Abstract Cauchy Problem Research Articles

Well-posedness of nonautonomous abstract Cauchy problems under dissipativity conditions expressed by metric-like functionals

Under a dissipativity condition expressed by a metric-like functional, the well-posedness of a nonautonomous abstract Cauchy problem is established. The main result not only extends Martin's result on the generation of an evolution operator in a class of uniformly convex Banach spaces but also can be applied to mixed problems for nonautonomous evolution equations of Kirchhoff type with which Kato's quasilinear theory or the theory of quasi-contractive semigroups cannot directly deal. The main result also gives an affirmative answer to Kōmura's conjecture that an evolution operator of linear operators with moving domains can be generated even under a weak stability condition rather than Kato's stability condition, and the result provides an effective means for solving degenerate wave equations.

Perturbations of non-autonomous second-order abstract Cauchy problems

In this paper we present time-dependent perturbations of second-order non-autonomous abstract Cauchy problems associated to a family of operators with constant domain. We make use of the equivalence to a first-order non-autonomous abstract Cauchy problem in a product space, which we elaborate in full detail. As an application we provide a perturbed non-autonomous wave equation.

Local K-Convoluted C-groups and Abstract Cauchy Problem

We first present a new form of a local K-convoluted C-group on a Banach space X, and then deduce some basic properties of a nondegenerate local K-convoluted C-group on X and some generation theorems of local K-convoluted C-groups, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local K-convoluted C-group on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem ACP(A,f,x).

The accuracy estimates of the Cayley transform method for the abstract Cauchy problem

We obtain the accuracy estimates of the Cayley transform method for solving the initial value problem for a homogeneous first-order differential equation with an unbounded operator coefficient in a Hilbert space. In the case of finite (in some sense) smoothness of the initial vector, our method has a power-law rate of convergence and, moreover, the rate automatically depends on this regularity (i.e. the Cayley transform method is a method without saturation of accuracy). If the initial vector is infinitely smooth, then our method is exponentially convergent. In addition, we substantiate that the estimates are unimprovable in the order of N (the discretization parameter N characterizes the number of summands in the partial sum of the approximate solution).

Novel interpolation spaces and maximal‐weighted Hölder regularity results for the fractional abstract Cauchy problem

AbstractIn this paper, the maximal‐weighted Hölder regularity for the fractional abstract Cauchy problem is presented. First, two classes of novel interpolation spaces constructed by resolvent families are portrayed. Afterward, by applying the resolvent method and the properties of interpolation spaces, the maximal‐weighted Hölder regularity of the mild solution is established in weighted Hölder space (, ). Eventually, an instance for the fractional differential equation of 2m‐order multidimensional parabolic type is provided to verify the reasonability of main conclusions.

On Exponential Splitting Methods for Semilinear Abstract Cauchy problems

Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than frac{pi }{2} (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of Kato (Math Z 187(4):471–480, 1984) to the local solvability of the Navier–Stokes equation, where the textrm{L}^p - textrm{L}^r-smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies.

Exponentially Convergent Numerical Method for Abstract Cauchy Problem with Fractional Derivative of Caputo Type

We present an exponentially convergent numerical method to approximate the solution of the Cauchy problem for the inhomogeneous fractional differential equation with an unbounded operator coefficient and Caputo fractional derivative in time. The numerical method is based on the newly obtained solution formula that consolidates the mild solution representations of sub-parabolic, parabolic and sub-hyperbolic equations with sectorial operator coefficient A and non-zero initial data. The involved integral operators are approximated using the sinc-quadrature formulas that are tailored to the spectral parameters of A, fractional order α and the smoothness of the first initial condition, as well as to the properties of the equation’s right-hand side f(t). The resulting method possesses exponential convergence for positive sectorial A, any finite t, including t=0 and the whole range α∈(0,2). It is suitable for a practically important case, when no knowledge of f(t) is available outside the considered interval t∈[0,T]. The algorithm of the method is capable of multi-level parallelism. We provide numerical examples that confirm the theoretical error estimates.

Blow-up of Solutions and Local Solvability of an Abstract Cauchy Problem of Second Order with a Noncoercive Source

Blow-up of Solutions and Local Solvability of an Abstract Cauchy Problem of Second Order with a Noncoercive Source

Blow-up of Solutions and Local Solvability of an Abstract Cauchy Problem of Second Order with a Noncoercive Source

An abstract Cauchy problem for a second-order differential equation with nonlinear operator coefficients is considered. The local solvability of the problem in suitable spaces of abstract continuous and differentiable functions is proved. Sufficient conditions for finite-time blow-up of its solutions are obtained.

On abstract Cauchy problems in the frame of a generalized Caputo type derivative

In this paper, we consider a class of abstract Cauchy problems in the framework of a generalized Caputo type fractional. We discuss the existence and uniqueness of mild solutions to such a class of fractional differential equations by using properties found in the related fractional calculus, the theory of uniformly continuous semigroups of operators and the fixed point theorem. Moreover, we discuss the continuous dependence on parameters and Ulam stability of the mild solutions. At the end of this paper, we bring forth some examples to endorse the obtained results

A characterization of well‐posedness for the second order abstract Cauchy problems with finite delay

AbstractIn this work, we introduce a strongly continuous one‐parameter family of bounded linear operators that completely describes the well‐posedness of a second order abstract differential delay equation in the initial history space , . This family, which satisfies a specific functional equation is applied to characterize the mild solution of the considered second order delay equation.

