Regularity of solutions of evolution integrodifferential equations with deviating argument

In this paper we prove the existence of mild and classical solutions of delay integrodifferential equation and delay integrodifferential evolution equations with nonlocal condition in Banach spaces. The regularity solutions of integrodifferential evolution equations interconnected with viscoelastic material is derived to guarantee the stabilization. The results are established by using the resolvent operators and the fixed point principles. Finally, an example is given to show the potential of the proposed techniques.     

Integrodifferential equations with applications to a plate equation with memory

In this paper we study local existence, uniqueness, and continuous dependence of an abstract integrodifferential equation. We also present a result on unique continuation and a blow‐up alternative for mild solutions of the integrodifferential equation. Finally, we apply our results to an interesting strongly damped plate equation with memory.     

Study of degenerate evolution equations with memory by operator semigroup methods

We reduce the problem with some history prescribed for an integrodifferential equation in a Banach space including memory effect to the Cauchy problem for some evolution system with a constant operator in a larger space that possesses a resolvent (C₀)-semigroup. This enables us to state conditions for the existence of a unique classical solution to the original problem. We use the results to study the unique solvability of problems with history prescribed for degenerate linear evolution equations with memory in Banach spaces. We show that the initial-boundary value problem for the linearized integrodifferential Oskolkov system describing the dynamics of Kelvin–Voigt fluids in linear approximation belongs to this class of problems.     

Controllability for some integrodifferential equations driven by vector measures

In this work, we consider the question of controllability of a class of integrodifferential equations on Hilbert space with measures as controls. We assume that the linear part has a resolvent operator in the sense given by R. Grimmer. We generalize the original work of N. Ahmed on vector measures, and we use it to develop necessary and sufficient conditions for weak and the exact controllability of the integrodifferential equation. Using the latter, we prove that exact controllability of the integrodifferential equation implies exact controllability of a perturbed integrodifferential equation. Controllability problem for the perturbed system is formulated fixed point problem in the space of vector measures. Our results cover impulsive controls as well as regular controls. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and regularity of solutions for neutral partial functional integrodifferential equations

In this paper, we study a class of neutral partial functional integrodifferential equations with finite delay by using the theory of resolvent operators. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. As an application, we also consider a diffusive neutral partial functional integrodifferential equation.     

Second order evolution equations with time-dependent subdifferentials

We study an abstract second order nonlinear evolution equation in a real Hilbert space. We consider time-dependent convex functions and their subdifferentials operating on the first derivative of the unknown function. Introducing appropriate assumptions on the convex functions and other data, we prove the existence and uniqueness of a strong solution, and give some applications of the abstract theorem to hyperbolic variational inequalities with time-dependent constraints.      

Convergence and optimal control problems of nonlinear evolution equations governed by time-dependent operator

We study an abstract nonlinear evolution equation governed by a time-dependent operator of subdifferential type in a real Hilbert space. In this paper we discuss the convergence of solutions to our evolution equations. Also, we investigate the optimal control problems of nonlinear evolution equations. Moreover, we apply our abstract results to a quasilinear parabolic PDE with a mixed boundary condition.     

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.     

Problem on small motions of ideal rotating relaxing fluid

We study an evolution problem on small motions of the ideal rotating relaxing fluid in bounded domains. We begin from the problem posing. Then we reduce the problem to a second-order integrodifferential equation in a Hilbert space. Using this equation, we prove a strong unique solvability problem for the corresponding initial-boundary value problem.     

Hyperbolic integro-differential equations of convolution type

Making use of the theory of Wiener-Hopf operators in the scale of abstract Krein spaces, we prove existence and uniqueness of unbounded solutions for the linear hyperbolic integrodifferential equation (P_o). We extend herewith results obtained in [8] for hyperbolic evolution equations, where the convolution integral was absent. The method utilizes Dunford's functional calculus and permits thus a constructive existence proof for solutions exhibiting an exponential growth rate when time increases. Our approach bases upon the fundamental hypothesis that the spectrum of the time-independent mapping -A shows a parabolic condensation along the negative real axis. This condition completely determines the admissible geometry of the spectral set of the convolution integral operator, and a fortiori the magnitude of the exponential growth rate. The theory works in arbitrary reflexive Banach spaces.     

Symmetry analysis and exact solutions of some Ostrovsky equations

We apply the classical Lie method and the nonclassical method to a generalized Ostrovsky equation (GOE) and to the integrable Vakhnenko equation (VE), which Vakhnenko and Parkes proved to be equivalent to the reduced Ostrovsky equation. Using a simple nonlinear ordinary differential equation, we find that for some polynomials of velocity, the GOE has abundant exact solutions expressible in terms of Jacobi elliptic functions and consequently has many solutions in the form of periodic waves, solitary waves, compactons, etc. The nonclassical method applied to the associated potential system for the VE yields solutions that arise from neither nonclassical symmetries of the VE nor potential symmetries. Some of these equations have interesting behavior such as “nonlinear superposition.”     

