Finite element methods for semilinear elliptic stochastic partial differential equations

We study finite element methods for semilinear stochastic partial differential equations. Error estimates are established. Numerical examples are also presented to examine our theoretical results. This research is supported by Air Force Office of Scientific Research under the grant number FA9550-05-1-0133 and 985 Project of Jilin University.     

Trichotomies for abstract semilinear differential equations

In this paper we present some results concerning the existence and uniqueness of mild solutions to certain abstract semilinear differential equations and the asymptotic behavior of these solutions. The basic techniques used are the iterative method and the fixed point theory for differential equations in Banach space. However, the most pleasant here is that it can be applied to nonlinear equations without assuming the eigenvalues of the differential operator in the linear parts of the differential equation has non-zero real part.     

Regularities for semilinear stochastic partial differential equations

In this paper, we study the regularities of solutions to semilinear stochastic partial differential equations in general settings, and prove that the solution can be smooth arbitrarily when the data is sufficiently regular. As applications, we also study several classes of semilinear stochastic partial differential equations on abstract Wiener space, complete Riemannian manifold as well as bounded domain in Euclidean space.     

Local solvability of semilinear partial differential equations

Summary It is proved an abstract theorem for local solvability of semilinear equations with complex coefficients. This result is illustrated by nonlinear perturbations of complex principal type and multiple complex characteristics operators.

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Riassunto Si dimostra un teorema astratto per la risolvibilità locale di equazioni semilineari di coefficienti complessi. Il risultato ottenuto viene illustrato mediante perturbationi nonlineari di operatori di tipo principale complesso e caratteristiche compless multiple.

     

Solvability of semilinear differential equations with singularity

We prove local theorems on the existence of solutions of the Cauchy problem for singular equations of the form {fx262-01} in Banach spaces. Solvability conditions depend on the type of the singularity of the pencil λA + B of closed linear operators A and B. Examples and applications to finite-dimensional differential-algebraic equations, infinite systems of differential equations, and partial differential equations of the non-Kovalevskaya type are presented. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 2, pp. 225–239, February, 2008.     

Solving chemical kinetics equations by explicit methods

The paper presents two versions of numerical methods for solving systems of chemical kinetics which are based on quasi-steady state and partial equilibrium approximations. These methods are explicit and guarantee that the solutions are positive and bounded for an arbitrary period of time. In contrast to the existing methods, they ensure the conservation of mass and the fulfillment of stoichiometric relations, which enables us to increase the timestep considerably while keeping the same accuracy.     

Strong solutions of semilinear stochastic partial differential equations

We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives, we consider the equation in the context of power scale generated by a strongly elliptic differential operator. Application of semigroup arguments then yields the existence of a continuous strong solution.     

Stability results of random impulsive semilinear differential equations

In this paper, we study the existence, uniqueness, continuous dependence, Ulam stabilities and exponential stability of random impulsive semilinear differential equations under sufficient condition. The results are obtained by using the contraction mapping principle. Finally an example is given to illustrate the applications of the abstract results.     

Local solvability for semilinear anisotropic partial differential equations

In this paper we introduce a class of Gevrey-Sobolev anisotropic spaces and, by means of the application of the Banach fixed point Theorem, we obtain the solvability of a related semilinear equation. Entrata in Redazione il 21 luglio 1999.     

Anti-periodic mild solutions to semilinear fractional differential equations

In this paper, by using the \(\alpha \)-resolvent family theory, Banach contraction mapping principle and Schauder’s fixed point theorem, we investigate the existence of anti-periodic mild solutions to the semilinear fractional differential equations \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t)),\ t\in R,1 \le \alpha \le 2 \) and \(D^{\alpha }_{t}u(t) = Au(t) +f(t,u(t),u'(t)),\ t\in R,1 < \alpha < 2\), where \(A : D(A)\subset X \rightarrow X\) is the infinitesimal generator of an \(\alpha \)-resolvent family defined on a Banach space \(X\) and \(f\) is a suitable function. Furthermore, an example is given to illustrate our results.     

Periodic solutions of nonlocal semilinear fourth-order differential equations

In this paper we study the existence of periodic solutions with prescribed wavelength for two classes of nonlocal fourth-order nonautonomous differential equations. Existence of nontrivial solutions for the first equation is proved using Clark’s theorem. Existence of nontrivial solutions for the second equation is proved using the symmetric mountain-pass theorem.     

Blow-up of solutions of semilinear parabolic differential equations

For a semilinear parabolic initial boundary value problem we establish criterions on blow-up of the solution in finite time and give bounds for the blow-up time. We treat several applications in both finite and infinite domains. For comparison, sufficient conditions are also given for the existence of global solutions.

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Zusammenfassung Für ein semilineares parabolisches Rand- und Anfangswertproblem stellen wir Kriterien für die Explosion der Lösung in endlicher Zeit auf und geben Schranken für die Explosionszeit an. Einige Anwendungen in beschränkten und unbeschränkten Gebieten werden untersucht, wobei wir als Gegenüberstellung auch hinreichende Bedingungen für die Existenz globaler Lösungen angeben.

     

Optimal control problems for semilinear non-well-posed parabolic differential equations

This paper deals with necessary conditions for optimal control problem governed by some semilinear parabolic differential equation which may be non-well-posed. State constrained problem is considered. Finally, under some suitable assumptions, we obtain the existence of optimal pairs.     

On quasiperiodic solutions of semilinear systems of differential equations

We prove existence theorems for analytic quasi-periodic solutions for analytic systems of differential equations in a Banach space by the method of accelerated convergence. The results obtained are new even in the finite-dimensional case. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

Approximating the coefficients in semilinear stochastic partial differential equations

We investigate, in the setting of UMD Banach spaces E, the continuous dependence on the data A, F, G and ξ of mild solutions of semilinear stochastic evolution equations with multiplicative noise of the form

The substitution theorem for semilinear stochastic partial differential equations

In this article we establish a substitution theorem for semilinear stochastic evolution equations (see's) depending on the initial condition as an infinite-dimensional parameter. Due to the infinite-dimensionality of the initial conditions and of the stochastic dynamics, existing finite-dimensional results do not apply. The substitution theorem is proved using Malliavin calculus techniques together with new estimates on the underlying stochastic semiflow. Applications of the theorem include dynamic characterizations of solutions of stochastic partial differential equations (spde's) with anticipating initial conditions and non-ergodic stationary solutions. In particular, our result gives a new existence theorem for solutions of semilinear Stratonovich spde's with anticipating initial conditions.