Analysis of Cauchy problem with fractal-fractional differential operators

Abstract Cauchy problems with fractal-fractional differential operators with a power law, exponential decay, and the generalized Mittag-Leffler kernels are considered in this work. We start with deriving some important inequalities, and then by using the linear growth and Lipchitz conditions, we derive the conditions under which these equations admit unique solutions. A numerical scheme was suggested for each case to derive a numerical solution to the equation. Some examples of fractal-fractional differential equations were presented, and their exact solutions were obtained and compared with the used numerical scheme. A nonlinear case was considered and solved, and numerical solutions were presented graphically for different values of fractional orders and fractal dimensions.

Two-parameter conformable fractional semigroups and abstract Cauchy problems

The goal of this work is to introduce the two-parameter conformable fractional semigroups and provide a definition of its infinitesimal generator. For such generators, we develop multiple results. In addition, we show that the two-parameter conformable fractional semigroups provide a solution for two-parameter conformable fractional abstract Cauchy problems.

A second-order Magnus-type integrator for evolution equations with delay

Abstract We rewrite abstract delay equations as nonautonomous abstract Cauchy problems allowing us to introduce a Magnus-type integrator for the former. We prove the second-order convergence of the obtained Magnus-type integrator. We also show that if the differential operators involved admit a common invariant set for their generated semigroups, then the Magnus-type integrator will respect this invariant set as well, allowing for much weaker assumptions to obtain the desired convergence. As an illustrative example we consider a space-dependent epidemic model with latent period and diffusion.

The Abstract Cauchy Problem with Caputo–Fabrizio Fractional Derivative

Given an injective closed linear operator A defined in a Banach space X, and writing CFDtα the Caputo–Fabrizio fractional derivative of order α∈(0,1), we show that the unique solution of the abstract Cauchy problem (∗)CFDtαu(t)=Au(t)+f(t),t≥0, where f is continuously differentiable, is given by the unique solution of the first order abstract Cauchy problem u′(t)=Bαu(t)+Fα(t),t≥0;u(0)=−A−1f(0), where the family of bounded linear operators Bα constitutes a Yosida approximation of A and Fα(t)→f(t) as α→1. Moreover, if 11−α∈ρ(A) and the spectrum of A is contained outside the closed disk of center and radius equal to 12(1−α) then the solution of (∗) converges to zero as t→∞, in the norm of X, provided f and f′ have exponential decay. Finally, assuming a Lipchitz-type condition on f=f(t,x) (and its time-derivative) that depends on α, we prove the existence and uniqueness of mild solutions for the respective semilinear problem, for all initial conditions in the set S:={x∈D(A):x=A−1f(0,x)}.

Tensor product and inverse fractional abstract Cauchy problem

Tensor product and inverse fractional abstract Cauchy problem

The abstract Cauchy problem in locally convex spaces

We derive necessary and sufficient criteria for the uniqueness and existence of solutions of the abstract Cauchy problem in locally convex Hausdorff spaces. Our approach is based on a suitable notion of an asymptotic Laplace transform and extends results of Langenbruch beyond the class of Fréchet spaces.

C-regularized solutions of ill-posed problems defined by strong strip-type operators

Abstract Cauchy problems of the form d ⁢ u d ⁢ t = q ⁢ ( A ) ⁢ u + h ⁢ ( t ) {\frac{du}{dt}=q(A)u+h(t)} , 0 < t < T {0<t<T} , u ⁢ ( 0 ) = φ {u(0)=\varphi} , are studied in a Banach space X where A is a strong strip-type operator and q ⁢ ( A ) {q(A)} is a complex polynomial in A. In this case, the spectrum of A lies within a horizontal strip of height θ, and so potentially neither A nor - A {-A} generates a strongly continuous semigroup on X. Therefore, depending on the definition of q ⁢ ( A ) {q(A)} , the original problem may be severely ill-posed. We utilize a functional calculus for strip-type operators in order to define an approximate operator f β ⁢ ( A ) {f_{\beta}(A)} such that f β ⁢ ( A ) {f_{\beta}(A)} is bounded for each β > 0 {\beta>0} and f β ⁢ ( A ) ⁢ χ → q ⁢ ( A ) ⁢ χ {f_{\beta}(A)\chi\rightarrow q(A)\chi} as β → 0 {\beta\rightarrow 0} for χ in a suitable domain. We show that this approximation gives rise to regularization for the original problem with respect to a graph norm related to C-regularized semigroups. We also fit the theory of the paper into a special case where iA generates a bounded, strongly continuous group on X. Under this assumption, which implies that A is a strong strip-type operator of height 0, results follow for a wide variety of ill-posed PDEs in L p {L^{p}} spaces.

Atomic Solution of Fractional Abstract Cauchy Problem of High Order in Banach Spaces

The Abstract Cauchy problem, which is a vector valued differential equation is an important equation in many branches of science. Authors usually study and discuss first and second order abstract Cauchy problem. In this paper, we try to find atomic solutions of the fractional abstract Cauchy problem with order 3α, where α ∈ (0, 1). It turned out that there are so many cases to consider in order to determine atomic solution.

Stability of mild solutions of the fractional nonlinear abstract Cauchy problem

<abstract><p>Since the first work on Ulam-Hyers stabilities of differential equation solutions to date, many important and relevant papers have been published, both in the sense of integer order and fractional order differential equations. However, when we enter the field of fractional calculus, in particular, involving fractional differential equations, the path that is still long to be traveled, although there is a range of published works. In this sense, in this paper, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of mild solutions for fractional nonlinear abstract Cauchy problem in the intervals $ [0, T] $ and $ [0, \infty) $ using Banach fixed point theorem.</p></abstract>