Existence of solutions for some functional integrodifferential equations with nonlocal conditions

In this paper, we discuss the existence of mild solution of functional integrodifferential equation with nonlocal conditions. To establish this results by using the resolvent operator theory and Sadovskii-Krasnosel'skii type of fixed point theorem and to show the usefulness and the applicability of our results to a broad class of functional integrodifferential equations, an example is given to illustrate the theory.     

Abstract time-dependent transport equations

We consider the abstract time-dependent linear transport equation as an initial-boundary value evolution problem in the Banach spaces L_p, 1 ⩽ p < ∞, or on a space of measures on a (possibly time-dependent) kinetic phase space. Existence, uniqueness, dissipativity, and positivity results are proved for very general, possibly time-dependent, transport operators and boundary conditions. When the phase space, boundary conditions, and transport operator are independent of time, corresponding results are obtained for the associated semigroup.     

Local existence for nonlinear Volterra integrodifferential equations with infinite delay

In this paper, we investigate the local existence and uniqueness of solutions to integrodifferential equations with infinite delay, which are more general than those in previous studies. We assume that the linear part of the equation is nondensely defined and satisfies a Hille–Yosida condition. Moreover, the continuity of solutions with respect to initial conditions is also studied. In order to illustrate our abstract results, we conclude this work with an example.     

On perturbed nonlinear Itô type stochastic integrodifferential equations

In this paper we consider the solution of the stochastic nonlinear integrodifferential equation of the Itô type with small perturbations, by comparing it with the solution of the corresponding unperturbed equation of the equal type. We investigate the closeness in the (2m)th moment sense of these solutions on finite fixed intervals or on intervals whose length tends to infinity as small perturbations tend to zero.     

Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations

The authors study integrodifferential equations in Hilbert space. The coefficients of the equations are unbounded and the principal part is an abstract hyperbolic equation perturbed by terms with Volterra integral operators. Such equations can be regarded as an abstract generalization of the Gurtin–Pipkin integrodifferential equation that describes heat transfer in materials with memory and has a number of other applications. Well-defined solvability of initial boundary value problems for such equations is established in weighted Sobolev spaces on the positive semi-axis. The authors examine spectral problems for operator-valued functions representing the symbols of the said equations and study the spectrum of the abstract Gurtin–Pipkin integrodifferential equation.     

Nonlinear generalized master equations and accounting for initial correlations

We develop a new method based on using a time-dependent operator (generally not a projection operator) converting a distribution function (statistical operator) of a total system into the relevant form that allows deriving new exact nonlinear generalized master equations (GMEs). The derived inhomogeneous nonlinear GME is a generalization of the linear Nakajima-Zwanzig GME and can be viewed as an alternative to the BBGKY chain. It is suitable for obtaining both nonlinear and linear evolution equations. As in the conventional linear GME, there is an inhomogeneous term comprising all multiparticle initial correlations. To include the initial correlations into consideration, we convert the obtained inhomogeneous nonlinear GME into the homogenous form by the previously suggested method. We use no conventional approximation like the random phase approximation (RPA) or the Bogoliubov principle of weakening of initial correlations. The obtained exact homogeneous nonlinear GME describes all evolution stages of the (sub)system of interest and treats initial correlations on an equal footing with collisions via the modified memory kernel. As an application, we obtain a new homogeneous nonlinear equation retaining initial correlations for a one-particle distribution function of the spatially inhomogeneous nonideal gas of classical particles. In contrast to existing approaches, this equation holds for all time scales and takes the influence of pair collisions and initial correlations on the dissipative and nondissipative characteristics of the system into account consistently with the adopted approximation (linear in the gas density). We show that on the kinetic time scale, the time-reversible terms resulting from the initial correlations vanish (if the particle dynamics are endowed with the mixing property) and this equation can be converted into the Vlasov-Landau and Boltzmann equations without any additional commonly used approximations. The entire process of transition can thus be followed from the initial reversible stage of the evolution to the irreversible kinetic stage.     

On identification of memory kernels in linear theory of heat conduction

A linear integrodifferential equation describing the heat flow in a material with memory is considered. This equation contains a pair of time-dependent convolution kernels that are unknown. Such kernels are determined as solutions of an optimal control problem by using additional data obtained from measurements of average temperature around some fixed points of the domain over some finite time interval. We show the existence of an optimal solution of this problem and derive optimality conditions for it.     

Construction of periodic solutions of integrodifferential equations of viscoelasticity

A theorem on the estimation of the periodic solutions of a linear integrodifferential equation and a theorem of existence and uniqueness of the periodic solution of a nonlinear integrodifferential equation are presented without proof.     

Existence results for partial neutral functional integrodifferential equations with unbounded delay

We prove the existence of mild solutions for a partial neutral functional integrodifferential equation with unbounded delay using the Leray-Schauder alternative.