Analytic Semigroup Research Articles

Stability radius and dichotomy radius of infinite-dimensional linear systems under unbounded perturbations via Yosida distance

In this paper, by using the concept of Yosida distance between two closed linear operators, we study the stability radius r(A) of linear systems u′(t)=Au(t), where A is the generator of an analytic semigroup, under unbounded perturbations in this class of generators. We show that r(A)=1/sups∈R‖R(is,A)‖L(X), so extending a classic result by Henrichsen and Pritchard to the infinite-dimensional case. A formula of the dichotomy radius is also established. Finally we give an estimate of the stability radius of general C0-semigroups. Two examples from a parabolic equation are given.

Existence of a mild solution for a fractional impulsive differential equation of the Sobolev type including deviating argument

The impulsive fractional differential equation of the Sobolev type, including deviating arguments, is the subject of the study. The analytic semigroup and fixed point approaches serve the purpose of determining the existence of the approximations. The fractional power of a closed linear operator concept is used to show how the approximation converges. To arrive at a unique approach, an approximation strategy is used. Our main conclusions are defined using an example.

Existence of Mild Solutions to Delay Diffusion Equations with Hilfer Fractional Derivative

Because of the prevalent time-delay characteristics in real-world phenomena, this paper investigates the existence of mild solutions for diffusion equations with time delays and the Hilfer fractional derivative. This derivative extends the traditional Caputo and Riemann–Liouville fractional derivatives, offering broader practical applications. Initially, we constructed Banach spaces required to handle the time-delay terms. To address the challenge of the unbounded nature of the solution operator at the initial moment, we developed an equivalent continuous operator. Subsequently, within the contexts of both compact and non-compact analytic semigroups, we explored the existence and uniqueness of mild solutions, considering various growth conditions of nonlinear terms. Finally, we presented an example to illustrate our main conclusions.

Non-autonomous fractional nonlocal evolution equations with superlinear growth nonlinearities

We carry out an analysis of the existence of solutions for a class of nonlinear fractional partial differential equations of parabolic type with nonlocal initial conditions. Sufficient conditions for the solvability of the desired problem are presented by transforming it into an abstract non-autonomous fractional evolution equation, and constructing two families of solution operators based on the Mittag-Leffler function, the Mainardi Wright-type function and the analytic semigroup generated by the closed densely defined operator −A(⋅). The discussions are based on the fractional power theory as well as the Banach fixed point theorem in the interpolation space Xpυ (0⩽υ<1, 1<p<∞).

Qualitative properties of fractional convolution elliptic and parabolic operators in Besov spaces

The maximal Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of a fractional convolution elliptic equation is studied. Particularly, it is proven that the operator generated by this nonlocal elliptic equation is sectorial in Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ B_{p,q}^{s}$$\\end{document} and also is a generator of an analytic semigroup. Moreover, well-posedeness of nonlocal fractional parabolic equation in Besov spaces is obtained. Then by using the Bp,qs\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$B_{p,q}^{s}$$\\end{document}-regularity properties of linear problem, the existence, uniqueness of maximal regular solution of corresponding fractional nonlinear equation is established.

Boundary values of analytic semigroups generated by fractional Laplacians

Boundary values of analytic semigroups generated by fractional Laplacians

Analysis on nonlinear differential equation with a deviating argument via Faedo–Galerkin method

This article focuses on the impulsive fractional differential equation (FDE) of Sobolev type with a nonlocal condition. Existence and uniqueness of the approximations are determined via analytic semigroup and fixed point method. Convergence’s approximation is demonstrated by the idea of fractional power of a closed linear operator. Using an approximation procedure, a novel approach is reached. An illustration is used to clarify our key findings.

Generation of analytic semigroups by elliptic operators with unbounded diffusion, drift and potential terms

Generation of analytic semigroups by elliptic operators with unbounded diffusion, drift and potential terms

Uniform bounds of families of analytic semigroups and Lyapunov Linear Stability of planar fronts

AbstractWe study families of analytic semigroups, acting on a Banach space, and depending on a parameter, and give sufficient conditions for existence of uniform with respect to the parameter norm bounds using spectral properties of the respective semigroup generators. In particular, we use estimates of the resolvent operators of the generators along vertical segments to estimate the growth/decay rate of the norm for the family of analytic semigroups. These results are applied to prove the Lyapunov linear stability of planar traveling waves of systems of reaction–diffusion equations, and the bidomain equation, important in electrophysiology.

