On a certain infinite system of semilinear evolution equations

Summary We consider a certain infinite system of semilinear evolution equations in a Banach space. There are proved the existence and uniqueness of solutions of the Cauchy problem for the above system. These results involve, as a particular case, a system of integro-differential evolution equations with functional arguments.     

A Sinc–Collocation method for the linear Fredholm integro-differential equations

A Sinc–Collocation method for solving linear integro-differential equations of the Fredholm type is discussed. The integro-differential equations are reduced to a system of algebraic equations and Q-R method is used to establish numerical procedures. The convergence rate of the method is O( e ^{- k?N} )O{\left( {e^{{ - k{\sqrt N }}} } \right)} . Numerical results are included to confirm the efficiency and accuracy of the method even in the presence of singularities and a comparison with the rationalized Haar wavelet method is made.     

Existence and uniqueness of mild solutions for semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays

In this paper, we establish the existence and uniqueness of mild solutions for a class of semilinear integro-differential equations of fractional order with nonlocal initial conditions and delays in α-norm. And the main tool is the fixed point theorem due to Sadovskii. Some known results are generalized.     

Decay estimates for second order evolution equations with memory

This paper develops a unified method to derive decay estimates for general second order integro-differential evolution equations with semilinear source terms. Depending on the properties of convolution kernels at infinity, we show that the energy of a mild solution decays exponentially or polynomially as t→+∞. Our approach is based on integral inequalities and multiplier techniques.These decay results can be applied to various partial differential equations. We discuss three examples: a semilinear viscoelastic wave equation, a linear anisotropic elasticity model, and a Petrovsky type system.     

Boundary characteristic point regularity for semilinear reaction-diffusion equations: towards an ode criterion

The classical problem of regularity of boundary characteristic points for semilinear heat equations with homogeneous Dirichlet conditions is considered. The Petrovskii ( 2?{loglog} ) \left( {2\sqrt {{\log \log }} } \right) criterion (1934) of the boundary regularity for the heat equation can be adapted to classes of semilinear parabolic equations of reaction–diffusion type and takes the form of an ordinary differential equation (ODE) regularity criterion. Namely, after a special matching with a boundary layer, the regularity problem reduces to a onedimensional perturbed nonlinear dynamical system for the first Fourier-like coefficient of the solution in an inner region. A similar ODE criterion, with an analogous matching procedures, is shown formally to exist for semilinear fourth order biharmonic equations of reaction-diffusion type. Extensions to regularity problems of backward paraboloid vertices in \mathbbR^N {\mathbb{R}^N} are discussed. Bibliography: 54 titles. Illustrations: 1 figure.     

A viability theorem of stochastic semilinear evolution equations

A viability theorem of stochastic semilinear evolution equations is discussed under a dissipative condition in terms of uniqueness functions and a stochastic subtangential condition. Our strategy is to interpret a stochastic viability problem into a characterization problem of evolution operators associated with stochastic semilinear evolution equations. The main theorem is a generalization of the results due to Aubin and Da Prato in the case of stochastic differential equations in ℝ ^d .     

On stability theory for solutions of semilinear dissipative equations of the Sobolev type

For abstract semilinear dissipative equations of the Sobolev type (1)–(8) the principle of linearized stability in the theory of exponential stability in the “strong” norm is investigated. Bibliography: 12 titles. To dear Olga Alexandrovna Ladyzhenskaya on the occasion of her jubilee Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 200, 1992, pp. 139–148. Translated by V. I. Ochkur.     

Backward SDEs and Cauchy problem for semilinear equations in divergence form

 We extend the definition of solutions of backward stochastic differential equations to the case where the driving process is a diffusion corresponding to symmetric uniformly elliptic divergence form operator. We show existence and uniqueness of solutions of such equations under natural assumptions on the data and show its connections with solutions of semilinear parabolic partial differential equations in Sobolev spaces. Received: 22 January 2002 / Revised version: 10 September 2002 / Published online: 19 December 2002 Research supported by KBN Grant 0253 P03 2000 19. Mathematics Subject Classification (2002): Primary 60H30; Secondary 35K55 Key words or phrases: Backward stochastic differential equation – Semilinear partial differential equation – Divergence form operator – Weak solution     

Comparison Method of Stability Analysis of Semilinear Time-Delay Stochastic Evolution Equation

Abstract

In this paper, we set up the comparison theorem between the mild solution of semilinear time-delay stochastic evolution equation with general time-delay variable and the solution of a class (1-dimension) deterministic functional differential equation, by using the Razumikhin–Lyapunov type functional and the theory of functional differential inequalities. By applying this comparison theorem, we give various types of the stability comparison criteria for the semilinear time-delay stochastic evolution equations. With the aid of these comparison criteria, one can reduce the stability analysis of semilinear time-delay stochastic evolution equations in Hilbert space to that of a class (1-dimension) deterministic functional differential equations. Furthermore, these comparison criteria in special case have been applied to derive sufficient conditions for various stability of the mild solution of semilinear time-delay stochastic evolution equations. Finally, the theories are illustrated with some examples.     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Numerical solution of Volterra integral and integro-differential equations with rapidly vanishing convolution kernels

Variable stepsize algorithms for the numerical solution of nonlinear Volterra integral and integro-differential equations of convolution type are described. These algorithms are based on an embedded pair of Runge–Kutta methods of order p=5 and p=4 proposed by Dormand and Prince with interpolation of uniform order p=4. They require O(N) number of kernel evaluations, where N is the number of steps. The cost of the algorithms can be further reduced for equations that have rapidly vanishing convolution kernels, by using waveform relaxation iterations after computing the numerical approximation by variable stepsize algorithm on some initial interval. AMS subject classification (2000)  65R20, 45L10, 93C22     

Optimal Control of Non-well-posed Heat Equations

This work is concerned with Pontryagin＇s maximum principle of optimal control problems governed by some non-well-posed semilinear heat equations. A type of approach to the non-well-posed optimal control problem is given.     

