On the cauchy problem for certain system of semilinear parabolic equations

In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations:

Successive approximation of infinite dimensional semilinear backward stochastic evolution equations with jumps

In this paper, we study the existence and uniqueness of mild solutions to semilinear backward stochastic evolution equations driven by the cylindrical I$I$-Brownian motion and the Poisson point process in a Hilbert space with non-Lipschitzian coefficients by the successive approximation.     

On stability for a class of semilinear stochastic evolution equations

Sufficient conditions for almost surely asymptotic stability with a certain decay function of sample paths, which are given by mild solutions to a class of semilinear stochastic evolution equations, are presented. The analysis is based on introducing approximating system with strong solution and using a limiting argument to pass on some properties of strong solution to our purposes. Several examples are studied to illustrate our theory. In particular, by means of the derived results we lose conditions of certain stochastic evolution systems from Haussmann (1978) to obtain the pathwise stability for mild solution with probability one.     

Existence of mild solutions of some semilinear neutral fractional functional evolution equations with infinite delay

We deal in this paper with the mild solution for fractional semilinear differential equations with infinite delay: with T>0 and 0<α<1. We prove the existence (and uniqueness) of solutions, assuming that A generates an α-resolvent family (S_α(t))_t?0 on a complex Banach space X by means of classical fixed points methods.     

Three-cylinder theorem for a certain class of semilinear parabolic equations

Translated from Matematicheskie Zametki, Vol. 51, No. 1, pp. 32–41, January, 1992.     

On a certain class of a nonlinear evolution partial differential equations

An existence result and a priori bound for the solution of a second-order nonlinear parabolic equation are established. Also a generalized tanh-function method is used for constructing exact travelling wave solutions for the nonlinear diffusion equation of Fisher type originated from the considered partial differential equation. And new multiple soliton solutions are obtained.     

Stability of semilinear stochastic evolution equations

Stability of moments of the mild solution of a semilinear stochastic evolution equation is studied and sufficient conditions are given for the exponential stability of the pth moment in terms of Liapunov function. Sufficient conditions for sample continuity of the solution are also obtained and the exponential stability of sample paths is proved. Three examples are given to illustrate the theory.     

Anti-periodic solutions for semilinear evolution equations

In this paper, we study the existence problem of anti-periodic solutions for the following first order evolution equation:

     

On noncoercive semilinear equations

In this article some noncoercive semilinear equations are investigated. Here, a subset of a right side for which the corresponding boundary value problems are uniquely solvable is described.     

Remarks on semilinear partial functional-differential equations with infinite delay

In this paper we will be concerned with the abstract semilinear equation x(Φ)(t) = Ax(Φ)(t) + F(t, x_t(Φ)), t ? τ, x_τ(Φ) = Φ, where A represents the infinitesimal generator of a linear C₀-semigroup in a Banach space and x_～(Φ) denotes the segment of the solution x due to the initial function Φ taken from ?∞ up to t. Conditions for existence, continuous dependence, and regularity of solutions are given which are independent of the topology of the phase space, as well as conditions which guarantee the compactness of bounded orbits, even if exp(At) is not compact for t > 0.     

Almost automorphic solutions of semilinear evolution equations

We are concerned with the semilinear differential equation in a Banach space ,

where generates an exponentially stable -semigroup and is a function of the form . Under appropriate conditions on and , and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation.

     

Discretization of backward semilinear stochastic evolution equations

The study on discretization and convergence of BSDEs rapidly developed in recent years. We especially mention the work of Ph. Briand, B. Delyon and J. Mémin [Donsker-type Theorem for BSDEs, Electron. Comm. Probab. 6 (2001) 1–14 (electronic)]. They got the convergence of the sequence Yⁿ$Y^n$ and pointed out that the weak convergence of filtrations was a powerful tool in this topic. In this paper, we first study the weak convergence of filtrations in Hilbert space. Using this tool, we get the convergence about discretization of backward semilinear stochastic evolution equations (BSSEEs for short).     

On the nonlocal Cauchy problem for semilinear fractional order evolution equations

In this paper, we develop the approach and techniques of [Boucherif A., Precup R., Semilinear evolution equations with nonlocal initial conditions, Dynam. Systems Appl., 2007, 16(3), 507–516], [Zhou Y., Jiao F., Nonlocal Cauchy problem for fractional evolution equations, Nonlinar Anal. Real World Appl., 2010, 11(5), 4465–4475] to deal with nonlocal Cauchy problem for semilinear fractional order evolution equations. We present two new sufficient conditions on existence of mild solutions. The first result relies on a growth condition on the whole time interval via Schaefer fixed point theorem. The second result relies on a growth condition splitted into two parts, one for the subinterval containing the points associated with the nonlocal conditions, and the other for the rest of the interval via O’Regan fixed point theorem.