Asymptotic properties of solutions to evolution equations

1992 ◽  
pp. 53-62
Author(s):  
Rodica Luca ◽  
Gheorghe Moroşanu
Keyword(s):  
Evolution Equations ◽  
Asymptotic Properties

scholarly journals Asymptotic properties of mild solutions of nonautonomous evolution equations with applications to retarded differential equations

1999 ◽  
Vol 4 (3) ◽  
pp. 169-194 ◽  
Author(s):  
Gabriele Gühring ◽  
Frank Räbiger
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Evolution Equation ◽  
Unique Solution ◽  
Evolution Equations ◽  
Bounded Solution ◽  
Mild Solutions ◽  
Asymptotic Properties ◽  
Yosida Operator ◽  
Retarded Differential Equations

We investigate the asymptotic properties of the inhomogeneous nonautonomous evolution equation(d/dt)u(t)=Au(t)+B(t)u(t)+f(t),t∈ℝ, where(A,D(A))is a Hille-Yosida operator on a Banach spaceX,B(t),t∈ℝ, is a family of operators inℒ(D(A)¯,X)satisfying certain boundedness and measurability conditions andf∈L loc 1(ℝ,X). The solutions of the corresponding homogeneous equations are represented by an evolution family(UB(t,s))t≥s. For various function spacesℱwe show conditions on(UB(t,s))t≥sandfwhich ensure the existence of a unique solution contained inℱ. In particular, if(UB(t,s))t≥sisp-periodic there exists a unique bounded solutionusubject to certain spectral assumptions onUB(p,0),fandu. We apply the results to nonautonomous semilinear retarded differential equations. For certainp-periodic retarded differential equations we derive a characteristic equation which is used to determine the spectrum of(UB(t,s))t≥s.

scholarly journals Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Serge Nicaise
Keyword(s):  
Differential Equations ◽  
Evolution Equations ◽  
Dimensional Space ◽  
Hyperbolic Systems ◽  
Asymptotic Properties ◽  
Strong Stability ◽  
Infinite Dimensional ◽  
Telegraph Equations ◽  
Sufficient And Necessary Conditions ◽  
Stability Results

<p style='text-indent:20px;'>In this paper, we obtain some stability results of systems corresponding to the coupling between a dissipative evolution equation (set in an infinite dimensional space) and an ordinary differential equation. Many problems from physics enter in this framework, let us mention dispersive medium models, generalized telegraph equations, Volterra integro-differential equations, and cascades of ODE-hyperbolic systems. The goal is to find sufficient (and necessary) conditions on the involved operators that garantee stability properties of the system, i.e., strong stability, exponential stability or polynomial one. We also illustrate our abstract statements for different concrete examples, where new results are achieved.</p>

scholarly journals Space-time fractional stochastic equations on regular bounded open domains

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vo V. Anh ◽  
Nikolai N. Leonenko ◽  
María D. Ruiz-Medina
Keyword(s):  
Evolution Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Weak Sense ◽  
Laplacian Operator ◽  
Mean Square ◽  
Fractional Polynomial ◽  
The Mean ◽  
Negative Laplacian ◽  
Definition Of

AbstractFractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic properties of the Mittag-Leffler function, and the asymptotic order of the eigenvalues of a fractional polynomial of the Dirichlet negative Laplacian operator on such bounded open domains.

scholarly journals Functional differential equations with nonlocal initial conditions

1997 ◽  
Vol 10 (2) ◽  
pp. 145-156 ◽  
Author(s):  
Sergiu Aizicovici ◽  
Yun Gao
Keyword(s):  
Banach Spaces ◽  
Continuous Dependence ◽  
Evolution Equations ◽  
Initial Conditions ◽  
Asymptotic Properties ◽  
Functional Differential Equations ◽  
Cauchy Problems ◽  
Accretive Operators ◽  
Nonlocal Initial Conditions ◽  
Functional Differential

We study the existence, uniqueness, asymptotic properties, and continuous dependence upon data of solutions to a class of abstract nonlocal Cauchy problems. The approach we use is based on the theory of m-accretive operators and related evolution equations in Banach spaces.

scholarly journals Quantization of energy and weakly turbulent profiles of solutions to some damped second-order evolution equations

2017 ◽  
Vol 8 (1) ◽  
pp. 902-927 ◽  
Author(s):  
Marina Ghisi ◽  
Massimo Gobbino ◽  
Alain Haraux
Keyword(s):  
Evolution Equations ◽  
Asymptotic Properties ◽  
Nonlinear Damping ◽  
Order Equation ◽  
Second Order ◽  
Simplified Model ◽  
Asymptotic Behavior Of Solutions ◽  
Damping Term ◽  
Linear Elastic ◽  
Elastic Part

Abstract We consider a second-order equation with a linear “elastic” part and a nonlinear damping term depending on a power of the norm of the velocity. We investigate the asymptotic behavior of solutions, after rescaling them suitably in order to take into account the decay rate and bound their energy away from zero. We find a rather unexpected dichotomy phenomenon. Solutions with finitely many Fourier components are asymptotic to solutions of the linearized equation without damping and exhibit some sort of equipartition of the total energy among the components. Solutions with infinitely many Fourier components tend to zero weakly but not strongly. We show also that the limit of the energy of the solutions depends only on the number of their Fourier components. The proof of our results is inspired by the analysis of a simplified model, which we devise through an averaging procedure, and whose solutions exhibit the same asymptotic properties as the solutions to the original equation.

Dichotomy theorems for evolution equations

2008 ◽  
Author(s):  
Alexandru Alin Pogan
Keyword(s):  
Evolution Equations ◽  
Dichotomy Theorems
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