Abstract Cauchy Problem Research Articles

Blow-up of solutions of an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity

We consider an abstract Cauchy problem for a formally hyperbolic equation with double non-linearity. Under certain conditions on the operators in the equation, we prove its local (in time) solubility and give sufficient conditions for finite-time blow-up of solutions of the corresponding abstract Cauchy problem. The proof uses a modification of a method of Levine. We give examples of Cauchy problems and initial-boundary value problems for concrete non-linear equations of mathematical physics.

Cauchy problems for some classes of linear fractional differential equations

Abstract Cauchy problems for a class of linear differential equations with constant coefficients and Riemann-Liouville derivatives of real orders, are analyzed and solved in cases when some of the real orders are irrational numbers and when all real orders appearing in the derivatives are rational numbers. Our analysis is motivated by a forced linear oscillator with fractional damping. We pay special attention to the case when the leading term is an integer order derivative. A new form of solution, in terms of Wright’s function for the case of equations of rational order, is presented. An example is treated in detail.

Applications of Dynamic Programming to Generalized Solutions for Hamilton — Jacobi Equations with State Constraints

Cauchy problem is considered for a Hamilton Jacobi equation with state constrains arising in molecular biology. This problem has no classical solution, and known concepts of generalized (minimax and/or viscosity) solution are not applicable to it. We introduce a new concept of a continuous generalized solution and propose a method for constructing this solution by solving an auxiliary optimal control problem.

The abstract Cauchy problem for dissipative operators with respect to metric-like functionals

This paper is devoted to a characterization of semigroups of Lipschitz operators on a closed subset D of a Banach space X and the abstract Cauchy problem for an operator A in X satisfying the following condition: There exists a proper lower semicontinuous functional φ from X into [0,∞] such that the effective domain of φ is D(A) and such that limn→∞⁡Axn=Ax in X for any x∈D(A) and any sequence {xn} in D(A) satisfying two conditions limn→∞⁡xn=x in X and limsupn→∞φ(xn)≤φ(x). The main result asserts that a semigroup of Lipschitz operators on D can be generated by an operator A satisfying the above-mentioned condition, a dissipative condition with respect to a metric-like functional and a subtangential condition. The Kirchhoff equations with acoustic boundary conditions are solved by the method based on the abstract result with the construction of suitable Liapunov functionals and the use of a metric-like functional on a suitable set.

Time asymptotic description of an abstract Cauchy problem solution and application to transport equation

In this paper, we study the time asymptotic behavior of the solution to an abstract Cauchy problem on Banach spaces without restriction on the initial data. The abstract results are then applied to the study of the time asymptotic behavior of solutions of an one-dimensional transport equation with boundary conditions in L 1-space arising in growing cell populations and originally introduced by M. Rotenberg, J. Theoret. Biol. 103 (1983), 181–199.

Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem

We firstly prove thatβ-times integratedα-resolvent operator function ((α,β)-ROF) satisfies a functional equation which extends that ofβ-times integrated semigroup andα-resolvent operator function. Secondly, for the inhomogeneousα-Cauchy problemcDtαu(t)=Au(t)+f(t),t∈(0,T),u(0)=x0,u'(0)=x1,ifAis the generator of an(α,β)-ROF, we give the relation between the functionv(t)=Sα,β(t)x0+(g1*Sα,β)(t)x1+(gα-1*Sα,β*f)(t)and mild solution and classical solution of it. Finally, for the problemcDtαv(t)=Av(t)+gβ+1(t)x,t>0,v(k)(0)=0,k=0,1,…,N-1,whereAis a linear closed operator. We show thatAgenerates an exponentially bounded(α,β)-ROF on a Banach spaceXif and only if the problem has a unique exponentially bounded classical solutionvxandAvx∈Lloc1(ℝ+,X).Our results extend and generalize some related results in the literature.

Riemann–Liouville abstract fractional Cauchy problem with damping

This paper is concerned with Riemann–Liouville abstract fractional Cauchy Problems with damping. The notion of Riemann–Liouville fractional (α,β,c) resolvent is developed, where 0<β<α≤1. Some of its properties are obtained. By combining such properties with the properties of general Mittag-Leffler functions, existence and uniqueness results of the strong solution of Riemann–Liouville abstract fractional Cauchy Problems with damping are established. As an application, a fractional diffusion equation with damping is presented.

Well-posedness of abstract distributed-order fractional diffusion equations

In this paper, based on distributed-order fractional diffusion equationwe propose the distributed-order fractional abstract Cauchy problem (DFACP)and study the well-posedness of DFACP.Using functional calculus technique, we prove that the general distributed-order fractional operatorgenerates a bounded analytic $\alpha$-times resolvent operator family or a $C_0$-semigroup under some suitable conditions.In addition, we reveal the relation between two $\alpha$-times resolvent familiesgenerated by the sectorial operator $A$ and the special distributed-order fractional operator,$p_1A^{\beta_1}+ p_2A^{\beta_2}+\ldots +p_nA^{\beta_n}$, respectively.

