Abstract Evolution Equations Research Articles

The effect of inhomogeneous terms for asymptotic profiles of solutions to second-order abstract evolution equations with time-dependent dissipation

The inhomogeneous problem of second-order evolution equations with a time-dependent dissipation term of the form u″+Au+b(t)u′=g(t) in a real Hilbert space H is considered, where A is a general nonnegative selfadjoint operator in H. The result is the explicit description of asymptotic profile of solutions of such a problem when the dissipation b provides a diffusive structure with a suitable restriction. The proof is a basic energy method with a scope of well-behaved quantity for the diffusive structure.

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations

Massera type theorems for abstract non-autonomous evolution equations

Well-posedness and stability for an abstract evolution equation with history memory and time delay in Hilbert space

Well-posedness and stability for an abstract evolution equation with history memory and time delay in Hilbert space

Long time decay analysis of complex-valued fractional abstract evolution equations with delay

The asymptotic stability and long time decay rates of solutions to linear Caputo time-fractional ordinary differential equations are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of time-fractional ordinary differential equations decay only polynomially, leading to the so-called Mittag-Leffler stability, which were already extended to linear fractional delay differential equations (FDDEs) with real constants by Čermák et al. (2017) [17]. This paper is devoted to the asymptotical stability and long time decay rates of the solutions for a class of fractional delay abstract evolution equations with complex-valued coefficients. By applying the spectral decomposition method, we reformulate the fractional delay abstract evolution equations to an infinite-dimensional FDDEs. Through the root locus technique, we develop a necessary and sufficient stability condition in a coefficient criterion for FDDEs. At the same time, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long time decay rates of the solutions for both linear FDDEs and fractional delay abstract evolution equations with complex-valued coefficients are O(t−α). Finally, several application models including both time and time-space fractional Schrödinger equations are presented to illustrate the effectiveness of our proposed criteria.

Multiscale theory

Abstract Boltzmann kinetic equation is put into the form of an abstract time evolution equation representing links connecting autonomous mesoscopic dynamical theories involving varying amount of details. In the chronological order we present results that led to the abstract time equation evolution in both state space and the space of vector fields. In the final section we list some open problems.

Singular limit problem of abstract second order evolution equations

Singular limit problem of abstract second order evolution equations

Cauchy Problem for an Abstract Evolution Equation of Fractional Order

In this paper, we define an operator function as a series of operators corresponding to the Taylor series representing the function of the complex variable. In previous papers, we considered the case when a function has a decomposition in the Laurent series with the infinite principal part and finite regular part. Our central challenge is to improve this result having considered as a regular part an entire function satisfying the special condition of the growth regularity. As an application, we consider an opportunity to broaden the conditions imposed upon the second term not containing the time variable of the evolution equation in the abstract Hilbert space.

Unique continuation results for abstract quasi-linear evolution equations in Banach spaces

Unique continuation properties for a class of evolution equations defined on Banach spaces are considered from two different point of views: the first one is based on the existence of conserved quantities, which very often translates into the conservation of some norm of the solutions of the system in a suitable Banach space. The second one is regarded to well-posed problems. Our results are then applied to some equations, most of them describing physical processes like wave propagation, hydrodynamics, and integrable systems, such as the $b-$; Fornberg-Whitham; potential and $\pi-$Camassa-Holm; generalised Boussinesq equations; and the modified Euler-Poisson system.

Ulam–Hyers stability and exponentially dichotomic equations in Banach spaces

For finite-dimensional linear differential systems with bounded coefficients, we prove that their exponential dichotomy on R is equivalent to their Ulam–Hyers stability on R with uniqueness. We also consider abstract non-autonomous evolution equations which are exponentially bounded and exponentially dichotomic and prove that Ulam–Hyers stability with uniqueness is maintained when perturbing them with a nonlinear term having a sufficiently small Lipschitz constant.

An explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation

We propose and justify an explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation for bounded linear operators on Banach space.

Abstract Evolution Equations with an Operator Function in the Second Term

In this paper, having introduced a convergence of a series on the root vectors in the Abel-Lidskii sense, we present a valuable application to the evolution equations. The main issue of the paper is an approach allowing us to principally broaden conditions imposed upon the second term of the evolution equation in the abstract Hilbert space. In this way, we come to the definition of the function of an unbounded non-selfadjoint operator. Meanwhile, considering the main issue we involve an additional concept that is a generalization of the spectral theorem for a non-selfadjoint operator.