The infinite-dimensional Pontryagin maximum principle for optimal control problems of fractional evolution equations with endpoint state constraints

&lt;abstract&gt;&lt;p&gt;In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $ \mathsf{X} $-valued left Caputo fractional evolution equation with the analytic semigroup, where $ \mathsf{X} $ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $ \mathsf{X} $. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $ \mathsf{X}^* $, the dual space of $ \mathsf{X} $. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.&lt;/p&gt;&lt;/abstract&gt;

Inverse problems for general parabolic systems and application to Ornstein-Uhlenbeck equation

We investigate the link between inverse problems and final state observability for a general class of parabolic systems. We generalize a stability result for initial data known for the case of self-adjoint dissipative operators. More precisely, we consider a system governed by an analytic semigroup. Under the assumption of final state observability, we prove a logarithmic stability estimate depending on the analyticity angle of the semigroup. This is done by using a general logarithmic convexity result. The abstract result is illustrated by considering the Ornstein–Uhlenbeck equation.

Well-posedness and properties of the flow for semilinear evolution equations

We derive conditions for well-posedness of semilinear evolution equations with unbounded input operators. Based on this, we provide sufficient conditions for such properties of the flow map as Lipschitz continuity, bounded-implies-continuation property, boundedness of reachability sets, etc. These properties represent a basic toolbox for stability and robustness analysis of semilinear boundary control systems. We cover systems governed by general C0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_0$$\\end{document}-semigroups, and analytic semigroups that may have both boundary and distributed disturbances. We illustrate our findings on an example of a Burgers’ equation with nonlinear local dynamics and both distributed and boundary disturbances.

Uniform complex time heat Kernel estimates without Gaussian bounds

Abstract The aim of this article is twofold. First, we study the uniform complex time heat kernel estimates of e − z ( − Δ ) α 2 {e}^{-z{\left(-\Delta )}^{\frac{\alpha }{2}}} for α &gt; 0 , z ∈ C + \alpha \gt 0,z\in {{\mathbb{C}}}^{+} . To this end, we establish the asymptotic estimates for P ( z , x ) P\left(z,x) with z z satisfying 0 &lt; ω ≤ ∣ θ ∣ &lt; π 2 0\lt \omega \le | \theta | \lt \frac{\pi }{2} followed by the uniform complex time heat kernel estimates. Second, we studied the uniform complex time estimates of the analytic semigroup generated by H = ( − Δ ) α 2 + V H={\left(-\Delta )}^{\tfrac{\alpha }{2}}+V , where V V belongs to higher-order Kato class.

Solving stochastic equations with unbounded nonlinear perturbations

This paper is interested in semilinear stochastic equations having unbounded nonlinear perturbations in the deterministic part and/or in the random part. Moreover, the linear part of these equations is governed by a not necessarily analytic semigroup. The main difficulty with these equations is how to define the concept of mild solutions due to the chosen type of unbounded perturbations. To overcome this problem, we first proved a regularity property of the stochastic convolution with respect to the domain of ‘admissible’ unbounded linear operators (not necessarily closed or closable). This is done using Yosida extensions of such unbounded linear operators. After proving the well-posedness of these equations, we also establish the Feller property for the corresponding transition semigroups. Several examples like heat equations and Schrödinger equations with nonlocal perturbations terms are given. Finally, we give an application to a general class of semilinear neutral stochastic equations.

Solvability and trajectory controllability of impulsive stochastic MHD equations with Rosenblatt process

This manuscript is concerned with the existence and trajectory controllability of impulsive stochastic magnetohydrodynamics equation governed by the Rosenblatt process. In this work, the study is made without imposing the compactness conditions in the generator S(τ) of an analytic semigroup. The noise terms rely on both velocity and magnetic fields. Initially, we reformulated the considered system in the Hilbert space by using the Stokes operator and the existence of a mild solution is established by using Banach fixed point theorem. The trajectory controllability of the proposed model is studied in the presence of impulses using Gronwall’s inequality. An example is illustrated for the developed theoretical concepts with graphical representations.

