Solutions Of Semilinear Evolution Equations Research Articles

Admissible, bounded and periodic solutions of semilinear evolution equations on the line

We investigate semilinear evolution equations of the form u(t)=U(t,s)u(s)+∫ ⁡stU(t,ξ)f(ξ,u(ξ))dξ for t≥s and s∈ℝ in Banach space X. Under the assumptions that the evolution family (U(t,s))t≥s has the exponential dichotomy and the function f:ℝ×X→X has the Carathéodory property, we show that the semilinear evolution equations on the line has a unique admissible solution, bounded solution, periodic solution when the function f satisfies the condition ϕ-Lipschitz and there exists a periodic solution when the function f satisfies the condition ∥f(t,x)∥≤φ(t)(1+∥x∥) for all x∈X and almost everywhere t∈ℝ.

Existence and uniqueness of (ω,c)-periodic solutions of semilinear evolution equations

In this work we study the existence and uniqueness of (ω, c)-periodic solutions for semilinear evolution equations in complex Banach spaces.

Existence and uniqueness of (ω,c)-periodic solutions of semilinear evolution equations

No AccessExistence and uniqueness of (ω,c)-periodic solutions of semilinear evolution equationsMakrina Agaoglou, Michal Fečkan and Angeliki P. PanagiotidouMakrina AgaoglouMathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia; Faculty of Engineering, Department of Mechanical Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece, Michal FečkanMathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia; Department of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia and Angeliki P. PanagiotidouDepartment of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia; School of Science and Technology, Hellenic Open University, 13-15 Tsamadou str., GR-26222 Patras, GreecePublished Online:March 18, 2020pp 149-166https://doi.org/10.1504/IJDSDE.2020.106027PDF ToolsAdd to FavouritesDownload CitationsTrack Citations Share this article on social mediaShareShare onFacebookTwitterLinkedInReddit AboutAbstractIn this work we study the existence and uniqueness of (ω, c)-periodic solutions for semilinear evolution equations in complex Banach spaces.Keywords(ω,c)-periodic solutions, semilinear evolution equations, nonresonance conditions Previous article Next article FiguresReferencesRelatedDetails Volume 10Issue 22020 ISSN: 1752-3583eISSN: 1752-3591 HistoryPublished onlineMarch 18, 2020 Copyright © 2020 Inderscience Enterprises Ltd.Keywords(ω,c)-periodic solutionssemilinear evolution equationsnonresonance conditionsAuthors and AffiliationsMakrina Agaoglou1 Michal Fečkan2 Angeliki P. Panagiotidou3 1. Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia; Faculty of Engineering, Department of Mechanical Engineering, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece2. Mathematical Institute of Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia; Department of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia3. Department of Mathematical Analysis and Numerical Mathematics, Comenius University in Bratislava, Mlynská dolina, 842 48 Bratislava, Slovakia; School of Science and Technology, Hellenic Open University, 13-15 Tsamadou str., GR-26222 Patras, GreecePDF download

Existence of asymptotically periodic solutions for semilinear evolution equations with nonlocal initial conditions

AbstractIn this paper we study a class of semilinear evolution equations with nonlocal initial conditions and give some new results on the existence of asymptotically periodic mild solutions. As one would expect, the results presented here would generalize and improve some results in this area.

Lipschitz continuity and convexity preserving for solutions of semilinear evolution equations in the Heisenberg group

In this paper we study viscosity solutions of semilinear parabolic equations in the Heisenberg group. We show uniqueness of viscosity solutions with exponential growth at space infinity. We also study Lipschitz and horizontal convexity preserving properties under appropriate assumptions. Counterexamples show that in general such properties that are well-known for semilinear and fully nonlinear parabolic equations in the Euclidean spaces do not hold in the Heisenberg group.

Global existence and stability for functional evolution equations with state-dependent delay

In this paper we prove some results on the global existence and attractivity of mild solutions for semilinear evolution equations with state-dependent delay in Banach space.

Existence and exponential stability of Stepanov-like almost automorphic mild solutions for semilinear evolution equations

This work is concerned with the existence and exponential stability of Stepanov-like almost automorphic mild solutions for the following semilinear evolution equations
 x'(t) = Ax(t) + F(t, x(t)), t ∈ ℝ,
 where A is the infinitesimal generator of a C0-semigroup of bounded linear operator on a Banach space X and F: ℝ × X → X is a Stepanov-like almost automorphic function in t uniformly with respect to the second argument x. By applying the Banach contraction mapping principle (when F satisfies Lipschitz type conditions), and the Schauder's fixed point theorem (when F does not necessarily satisfy Lipschitz type conditions), we obtain the existence and exponential stability of Stepanov-like almost automorphic mild solutions for the semilinear evolution equations. Moreover, as application, two examples are given to illustrate our abstract results.

GLOBAL EXISTENCE AND ASYMPTOTIC BEHAVIOR FOR FUNCTIONAL EVOLUTION EQUATIONS

In this paper we present some results of global existence and attrac-tivity of mild solutions for semilinear evolution equations with innitedelay in a Banach space. The considerations of this paper are basedon the Schauder's xed point theorem and the theory of evolutionsystem.

Weighted pseudo almost automorphic functions and WPAA solutions to semilinear evolution equations

In this paper, we reveal several basic properties about nonlinear vector-valued weighted pseudo almost automorphic functions, including equivalence, completeness, translation invariance, composition theorem, and convolution theorem of these functions. We also give some concrete examples to illustrate our results. Finally, we obtain a new existence theorem of nonlinear weighted pseudo almost automorphic solutions for semilinear evolution equations in Banach spaces.