Exact null-controllability of interconnected abstract evolution equations with unbounded input operators

<p style='text-indent:20px;'>The exact null-controllability problem in the class of smooth controls with applications to interconnected systems was considered in [<xref ref-type="bibr" rid="b23">23</xref>] for the case of bounded input operators appearing in systems with distributed controls. The current paper constitutes an extension of the [<xref ref-type="bibr" rid="b23">23</xref>] for the case of unbounded input operators (with more emphasis on the controllability of interconnected systems). The proofs of the results of [<xref ref-type="bibr" rid="b23">23</xref>] for the case of bounded input operators are adopted for the case of unbounded input operators.</p>

Stability of solutions to abstract evolution equations in Banach spaces under nonclassical assumptions

The stability of the solution to the equation $(*)\dot{u} = F(t,u)+f(t)$, $t\ge 0$, $u(0)=u_0$ is studied. Here $F(t,u)$ is a nonlinear operator in a Banach space $\mathcal{X}$ for any fixed $t\ge 0$ and $F(t,0)=0$, $\forall t\ge 0$. We assume that the Fr\'echet derivative of $F(t,u)$ is H\"{o}lder continuous of order $q>0$ with respect to $u$ for any fixed $t\ge 0$, i.e., $\|F'_u(t,w) - F'_u(t,v)\|\le \alpha(t)\|v - w\|^{q}$, $q>0$. We proved that the equilibrium solution $v=0$ to the equation $\dot{v} = F(t,v)$ is Lyapunov stable under persistently acting perturbation $f(t)$ if $\sup_{t\ge 0}\int_0^t \alpha(\xi)\|U(t,\xi)\|\, d\xi<\infty$ and $\sup_{t\ge 0}\|U(t)\|<\infty$. Here, $U(t):=U(t,0)$ and $U(t,\xi)$ is the solution to the equation $\frac{d}{dt}{U}(t,\xi) = F'_u(t,0)U(t,\xi)$, $t\ge \xi$, $U(\xi,\xi)=I$, where $I$ is the identity operator in $\mathcal{X}$. Sufficient conditions for the solution $u(t)$ to equation (*) to be bounded and for $\lim_{t\to\infty}u(t) = 0$ are proposed and justified. Stability of solutions to equations with unbounded operators in Hilbert spaces is also studied.

Indirect stabilization of coupled abstract evolution equations with memory: Different propagation speeds

We consider the Cauchy problem for coupled second‐order evolution equations in a Hilbert space with indirect memory damping, which arises from the theory of viscoelasticity. Here, the coefficients of the main operators in the system are non‐equal (the case of different speeds of propagation). Under much weaker assumptions on memory kernels, we obtain a polynomial decay rate, whereas it is known that exponential stability does not hold. This work essentially generalizes and improves the related ones in the literature. As an application of our abstract results, we also establish for the Timoshenko system the so far best decay rate under such weak assumptions on the memory kernel in the case of different speeds of propagation.

Exponential decay for nonlinear abstract evolution equations with a countably infinite number of time‐dependent time delays

In this paper, we analyze a nonlinear abstract evolution equation with an infinite number of time‐dependent time delays and a Lipschitz continuous nonlinear term. By using a fixed point argument, we prove the existence of a mild solution. This allows us to take into account also nonnegative time delays. Furthermore, by using Gronwall estimates, exponential decay of the solution is also proven under some smallness assumptions on the parameters appearing in the system and on the initial data. Finally, some examples are illustrated.

Maximal L-regularity for an abstract evolution equation with applications to closed-loop boundary feedback control problems

In this paper we present an abstract maximal $L^p$-regularity result up to $T = \infty$, that is tuned to capture (linear) Partial Differential Equations of parabolic type, defined on a bounded domain and subject to finite dimensional, stabilizing, feedback controls acting on (a portion of) the boundary. Illustrations include, beside a more classical boundary parabolic example, two more recent settings: (i) the $3d$-Navier-Stokes equations with finite dimensional, localized, boundary tangential feedback stabilizing controls as well as Boussinesq systems with finite dimensional, localized, feedback, stabilizing, Dirichlet boundary control for the thermal equation.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p &lt; ∞ 2\le p\lt \infty .

Stability of solutions to some abstract evolution equations with delay

The global existence and stability of the solution to the delay differential equation (*)$\dot{u} = A(t)u + G(t,u(t-\tau)) + f(t)$, $t\ge 0$, $u(t) = v(t)$, $-\tau \le t\le 0$, are studied. Here $A(t):\mathcal{H}\to \mathcal{H}$ is a closed, densely defined, linear operator in a Hilbert space $\mathcal{H}$ and $G(t,u)$ is a nonlinear operator in $\mathcal{H}$ continuous with respect to $u$ and $t$. We assume that the spectrum of $A(t)$ lies in the half-plane $\Re \lambda \le \gamma(t)$, where $\gamma(t)$ is not necessarily negative and $\|G(t,u)\| \le \alpha(t)\|u\|^p$, $p>1$, $t\ge 0$. Sufficient conditions for the solution to the equation to exist globally, to be bounded and to converge to zero as $t$ tends to $\infty$, under the non-classical assumption that $\gamma(t)$ can take positive values, are proposed and justified.

Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping

We consider an abstract second order non-autonomous evolution equation in a Hilbert space H : u″ + Au + γ(t)u′ + f(u) = 0, where A is a self-adjoint and nonnegative operator on H, f is a conservative H-valued function with polynomial growth (not necessarily to be monotone), and γ(t)u′ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient γ(t) and the exponent associated with the nonlinear term f? There seems to be little development on the study of such problems, with regard to non-autonomous equations, even for strongly positive operator A. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of γ(t) and f. As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when f is a monotone operator.