper deals in parallel with an abstract framework and its illustration by systems of second order parabolic equations. Some fundamental aspects of parabolic equations (such as analyticity of the semigroup, the availability of a “scale” of spaces) are more or less inserted as part of the abstract frame. Nevertheless, the author succeeds in his attempt to extract a number of “parabolic”-independent properties (notably, with his abstract “boundary” spaces \(\partial W\), the derivation of a “generalized” Green formula from a “formal” one, the notion of retraction in place of the Sobolev lifting property...). A key step lies in the derivation for the “linear” initial value problem [\({\mathcal B}(t)u=g(t)]\) of weak formulation in an and hoc space. Using this representation, the author proves an existence, uniqueness and continuity result for equation (1) (Theorem 12.1). Global existence results are then obtained assuming “abstract” growth conditions on f and g (Proposition 12.6), as well as a blow up result (Theorem 12.10). The paper is divided into 4 main parts: the first two are preparatory; the third one entitled “Abstract Parabolic Initial Boundary Value Problems” covers sections 8 to 12. The main existence results are in section 12. Part IV deals with applications to second order parabolic equations, culminating in theorem 15.2. This last theorem states a global existence result and precompactness of bounded orbits under “polynomial” growth conditions of f and g (conditions which, according to an example by A. Friedman and B. McLeod quoted in this paper are optimal). The interest and importance of this work are underlined by the appreciable number of contributions which have been made on specific related problems (see references there in). This is especially true in the area of population dynamics (which is the reviewer’s speciality). The only objection in that case is that the abstract framework of this paper is not large enough to include the general problems set in this domain.
Reviewer: O.Arino

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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