It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction-diffusion systems which are — on the one h and — typical for the whole class of systems to which the general theory applies and — on the other h and — still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.

Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems

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Dynamic theory of quasilinear parabolic equations. II. Reaction-diffusion systems

Part of the Pitman Research Notes in Mathematics series, this text covers: linear evolution equations of parabolic type; semilinear evolution equations of parabolic type; evolution equations and positivity; semilinear periodic evolution equations; and applications.

Abstract Evolution Equations, Periodic Problems and Applications

We study abstract non linear parabolic equations by linearization and give examples to partial differential equations.

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Equations d'évolution abstraites non linéaires de type parabolique

A variety of problems in differential equations ((abstract) functional differential equations, age-dependent population models (with and without delay), evolution equations with boundary conditions e.g.) can be written as semilinear Cauchy problems with a Lipschitz perturbation of a closed linear operator which is not non-densely defined but satisfies the estimates of the Hille&Yosida theorem. A natural generalized notion of solution is provided by the integral solutions in the sense of Da Prato&Sinestrari. Ideas from ‘integrated semigroup’ theory yield a variation of constants formula which allows to transform the integral solutions of the evolution equation to solutions of an abstract semilinear Volterra integral equation. The latter can be used to find integral solutions to the Cauchy problem; moreover one finds sufficient and necessary conditions for the (forward) invariance of closed convex sets under the solution flow. The solution flow can be shown to form a dynamical system. Conditions for the regularity of the flow in time and initial state are derived. The steady states of the flow are characterized and sufficient conditions for local stability and instability are found. Finally the problems mentioned at the beginning are fitted into the general framework.

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Semiflows generated by lipschitz perturbations of non-densely defined operators

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Differential operators with non dense domain

Cα-regularity for non-autonomous linear integrodifferential equations of parabolic type

In this paper, a general initial-boundary value problem for a higher order linear parabolic equation with time dependent coefficients and nonhomogeneous data is studied via the abstract results obtained from a generalization of the classical theory of analytic semigroups of linear operators.

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A semigroup approach to the time dependent parabolic initial-boundary value problem

In this paper we consider a Cauchy problem in a Banach spaceE:u′(t)=A(t)u(t)+f(t), t∈[t 0, T], u(t0)=u0, whereA(·) is a family of linear operators inE which satisfy all the requirements of Kato's semigroup approach to the non autonomous hyperbolic equations except for the density of the common domains ofA(t). An application is given to a hyperbolic partial differential equation with discontinuous coefficients.

Non autonomous evolution operators of hyperbolic type

On etudie des equations integrodifferentielles non lineaires dans un espace de Banach qui peuvent etre considerees comme la version abstraite d'equations integrodifferentielles partielles non lineaires. On considere un probleme aux valeurs limites et initiales

Eugenio Sinestrari

Papers

Differential operators with non dense domain

Cα-regularity for non-autonomous linear integrodifferential equations of parabolic type

A semigroup approach to the time dependent parabolic initial-boundary value problem

Non autonomous evolution operators of hyperbolic type

Fully nonlinear integrodifferential equations in general Banach space