Nonautonomous Evolution Equations Research Articles

Existence of mild solutions for fractional nonautonomous evolution equations of Sobolev type with delay

In this paper, we deal with a class of nonlinear fractional nonautonomous evolution equations with delay by using Hilfer fractional derivative, which generalizes the famous Riemann-Liouville fractional derivative. The definition of mild solutions for the studied problem was given based on an operator family generated by the operator pair (A,B) and probability density function. Combining the techniques of fractional calculus, measure of noncompactness, and fixed point theorem with respect to k-set-contractive, we obtain a new existence result of mild solutions. The results obtained improve and extend some related conclusions on this topic. At last, we present an application that illustrates the abstract results.

Suppression of decoherence of a spin–boson system by time-periodic control

We consider a finite-dimensional quantum system coupled to the bosonic radiation field and subject to a time-periodic control operator. Assuming the validity of a certain dynamic decoupling condition, we approximate the system’s time evolution with respect to the non-interacting dynamics. For sufficiently small coupling constants g and control periods T, we show that a certain deviation of coupled and uncoupled propagator may be estimated by $$\mathcal {O}(gt \, T)$$ . Our approach relies on the concept of Kato stability and general theory on non-autonomous linear evolution equations.

Asymptotic behavior of nonautonomous monotone and subgradient evolution equations

In a Hilbert setting $ H$, we study the asymptotic behavior of the trajectories of nonautonomous evolution equations $ \dot x(t)+A_t(x(t))\ni 0$, where for each $ t\geq 0$, $ A_t:H\rightrightarrows H$ denotes a maximal monotone operator. We provide general conditions guaranteeing the weak ergodic convergence of each trajectory $ x(\cdot )$ to a zero of a limit maximal monotone operator $ A_\infty $ as the time variable $ t$ tends to $ +\infty $. The crucial point is to use the Brezis-Haraux function, or equivalently the Fitzpatrick function, to express at which rate the excess of $ \mathrm {gph} A_\infty $ over $ \mathrm {gph} A_t$ tends to zero. This approach gives a sharp and unifying view of this subject. In the case of operators $ A_t= \partial \phi _t$ which are subdifferentials of proper closed convex functions $ \phi _t$, we show convergence results for the trajectories. Then, we specialize our results to multiscale evolution equations and obtain asymptotic properties of hierarchical minimization and selection of viscosity solutions. Illustrations are given in the field of coupled systems and partial differential equations.

On convergence rate estimates for approximations of solution operators for linear non-autonomous evolution equations

We improve some recent convergence rate estimates for approximations of solution operators of linear non-autonomous evolution equations. The approximation results from the Trotter product formula which is proved to converge in the operator-norm and its convergence can be estimated. The result is applied to a diffusion perturbed by a time-dependent potential.

Study on fractional non-autonomous evolution equations with delay

In this paper, we deal with a class of nonlinear time fractional non-autonomous evolution equations with delay by introducing the operators ψ(t,s), φ(t,η) and U(t), which are generated by the operator −A(t) and probability density function. The definition of mild solutions for studied problem was given based on these operators. Combining the techniques of fractional calculus, operator semigroups, measure of noncompactness and fixed point theorem with respect to k-set-contractive, we obtain new existence result of mild solutions with the assumptions that the nonlinear term satisfies some growth condition and noncompactness measure condition and the closed linear operator −A(t) generates an analytic semigroup for every t>0. The results obtained in this paper improve and extend some related conclusions on this topic. At last, by utilizing the abstract result obtained in this paper, the existence of mild solutions for a class of nonlinear time fractional reaction–diffusion equation introduced in Ouyang (2011) is obtained.

Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions

Existence of mild solutions for a class of non-autonomous evolution equations with nonlocal initial conditions

Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity

This paper is devoted to studying a non-autonomous stochastic linear evolution equation in Banach spaces of martingale type 2. We construct unique strict solutions to the equation and show their maximal regularity. The abstract results are then applied to stochastic diffusion equations.

