Integral manifolds for semilinear evolution equations and admissibility of function spaces

We prove the existence of integral (stable, unstable, and center) manifolds for the solutions to a semilinear integral equation in the case where the evolution family (U(t, s)) t≥s has an exponential trichotomy on a half line or on the whole line, and the nonlinear forcing term f satisfies the φ-Lipschitz conditions, i.e., where φ(t) belongs to some classes of admissible function spaces. Our main method is based on the Lyapunov–Perron methods, rescaling procedures, and the techniques of using the admissibility of function spaces.Доведено iснування iнтегральних (стiйких, нестiйких, центральних) многовидiв для розв’язкiв напiвлiнiйного iнтегрального рiвняння у випадку, коли сiм’я еволюцiй (U(t,s))tleqs має експоненцiальну трихотомiю на пiвосi або на всiй осi, а нелiнiйний збурюючий член f задовольняє φ-лiпшицевi умови, тобто належить до деяких класiв допустимих просторiв функцiй. Наш основний метод базується на методах Ляпунова – Перрона, процедурах перемасштабування та технiцi застосування допустимостi просторiв функцiй

Functional Partial Differential Equations and Evolution Semigroups

Thema dieser Dissertation ist die Wohlgestelltheit und Asymptotik von nichtautonomen funktionalen Partielle-Differentialgleichungen Wesentliches Hilfsmittel zur Diskussion sind Evolutionshalbgruppen, die von der Evolutionsfamilie auf einer Halbgeraden erzeugt werden, sowie die Theorie der Randstörung eines Generators. In Kapitel 1 werden die grundlegenden Konzepte über Evolutionhalbguppen und Evolutionsfamilien auf einer Halbgeraden behandelt. Hier werden alle Hilfsmittel bereitgestellt, die wir später benötigen, einschliesslich der Ergebnisse über die exponentielle Dichotomie allgemeiner Evolutiongleichungen. In Kapitel 2 betrachten wir zuerst die Gleichung (DPDG) mit nichtautonomer Vergangenheit. Wir konstruieren eine stark stetige Halbgruppe, die die Gleichung löst. Dann benutzen wir die Charakterisierung der hyperbolischen Halbgruppe, um die Robustheit der exponentiellen Dichotomie der Lösungen zu erhalten. Am Ende des Kapitels studieren wir mit dem Methoden und Ergebnissen aus Kapitel 1 die Robustheit der exponentiellen Dichotomie der Lösungen der allgemeinen nichtautonomen . Wir bekommen ähnliche Ergebnisse für Delayoperatoren, die nur auf einem endlichem Intervall wirken. In Kapitel 3 schlagen wir eine Halbgruppenbehandlung zu autonomer (NPDG) vor. Durch Anwendung die Theorie der St örungen des Operator auf dem Rand können wir eine Lösungshalbgruppe für die Gleichung unter Bedingungen an den Differenzoperator konstruieren und die Wohlgestelltheit der Gleichung zeigen. In Kapitel 4 we bekommen die Wohlgestellheit und die Robustheit der exponentiellen Stabilität der Lösungen der (NPDG) mit nichtautonomen Vergangenheit zu bekommen. In Kapitel 5, wir betrachten allgemeine nichtautonome (NPDG). Wir erhalten ähnliche Resultate für Delayoperatoren und Differenzoperatoren, die auf endlichem Intervall wirken.This thesis deals with the well-posedness and asymptotic behavior of non-autonomous functional partial differential equations. The necessary tools for the disscusion are evolution semigroups, which are generated by evolution families on a half-line, as well as the theory of perturbations of generators at the boundary. In Chapter 1, we briefly recall some basic concepts of evolution families and evolution semigroups on a half-line. We also include in this chapter the results on the asymptotic behavior of the solutions to general evolution equations on a half-line. In Chapter 2, we first deal with the DPDE with non-autonomous past. We are able to construct a semigroup solving the above equations. We then study the robustness of stability and dichotomy of the solutions to this equations. At the end of Chapter 2, we study the robustness of exponential dichotomy of the solutions to general non-autonomous partial functional differential equations. We obtain similar results for delay operators acting on a finite interval. In Chapter 3, we propose a semigroup approach to linear autonomous neutral partial functional differential equations. We can construct a strongly continuous semigroup solving the above equation and obtain the well-posedness of this equation under appropriate conditions on the difference operator. In Chapter 4, we obtain the well-posedness and the robustness of exponential stability of partial neutral functional differential equations with non-autonomous past. Finally, in Chapter 5, we study general non-autonomous partial neutral functional differential equations and obtain the similar results for equations with delay and difference operators acting on a finite interval

Stability and periodicity of solutions to the Oldroyd-B model on exterior domains

Consider the Oldroyd-B system on exterior domains with nonzero external forces f. It is shown that this system admits under smallness assumptions on f a bounded, global solution (u; τ), which is stable in the sense that any other global solution to this system starting in a sufficiently small neighborhood of (u(0); τ (0)) is tending to (u; τ). In addition, if the outer force is T-periodic and small enough, the Oldroyd-B system admits a T-periodic solution. Note that no smallness condition on the coupling coefficient is assumed

DICHOTOMY AND POSITIVITY OF NEUTRAL EQUATIONS WITH NONAUTONOMOUS PAST

Consider the linear partial neutral functional differential equations with nonautonomous past of the form where the function u(·, ·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ we prove that the solution semigroup for this system of equations is hyperbolic (or admits an exponential dichotomy) provided that the backward evolution family U = (U (t, s)) t≤s≤0 generated by A(s) is uniformly exponentially stable and the operator B generates a hyperbolic semigroup (e tB ) t≥0 on X. Furthermore, under the positivity conditions on (e tB ) t≥0 , U, F and Φ we prove that the above-mentioned solution semigroup is positive and then show a sufficient condition for the exponential stability of this solution semigroup