Sharp Extensions for Convoluted Solutions of Abstract Cauchy Problems

In this paper we give sharp extension results for convoluted solutions of abstract Cauchy problems in Banach spaces. The main technique is the use of the algebraic structure (for the usual convolution product *) of these solutions which are defined by a version of the Duhamel formula. We define algebra homomorphisms from a new class of test-functions and apply our results to concrete operators. Finally, we introduce the notion of k-distribution semigroups to extend previous concepts of distribution semigroups.

Existence of solutions of the nonautonomous abstract Cauchy problem of second order

In this paper the existence of solutions of a nonautonomous abstract Cauchy problem of second order is considered. Assuming appropriate conditions on the operator of the equation, we establish the existence of mild solutions and, in some cases, we construct an evolution operator associated to the homogeneous equation. Using this evolution operator we obtain existence of solutions for the inhomogeneous equation. Finally, we apply our results to study the existence of solutions of the nonautonomous wave equation.

Solution of an abstract Cauchy problem with nonlinear and random perturbations in the Colombeau algebra

Solution of an abstract Cauchy problem with nonlinear and random perturbations in the Colombeau algebra

A convergence analysis of the Adomian decomposition method for an abstract Cauchy problem of a system of first-order nonlinear differential equations

In this article, we study some fundamental results concerning the convergence of the Adomian decomposition method (ADM) for an abstract Cauchy problem of a system of first-order nonlinear differential equations. Under certain conditions, we obtain upper estimates for the norm of solutions of this system. We also obtain results about the error estimates for the approximate solutions by the ADM and discuss their applications.

On general fractional abstract Cauchy problem

This paper is concerned with general fractional Cauchy problems oforder $0 < \alpha < 1$ and type $0 \leq \beta \leq 1$ ininfinite-dimensional Banach spaces. A new notion, named generalfractional resolvent of order $0 < \alpha < 1$ and type $0 \leq \beta \leq1$ is developed. Some of its properties are obtained.Moreover, some sufficient conditions are presented to guarantee thatthe mild solutions and strong solutions of homogeneous and inhomogeneous general fractional Cauchy problem exist.An illustrative example is presented.

Abstract Cauchy problem for fractional differential equations

In this paper, a generalized Darbo’s fixed-point theorem associated with Hausdorff measure of noncompactness is established. Then we apply this new variant fixed-point theorem to study some fractional differential equations in Banach spaces via the technique of measure of noncompactness. Many novel existence and uniqueness results for solutions are obtained under the more general conditions.

Exponential Stability and Uniform Boundedness of Solutions for Nonautonomous Periodic Abstract Cauchy Problems. An Evolution Semigroup Approach

Let u μ, x, s (., 0) be the solution of the following well-posed inhomogeneous Cauchy Problem on a complex Banach space X $$\left \{\begin{array}{ll}\dot{u}(t) = A(t)u(t) + e^{i\mu t}x, \quad t > s \\ u(s) = 0. \end{array} \right.$$ Here, x is a vector in X, μ is a real number, q is a positive real number and A(·) is a q-periodic linear operator valued function. Under some natural assumptions on the evolution family \({\mathcal{U} = \{U(t, s): t \geq s\}}\) generated by the family {A(t)}, we prove that if for each μ, each s ≥ 0 and every x the solution u μ, x, s (·, 0) is bounded on R + by a positive constant, depending only on x, then the family \({\mathcal{U}}\) is uniformly exponentially stable. The approach is based on the theory of evolution semigroups.

On some examples in the theory of abstract Cauchy problems

We give examples of abstract Cauchy problems proving that inclusions in a chain of functional classes are strict and disproving a sufficient condition for the solvability of the Cauchy problem published earlier.

On the abstract Cauchy problem for operators in locally convex spaces

We provide a complete solution of the abstract Cauchy problem for operator valued Laplace distributions or hyperfunctions on complete ultrabornological locally convex spaces (like spaces of smooth functions and distributions). This extends results of Komatsu for operators on Banach spaces. Concrete examples are provided. The crucial tools for our solution are a general notion of a resolvent for operators on locally convex spaces and the theory of Laplace transform for Laplace hyperfunctions valued in a complete locally convex space X developed earlier by the authors.

Time asymptotic behavior of the solution of an abstract Cauchy problem given by a one-velocity transport operator with Maxwell boundary condition

This paper is devoted to describe the time asymptotic behavior of the solution of a one-velocity transport operator with Maxwell boundary condition on L 1-spaces. A practical way to study the behavior of the solution without restriction on the initial data is given.

A novel characteristic of solution operator for the fractional abstract Cauchy problem

Motivated by an equality of the Mittag–Leffler function proved recently by the authors, this paper develops an operator theory for the fractional abstract Cauchy problem (FACP) with order α ∈ ( 0 , 1 ) . The notion of fractional semigroup is introduced. It is proved that a family of bounded linear operator is a solution operator for (FACP) if and only if it is a fractional semigroup. Moreover, the well-posedness of the problem (FACP) is also discussed. It is shown that the problem (FACP) is well-posed if and only if its coefficient operator generates a fractional semigroup.

Convergence of a semi-discrete finite difference scheme applied to the abstract Cauchy problem on a scale of Banach spaces

We show convergence of a finite difference scheme to solve the abstract Cauchy problem on a scale of Banach spaces which includes that for Kowalevskaya’s system. We show convergence of consistent difference schemes even for unstable cases.