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1.
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. 相似文献
2.
A hyperbolic Stefan problem based on the linearized Gurtin—Pipkin heat conduction law is considered. The temperature and
free boundary are controlled by a thermostat acting on the boundary. This feedback control is based on temperature measurements
performed by real thermal sensors located within the domain containing the two-phase system and/or at its boundary. Three
different types of thermostats are analyzed: simple switch, relay switch, and a Preisach hysteresis operator. The resulting
models lead to integrodifferential hyperbolic Stefan problems with nonlinear and nonlocal boundary conditions. Existence results
are proved in all the cases. Uniqueness is also shown, except in the situation corresponding to the ideal switch.
Accepted 27 May 1997 相似文献
3.
Nasser–eddine Tatar 《Mediterranean Journal of Mathematics》2012,9(1):199-210
We consider an abstract second order semilinear integrodifferential equation involving fractional time derivatives of order
between 1 and 2. Well–posedness is established under appropriate conditions on the initial data and the nonlinearity which
is itself a fractional integral. These conditions will determine the exact underlying space where to look for solutions. 相似文献
4.
Under consideration is the first order linear inhomogeneous differential equation in an abstract Banach space with a degenerate
operator at the derivative, a relatively p-radial operator at the unknown function, and a continuous delay operator. We obtain conditions of unique solvability of the
Cauchy problem and the Showalter problem by means of degenerate semigroup theory methods. These general results are applied
to the initial boundary value problems for systems of integrodifferential equations of the type of phase field equations. 相似文献
5.
1IntroductionInpractice,manydifferentialequationsandintegrodifferentialeqllationscanbeformulatedasabstractCauchyproblemswhicharedependentonmulti-parameters[l,2,3]whereA(E)areclosedoperatorsonaBanachspaceandEisamulti-parameter.InordertostudydifferentiabilitywithrespecttotheparameterEoftheso-lutionof(l.1),severalstudieshavebeendevotedtothedifferentiabilitywithrespecttoparametersofCO--semigroupwhichisgeneratedbytheoperatorA(e)[l,5,6,7].AninterestingquestionisraisedwhenwestudVthefollowingabstr… 相似文献
6.
We study the well-posed solvability of initial value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space. These equations are an abstract form of linear partial integro-differential equations that arise in the theory of viscoelasticity and have a series of other important applications. We obtain results on the wellposed solvability of the considered integro-differential equations in weighted Sobolev spaces of vector functions defined on the positive half-line and ranging in a Hilbert space. 相似文献
7.
在这篇文章内,研究一类含参数线性一算子族A(t)和线性算子B的积-微分方程,应用积分半群方法证明了该方程存在指数有界解,和含A(t),B的二阶微分方程存在指数有界和分解。这里A(t),B不必满足Hille-Yosida条件。 相似文献
8.
Qiang Tao Hang Gao Bo Zhang Zheng'an Yao 《Mathematical Methods in the Applied Sciences》2014,37(15):2236-2244
In this paper, we obtain the approximate controllability of a parabolic integrodifferential equation with interior controls. The proof relies on the linear operator semigroup theory and the controllability theory for abstract equations. Copyright © 2013 John Wiley & Sons, Ltd. 相似文献
9.
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. 相似文献
10.
In this article, we study positivity properties of exponential Runge–Kutta methods for abstract evolution equations. Our problem
class includes linear ordinary differential equations with a time-dependent inhomogeneity. We show that the order of a positive
exponential Runge–Kutta method cannot exceed two. On the other hand there exist second-order methods that preserve positivity
for linear problems. We give some examples for the latter. 相似文献
11.
Results providing sufficient conditions for the approximate controllability of a class of second-order abstract functional evolution equations governed by the generator of a strongly continuous cosine family of linear operators, together with nonlocal initial conditions, are developed. This work extends results included in recent work by Naito, Park, Zhou, and others. The abstract theory is then applied to a parabolic partial integrodifferential equation. 相似文献
12.
The deformation of an elastic plane with an elliptic hole in a uniform stress field is considered, taking into account the surface elasticity and the residual surface tension. The solution of the problem, based on the use of the linearized Gurtin–Murdoch surface elasticity relations and the complex Goursat–Kolosov potentials, is reduced to a singular integrodifferential equation. Using the example of a circular hole, for which an exact solution of the equation is obtained in closed form, the effect of the residual surface tension and the surface elasticity on the stress state close to and on the boundary of a nanohole is analysed for uniaxial tension. It is shown that the effect of the residual surface stress and the surface tension, due to deformation of the body, depends on the elastic properties of the surface, the value of the stretching load and the dimensions of the hole. 相似文献
13.
We consider an abstract Cauchy problem for a system of nonhomogeneous abstract differential equations in Hilbert spaces. The
“main” equation is of the second order and “boundary” equations are of the first order. Existence of a solution is proved.
Application to mixed (initial boundary-value) problems for one-dimensional second order hyperbolic equations and for fourth
order PDEs with the time derivative in boundary conditions has been shown.
The first author was partially supported by 60% funds of the University of Bologna and G.N.A.M.P.A. of INdAM; the second author
was supported by the Israel Ministry of Absorption. 相似文献
14.
Doklady Mathematics - For abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space, the correct solvability of initial value problems is studied and the... 相似文献
15.
M. N. Yakovlev 《Journal of Mathematical Sciences》1983,22(2):1270-1274
One proves theorems regarding the estimates of the solutions of systems of linear algebraic equations and inequalities. On the basis of these theorems one suggests a method for estimating the norm of the inverse matrix of a system of difference equations which approximates a boundary-value problem for an integrodifferential equation. The method allows us to eliminate the restrictions which are usually imposed on the coefficients of the integrodifferential equation in order to ensure the diagonal dominance in the system of the difference equations. One considers an application to nonlinear problems. 相似文献
16.
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. 相似文献
17.
Maurizio Grasselli Vittorino Pata 《Mathematical Methods in the Applied Sciences》2004,27(16):1917-1934
We consider a conserved phase‐field system of Caginalp type, characterized by the assumption that both the internal energy and the heat flux depend on the past history of the temperature and its gradient, respectively. The latter dependence is a law of Gurtin–Pipkin type, so that the equation ruling the temperature evolution is hyperbolic. Thus, the system consists of a hyperbolic integrodifferential equation coupled with a fourth‐order evolution equation for the phase‐field. This model, endowed with suitable boundary conditions, has already been analysed within the theory of dissipative dynamical systems, and the existence of an absorbing set has been obtained. Here we prove the existence of the universal attractor. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
18.
S. A. Kuzhel' 《Ukrainian Mathematical Journal》1999,51(4):492-507
We present the definition of ρ-perturbations of an abstract wave equation. As a special case, this definition involves perturbations
with compact support for the classical wave equation. We construct the scattering matrix for equations of such a type.
Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51,
No. 4, pp. 445–457, April, 1999. 相似文献
19.
《Journal of Mathematical Analysis and Applications》1999,234(2):417-456
We prove the well posedness of an inverse problem related to an abstract integrodifferential problem in two time variables with applications to ultraparabolic partial integrodifferential equations of Volterra type. 相似文献
20.
Of concern is a class of abstract semilinear integrodifferential equations with nonlocal initial conditions. Under some suitable hypotheses, we establish some new theorems about the existence of asymptotically almost automorphic solutions to the integrodifferential equations. Moreover, an example is given to illustrate our results. 相似文献