Semilinear Mixed Problems on Hilbert Complexes and Their Numerical Approximation

Arnold, Falk, and Winther recently showed (Bull. Am. Math. Soc. 47:281–354, 2010) that linear, mixed variational problems, and their numerical approximation by mixed finite element methods, can be studied using the powerful, abstract language of Hilbert complexes. In another recent article (arXiv:), we extended the Arnold–Falk–Winther framework by analyzing variational crimes (à la Strang) on Hilbert complexes. In particular, this gave a treatment of finite element exterior calculus on manifolds, generalizing techniques from surface finite element methods and recovering earlier a priori estimates for the Laplace–Beltrami operator on 2- and 3-surfaces, due to Dziuk (Lecture Notes in Math., vol. 1357:142–155, 1988) and later Demlow (SIAM J. Numer. Anal. 47:805–827, 2009), as special cases. In the present article, we extend the Hilbert complex framework in a second distinct direction: to the study of semilinear mixed problems. We do this, first, by introducing an operator-theoretic reformulation of the linear mixed problem, so that the semilinear problem can be expressed as an abstract Hammerstein equation. This allows us to obtain, for semilinear problems, a priori solution estimates and error estimates that reduce to the Arnold–Falk–Winther results in the linear case. We also consider the impact of variational crimes, extending the results of our previous article to these semilinear problems. As an immediate application, this new framework allows for mixed finite element methods to be applied to semilinear problems on surfaces.     

Nonlocal problems for the equations of motion of Kelvin-Voight fluids

For the equations of the motion of Kelvin-Voight fluids (0.1) and for the semilinear abstract differential Eqs. (0.11)–(0.17) arising in the theory of Sobolev equations (to which Eqs. (0.1) also belong), we study the four following nonlocal problems: 1) the solvability of the initial boundary problem for Eqs. (0.1) and the Cauchy problem for Eqs. (0.11)–(0.17) on the semiaxis 0<t≤∞; 2) the existence of periodic solutions of Eqs. (0.1) and Eqs. (0.11)–(0.17) with a free term periodic in t; 3) exponential stability theory for solutions of Eqs. (0.1) and Eqs. (0.11)–(0.17) as t→∞ and related problems; 4) attractor theory for Eqs. (0.1). Bibliography: 40 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 197, pp. 120–158, 1992.     

A Sinc–Collocation method for the linear Fredholm integro-differential equations

A Sinc–Collocation method for solving linear integro-differential equations of the Fredholm type is discussed. The integro-differential equations are reduced to a system of algebraic equations and Q-R method is used to establish numerical procedures. The convergence rate of the method is . Numerical results are included to confirm the efficiency and accuracy of the method even in the presence of singularities and a comparison with the rationalized Haar wavelet method is made.     

Wave equations with memory: The minimal state approach

Recently, in M. Conti et al. (2010) [6] and M. Fabrizio et al. (2010) [12], a new theoretical scheme has been developed in order to study equations with memory, the so-called minimal state approach. The aim of this work is to provide the technical body needed to study the asymptotic behavior of semilinear integro-differential equations of hyperbolic type in the novel framework.     

On high accuracy schemes of the method of straight lines for two-dimensional parabolic equations

The method of straight lines is applied to the first mixed problem for two-dimensional semilinear and linear parabolic equations defined in a domain with rectangular base. For a semilinear equation, a scheme of accuracy O(h³) is obtained. For a linear equation with coefficients that are functions only of t, a scheme of any given accuracy is obtained under the relevant accuracy of the approximation. Bibliography:7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, VOl. 219, 1994, pp. 81–93. This paper was supported by the International Foundation of Cultural Initiative and by the Russian Academy of Natural Sciences. Translated by N. S. Zabavnikova.     

Bifurcations in reaction-diffusion equations and associated dissipative structures

The paper considers semilinear parabolic equations with conditions dependent on a parameter. A stationary solution of the boundary-value problem is constructed in the neighborhood of the bifurcation value of the parameter. The evolution of the solutions of the Cauchy problem to the bifurcation solution—a spatially nonhomogeneous dissipative structure—is examined. Translated from Chislennye Metody i Vychislitel'nyi Eksperiment, Moscow State University, pp. 15–30, 1998.     

Bounded Mild Solutions for Semilinear Integro Differential Equations in Banach Spaces

In this paper we study the structure of various classes of spaces of vector-valued functions M(\mathbbR;X){\mathcal{M}(\mathbb{R};X)} ranging between periodic functions and bounded continuous functions. Some of these functions are introduced here for the first time. We propose a general operator theoretical approach to study a class of semilinear integro-differential equations. The results obtained are new and they recover, extend or improve variety of recent works.