The paper extends well-posedness results of a previously explored class of time-shift invariant evolutionary problems to the case of non-autonomous media. The Hilbert space setting developed for the time-shift invariant case can be utilized to obtain an elementary approach to non-autonomous equations. The results cover a large class of evolutionary equations, where well-known strategies like evolution families may be difficult to use or fail to work. We exemplify the approach with an application to a Kelvin-Voigt-type model for visco-elastic solids.

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We study an abstract class of autonomous differential inclusions in Hilbert spaces and show the well-posedness and causality, by establishing the operators involved as maximal monotone operators in time and space. Then the proof of the well-posedness relies on a well-known perturbation result for maximal monotone operators. Moreover, we show that certain types of nonlinear boundary value problems are covered by this class of inclusions and we derive necessary conditions on the operators on the boundary in order to apply the solution theory. We exemplify our findings by two examples.

Abstract. A well-posedness result for a time-shift invariant class of evolutionary operator equations involving material laws with fractional time-integrals of order α ∈ ]0, 1[ is considered and exemplified by an application to a Kelvin-Voigt type model.

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We give an approach to exponential stability within the framework of evolutionary equations due to Picard [Math. Methods Appl. Sci. 32(14) (2009), 1768-1803]. We derive sufficient conditions for exponential stability in terms of the material law operator which is defined via an analytic and bounded operator-valued function and give an estimate for the expected decay rate. The results are illustrated by three examples: differential-algebraic equations, partial differential equations with finite delay and parabolic integro-differential equations.S. Trostorff / Exponential stability for linear evolutionary equations orem 2.4 of this article). Moreover, the question of causality, which can be seen as a characterizing property for evolutionary processes, was addressed, which leads to additional constraints on the operator M, namely that M is defined via the Fourier-Laplace transformation of an analytic and bounded function M : B C (r, r) → L(H) (for more details see Section 2). Especially, the analyticity of M is crucial for the causality, due to the correlation of supports of L 2 -functions and the analyticity of their Laplace transforms by the Paley-Wiener theorem (see [21, Theorem 19.2]). Later on, these results were generalized to the case of A being a maximal monotone relation in [22,24].In this work we give sufficient criteria for the exponential stability of the evolutionary problem in terms of the function M . The study of stability issues for differential equations, which goes back to Lyapunov (see [14] for a survey), has become a very active field of research for many decades and there exist numerous works dealing with this topic. We just like to mention some standard approaches to exponential stability. The first strategy goes back to Lyapunov. The aim is to find a suitable function (a so-called Lyapunov function) yielding a certain differential inequality which allows to derive statements about the asymptotic behavior of solutions of the differential equation. A second approach, which applies to linear differential equations, is based on the theory of semigroups. In this framework, different criteria for exponential stability were derived in terms of the semigroup or its generator, e.g. Datko's lemma ([8] or [9, p. 300]), Gearharts theorem ([11] or [9, p. 302]) or the Spectral Mapping Theorem (see [9, p. 302, Theorem 1.10]). A third approach uses the Fourier or the Laplace transform of a solution to derive statements of their asymptotics. These methods are sometimes referred to as Frequency Domain Methods. In our framework it seems to be appropriate to employ the last approach, since, by the definition of M, methods of vector-valued complex analysis are at hand through the Fourier-Laplace transformation. Note that, due to the general structure of evolutionary equations, the results apply to a broad class of differential equations, such as differential-algebraic equations, equations with memory effects or integro-differential equations, where semigroup methods may be difficult to apply.The article is s...

We study integro-differential inclusions in Hilbert spaces with operator-valued kernels and give sufficient conditions for the well-posedness. We show that several types of integro-differential equations and inclusions are covered by the class of evolutionary inclusions, and we therefore give criteria for the well-posedness within this framework. As an example, we apply our results to the equations of visco-elasticity and to a class of nonlinear integro-differential inclusions describing phase transition phenomena in materials with memory.