Corrigendum on “Almost automorphic solutions of semilinear evolution equations”

Corrigendum on “Almost automorphic solutions of semilinear evolution equations”

Alternative Theorems for Pseudo Almost Automorphic Problems

This paper gives alternative theorems to investigate the existence of pseudo almost automorphic solutions for semilinear evolution equations. Moreover, we consider a parabolic equation to illustrate the theorem.

Anti-periodic solutions for semilinear evolution equations in Banach spaces

In this paper, we study the anti-periodic problem for semi-linear evolution equations in reflexive Banach spaces. Several existence results are obtained under suitable conditions.

Existence of anti-periodic mild solutions for semilinear evolution equations

In this paper we study the existence of anti-periodic mild solutions for a class of semilinear evolution equations in Banach spaces and extend some related results in this direction. An example is given to illustrate our results.

A composition theorem for weighted pseudo-almost automorphic functions and applications

In this paper, we establish a composition theorem for weighted pseudo-almost automorphic functions under a weaker Lipschitz condition. Our composition theorem generalizes some known results. Moreover, the existence and uniqueness of pseudo-almost automorphic solutions for abstract semilinear evolution equations are studied.

A class of semilinear evolution equations with impulses at variable times on Banach spaces

In this paper, a general class of semilinear evolution equations with impulses at variable times on infinite-dimensional Banach spaces is considered. Based on a discussion of the relationship between a curve Γ y on [ 0 , + ∞ ) × X and the semiflow W generated by the semilinear evolution system on [ 0 , + ∞ ) × X and analysis about all case of impulses, we introduce a reasonable P C -mild solution of semilinear evolution equations with impulses at variable times and prove the local and global existence of the P C -mild solution and study the maximal existence interval of the solution.

Some properties of Stepanov-like almost automorphic functions and applications to abstract evolution equations

This article is concerned with some properties of Stepanov-like almost automorphic (S p -a.a.) functions. We establish a composition theorem about S p -a.a. functions, and with its help, study the existence and uniqueness of almost automorphic solutions for semilinear evolution equations in Banach spaces. Moreover, integration and differentiation of S p -a.a. functions are discussed. Some theorems extend earlier results.

Almost automorphic and weighted pseudo almost automorphic solutions of semilinear evolution equations

In this paper, applying the theory of semigroups of operators to evolution family and Banach fixed point theorem, we prove the existence and uniqueness of an (a) almost automorphic (weighted pseudo almost automorphic) mild solution of the semilinear evolution equation x ′ ( t ) = A ( t ) x ( t ) + f ( t , x ( t ) ) in Banach space under conditions.

Existence of mild solutions for semilinear evolution equations with non-local initial conditions

In this paper we study the existence of mild solutions for a class of semilinear evolution equations with non-local initial conditions and extend some related results in this direction.

ON APPROXIMATE SOLUTIONS OF SEMILINEAR EVOLUTION EQUATIONS II: GENERALIZATIONS, AND APPLICATIONS TO NAVIER–STOKES EQUATIONS

In our previous paper [12], a general framework was outlined to treat the approximate solutions of semilinear evolution equations; more precisely, a scheme was presented to infer from an approximate solution the existence (local or global in time) of an exact solution, and to estimate their distance. In the first half of the present work, the abstract framework of [12] is extended, so as to be applicable to evolutionary PDEs whose nonlinearities contain derivatives in the space variables. In the second half of the paper, this extended framework is applied to the incompressible Navier–Stokes equations, on a torus Tdof any dimension. In this way, a number of results are obtained in the setting of the Sobolev spaces ℍn(Td), choosing the approximate solutions in a number of different ways. With the simplest choices we recover local existence of the exact solution for arbitrary data and external forces, as well as global existence for small data and forces. With the supplementary assumption of exponential decay in time for the forces, the same decay law is derived for the exact solution with small (zero mean) data and forces. The interval of existence for arbitrary data, the upper bounds on data and forces for global existence, and all estimates on the exponential decay of the exact solution are derived in a fully quantitative way (i.e. giving the values of all the necessary constants; this makes a difference with most of the previous literature). Next, the Galerkin approximate solutions are considered and precise, still quantitative estimates are derived for their ℍndistance from the exact solution; these are global in time for small data and forces (with exponential time decay of the above distance, if the forces decay similarly).

Global solutions and smoothing effects for semi-linear evolution equations in circular domains

The paper deals with global-in-time solutions of semi-linear evolution equations in circular domains. These solutions are constructed by means of the series of eigenfunctions of the Laplace operator in the corresponding region. A forced Cahn–Hilliard-type equation in a unit disc Ω is considered as an example. The current work focuses on revealing the mechanism of nonlinear smoothing, i.e., on tracing the influence of smoothness of the source term on the regularity of solutions of the nonlinear mixed problem. To this end convolutions of Rayleigh functions with respect to the Bessel index are employed. The study of this new family of special functions was initiated in [V. Varlamov, Convolution of Rayleigh functions with respect to the Bessel index, J. Math. Anal. Appl. 306 (2005) 413–424; V. Varlamov, Convolutions of Rayleigh functions and their application to semi-linear equations in circular domains, J. Math. Anal. Appl. 327 (2007) 1461–1478]. In order to reveal the effect of nonlinear smoothing, anisotropic Sobolev spaces H s , α ( Ω ) are introduced. They are based on the Sobolev spaces H s ( Ω ) “weighted” by tangential derivatives, so that the index α is responsible for the smoothness in θ and s is the usual Sobolev index. Global-in-time solutions of the mixed problem in question are constructed, and additional smoothness with respect to the angular coordinate is established in the anisotropic Sobolev spaces. The above mentioned special functions are essentially used for this purpose.