Feedback Stabilization to Nonstationary Solutions of a Class of Reaction Diffusion Equations of FitzHugh--Nagumo Type

Stabilization to a trajectory for the monodomain equations, a coupled nonlinear PDE-ODE system, is investigated. The results rely on stabilization of linear first-order in time nonautonomous evolution equations combined with stabilizability results for the linearized monodomain equations and a fixed point argument to treat local stabilizability of the nonlinear system. Numerical experiments for feedback stabilization of reentry phenomena are included.

A class of dissipative nonautonomous nonlocal second-order evolution equations

In this paper we consider the following nonlinear and spatially nonlocal second-order evolution equation from nonlocal theory of continuum mechanics where is a bounded smooth domain in , , , and a is a bounded continuous function. Here, the kernel J is a nonnegative, symmetric bounded function with bounded derivative, satisfying certain growth conditions. We deduce an energy functional associated to these problem, and we study the local and global well posedness, boundedness and asymptotic behavior of its solutions. Additionally we study the stability of the trivial solution associated to these problem.

Lie Symmetries of (1+2) Nonautonomous Evolution Equations in Financial Mathematics

We analyse two classes of ( 1 + 2 ) evolution equations which are of special interest in Financial Mathematics, namely the Two-dimensional Black-Scholes Equation and the equation for the Two-factor Commodities Problem. Our approach is that of Lie Symmetry Analysis. We study these equations for the case in which they are autonomous and for the case in which the parameters of the equations are unspecified functions of time. For the autonomous Black-Scholes Equation we find that the symmetry is maximal and so the equation is reducible to the ( 1 + 2 ) Classical Heat Equation. This is not the case for the nonautonomous equation for which the number of symmetries is submaximal. In the case of the two-factor equation the number of symmetries is submaximal in both autonomous and nonautonomous cases. When the solution symmetries are used to reduce each equation to a ( 1 + 1 ) equation, the resulting equation is of maximal symmetry and so equivalent to the ( 1 + 1 ) Classical Heat Equation.

A generalisation of a theorem of O. Perron for semilinear evolution equations

AbstractWe generalise a well-known result of O. Perron from the 30s that connects the asymptotic behavior of a linear homogeneous differential equation with the response of the inhomogeneous associated equation to a certain class of inhomogeneities (for this reason, Perron's result is also referred to as “input-output method”, “test function method” or “admissibility”).Our extension is twofold, on the one hand, through the means of a (non)linear evolution family, we deal with the mild solution of a nonautonomous semilinear evolution equation and on the other hand, we collect a very general class of inhomogeneities, eligible for a Perron-type approach in this case.From a technical point of view, the Perron input-output scenario is achieved here by using the Green operator.

The Existence and Stability of Synchronizing Solution of Non-Autonomous Equations with Multiple Delays

In this paper, we consider an abstract non-autonomous evolution equation with multiple delays in a Hilbert space H: u'(t) + Au(t) = F(u(t-r1),...,u((t-rn)) + g(t), where A: D(A)?H→H is a positive definite selfadjoint operator, F: Hna → H is a nonlinear mapping, r1,...,rn are nonnegative constants, and g(t)∈ C(□;H) is bounded. Motivated by [1] [2], we obtain the existence and stability of synchronizing solution under some convergence condition. By this result, we provide a general approach for guaranteeing the existence and stability of periodic, quasiperiodic or almost periodic solution of the equation.

Topological decoupling and linearization of nonautonomous evolution equations

Topological linearization results typically require solution flows rather than merely semiflows. An exception occurs when the linearization fulfills spectral assumptions met e.g. for scalar reaction-diffusion equations. We employ tools from the geometric theory of nonautonomous dynamical systems in order to extend earlier work by Lu [12] to time-variant evolution equations under corresponding conditions on the Sacker-Sell spectrum of the linear part. Our abstract results are applied to nonautonomous reaction-diffusion and convection equations.

Periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations in Banach spaces

In this paper, we are concerned with the periodicity of solutions to the Cauchy problem for nonautonomous impulsive delay evolution equations with periodic inhomogenous terms in Banach spaces, where the operators in the linear part (possibly unbounded) depend on the time [Formula: see text] and generate an evolution family of linear operators. We first establish two new Gronwall–Bellman-type inequalities, and then prove a new and general existence theorem for periodic mild solutions to the nonautonomous impulsive delay evolution equations, which extends essentially some existing results even for the autonomous case as well as for the case when impulsive perturbations or delays are absent.

Weighted pseudo asymptotically periodic mild solutions of evolution equations

In this paper, we propose a new class of functions called weighted pseudo S-asymptotically periodic function in the Stepanov sense and explore its properties in Banach space including composition theorems. Furthermore, the existence, uniqueness of the weighted pseudo S-asymptotically periodic mild solutions to partial evolution equations and nonautonomous semilinear evolution equations are investigated. Some interesting examples are presented to illustrate the main findings.

Pseudo almost periodic and automorphic mild solutions to nonautonomous neutral partial evolution equations

AbstractIn this work, we present a new concept of Stepanov weighted pseudo almost periodic and automorphic functions which is more generale than the classical one, and we obtain a new existence result of μ-pseudo almost periodic and μ-pseudo almost automorphic mild solutions for some nonautonomous evolution equations with Stepanov μ-pseudo almost periodic terms. An example is shown to illustrate our results.

Maximal regularity for non-autonomous evolution equations governed by forms having less regularity

We consider the maximal regularity problem for non-autonomous evolution equations $$\begin{array}{l}{u'(t) + A(t)\,u(t) = f(t), \quad t \in (0, \tau]}\\ {u(0) = u_0.}\end{array}$$ (0.1) Each operator A(t) is associated with a sesquilinear form a(t) on a Hilbert space H. We assume that these forms all have the same domain V. It is proved in Haak and Ouhabaz (Math Ann, doi:10.1007/s00208-015-1199-7, 2015) that if the forms have some regularity with respect to t (e.g., piecewise α-Hölder continuous for some α > ½) then the above problem has maximal L p -regularity for all u 0 in the real-interpolation space \((H, \fancyscript{D}(A(0)))_{1-{1}/{p},p}\). In this paper we prove that the regularity required there can be improved for a class of sesquilinear forms. The forms considered here are such that the difference a(t;.,.) − a(s;.,.) is continuous on a larger space than the common domain V. We give three examples which illustrate our results.

Non Autonomous Semilinear Stochastic Evolution Equations

In this article, we first give a version with continuous paths for stochastic convolution ∫t0U(t, s)φ(s)dW(s) driven by a Wiener process W in a Hilbert space under weaker conditions. Based on the Picard approximation and the factorization method, we prove the existence, uniqueness and regularity of mild solutions for non-autonomous semilinear stochastic evolution equations with more general assumptions on the coefficients. As an application, we obtain the Feller property of the associated semigroup.

Maximal regularity for non-autonomous evolution equations

We consider the maximal regularity problem for non-autonomous evolution equations 1 Each operator $$A(t)$$ is associated with a sesquilinear form $${\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}$$ on a Hilbert space $$H$$ . We assume that these forms all have the same domain and satisfy some regularity assumption with respect to $$t$$ (e.g., piecewise $$\alpha $$ -Hölder continuous for some ). We prove maximal $$L_p$$ -regularity for all $$u_0 $$ in the real-interpolation space . The particular case where $$p = 2$$ improves previously known results and gives a positive answer to a question of Lions (Équations différentielles opérationnelles et problèmes aux limites, Springer, Berlin, 1961) on the set of allowed initial data $$u_0$$ .

Existence of anti-periodic mild solutions to semilinear nonautonomous evolution equations

In this paper, applying the theory of evolution family and Schauder's fixed point theorem, we prove the anti-periodic mild solutions for semilinear nonautonomous evolution equations in Banach space under conditions. Furthermore, an example is given to illustrate our results.