Off-diagonal bounds for the Dirichlet–to–Neumann operator on Lipschitz domains

Let Ω be a bounded domain of Rn+1 with n≥1. We assume that the boundary Γ of Ω is Lipschitz. Consider the Dirichlet–to–Neumann operator N0 associated with a system in divergence form of size m with real symmetric and Hölder continuous coefficients. We prove Lp(Γ)→Lq(Γ) off-diagonal bounds of the form‖1Fe−tN01Ef‖q≲(t∧1)nq−np(1+d(E,F)t)−1‖1Ef‖p for all measurable subsets E and F of Γ. If Γ is C1+κ for some κ>0 and m=1, we obtain a sharp estimate in the sense that (1+d(E,F)t)−1 can be replaced by (1+d(E,F)t)−(1+np−nq). Such bounds are also valid for complex time. For n=1, we apply our off-diagonal bounds to prove that the Dirichlet–to–Neumann operator associated with a system generates an analytic semigroup on Lp(Γ) for all p∈(1,∞). In addition, the corresponding evolution problem has Lq(Lp)-maximal regularity.

Wellposedness and controllability results of stochastic integrodifferential equations with noninstantaneous impulses and Rosenblatt process

The purpose of this work is to investigate a novel class of noninstantaneous impulsive stochastic integrodifferential equations (SIDEs) driven by Brownian motion and Rosenblatt process. We construct a new set of adequate assumptions for the existence and uniqueness of mild solutions using stochastic analysis, analytic semigroup theory, integral equation theory, and a fixed point methodology. Additionally, we study the asymptotic behavior of mild solutions and provide stochastic system controllability results. Finally, we include an example to illustrate the application of our main findings.

Quasilinear Fractional Order Equations and Fractional Powers of Sectorial Operators

The fractional powers of generators for analytic operator semigroups are used for the proof of the existence and uniqueness of a solution of the Cauchy problem to a first order semilinear equation in a Banach space. Here, we use an analogous construction of fractional powers Aγ for an operator A such that −A generates analytic resolving families of operators for a fractional order equation. Under the condition of local Lipschitz continuity with respect to the graph norm of Aγ for some γ∈(0,1) of a nonlinear operator, we prove the local unique solvability of the Cauchy problem to a fractional order quasilinear equation in a Banach space with several Gerasimov–Caputo fractional derivatives in the nonlinear part. An analogous nonlocal Lipschitz condition is used to obtain a theorem of the nonlocal unique solvability of the Cauchy problem. Abstract results are applied to study an initial-boundary value problem for a time-fractional order nonlinear diffusion equation.

Weighted estimates for square functions associated with operators

AbstractLet L be a non‐negative self‐adjoint operator on . Suppose that the kernels of the analytic semigroup satisfy the upper bound related to a critical function ρ but without any assumptions of smooth conditions on spacial variables. In this paper, we consider the weighted inequalities for square functions associated with L, which include the vertical square functions, the conical square functions and the Littlewood–Paley g‐functions. A new bump condition related to the critical function is given for the two‐weighted boundedness of square functions associated with L. Besides, we also prove the weighted inequalities for square functions associated with L on weighted variable Lebesgue spaces with new classes of weights considered in [5]. As applications, our results can be applied to magnetic Schrödinger operator, Laguerre operators.

On fractional semidiscrete Dirac operators of Lévy–Leblond type

AbstractIn this paper, we introduce a wide class of space‐fractional and time‐fractional semidiscrete Dirac operators of Lévy–Leblond type on the semidiscrete space‐time lattice (), resembling to fractional semidiscrete counterparts of the so‐called parabolic Dirac operators. The methods adopted here are fairly operational, relying mostly on the algebraic manipulations involving Clifford algebras, discrete Fourier analysis techniques as well as standard properties of the analytic fractional semidiscrete semigroup , carrying the parameter constraints and . The results obtained involve the study of Cauchy problems on .