Asymptotic smoothing effect of solutions to weakly dissipative Klein-Gordon-Schrödinger equations

In this paper the authors consider the Cauchy problem of weakly dissipative Klein-Gordon-Schrödinger equations through Yukawa coupling in . Making use of a Strichartz type inequality and a suitable decomposition of the solution semigroup they prove the asymptotic smoothing effect of the solutions.     

A dissipative singular Sturm-Liouville problem with a spectral parameter in the boundary condition

In the Hilbert space , we consider nonselfadjoint singular Sturm-Liouville boundary value problem (with two singular end points a and b) in limit-circle cases at a and b, and with a spectral parameter in the boundary condition. The approach is based on the use of the maximal dissipative operator, and the spectral analysis of this operator is adequate for boundary value problem. We construct a selfadjoint dilation of the maximal dissipative operator and its incoming and outgoing spectral representations, which make it possible to determine the scattering matrix of the dilation. We also construct a functional model of the maximal dissipative operator and define its characteristic function in terms of solutions of the corresponding Sturm-Liouville equation. On the basis of the results obtained regarding the theory of the characteristic function, we prove theorems on completeness of the system of eigenvectors and associated vectors of the maximal dissipative operator and Sturm-Liouville boundary value problem.     

Density results with linear combinations of translates of fundamental solutions

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain Ω by linear combinations of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie outside of . The domains under consideration may possess holes and they are required to satisfy a rather mild boundary regularity requirement, namely the segment condition. We study approximations with respect to the norms of the spaces and the spaces of uniformly Hölder continuous functions , and we establish density and non-density results for elliptic operators with constant coefficients. We also provide applications of our density results related to the method of fundamental solutions and to the theory of universal series.     

Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for “suitable weak solutions” of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are Hölder continuous up to the boundary provided that the scaled mixed norm with 3/p+2/q?2, 2<q?∞, (p,q)≠(3/2,∞) is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some additional integral conditions.     

On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations

We study an iterative method with order for solving nonlinear operator equations in Banach spaces. Algorithms for specific operator equations are built up. We present the received new results of the local and semilocal convergence, in case when the first-order divided differences of a nonlinear operator are Hölder continuous. Moreover a quadratic nonlinear majorant for a nonlinear operator, according to the conditions laid upon it, is built. A priori and a posteriori estimations of the method’s error are received. The method needs almost the same number of computations as the classical Secant method, but has a higher order of convergence. We apply our results to the numerical solving of a nonlinear boundary value problem of second-order and to the systems of nonlinear equations of large dimension.     

Asymptotic behavior and decay rate estimates for a class of semilinear evolution equations of mixed order

In this article we present a unified approach to study the asymptotic behavior and the decay rate to a steady state of bounded weak solutions of nonlinear, gradient-like evolution equations of mixed first and second order. The proof of convergence is based on the Lojasiewicz-Simon inequality, the construction of an appropriate Lyapunov functional, and some differential inequalities. Applications are given to nonautonomous semilinear wave and heat equations with dissipative, dynamical boundary conditions, a nonlinear hyperbolic-parabolic partial differential equation, a damped wave equation and some coupled system.     

Trotter products and reaction-diffusion equations

In this paper, we study a class of generalized diffusion-reaction equations of the form , where A is a pseudodifferential operator which generates a Feller semigroup. Using the Trotter product formula we give a corresponding discrete time integro-difference equation for numerical solutions.     

Approximation of the semigroup generated by the Robin Laplacian in terms of the Gaussian semigroup

We consider the Laplacian ΔR subject to Robin boundary conditions on the space , where Ω is a smooth, bounded, open subset of RN. It is known that ΔR generates an analytic contraction semigroup. We show how this semigroup can be obtained from the Gaussian semigroup on C₀(RN) via a Trotter formula. As the main ingredient, we construct a positive, contractive, linear extension operator Eβ from to C₀(RN) which maps an operator core for ΔR into the domain of the generator of the Gaussian semigroup.     

Strong trajectory attractors for dissipative Euler equations

The 2D Euler equations with periodic boundary conditions and extra linear dissipative term Ru, R>0 are considered and the existence of a strong trajectory attractor in the space is established under the assumption that the external forces have bounded vorticity. This result is obtained by proving that any solution belonging the proper weak trajectory attractor has a bounded vorticity which implies its uniqueness (due to the Yudovich theorem) and allows to verify the validity of the energy equality on the weak attractor. The convergence to the attractor in the strong topology is then proved via the energy method.     

Trapping regions for discontinuously coupled systems of evolution variational inequalities and application

We consider initial-boundary value problems for weakly coupled systems of parabolic equations under coupled nonlinear flux boundary condition. Both coupling vector fields and are assumed to be either of competitive or cooperative type, but may otherwise be discontinuous with respect to all their arguments. The main goal is to provide conditions for the vector fields f and g that allow the identification of regions of existence of solutions (so called trapping regions). To this end the problem is transformed to a discontinuously coupled system of evolution variational inequalities. Assuming a generalized outward pointing vector field on the boundary of a rectangle of the dependent variable space, the system of evolution variational inequalities is solved via a fixed point problem for some increasing operator in an appropriate ordered Banach space. The main tools used in the proof are evolution variational inequalities, comparison techniques, and fixed point results in ordered Banach spaces.     

Stationary solutions of SPDEs and infinite horizon BDSDEs

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.     

Second order differential operators with Feller-Wentzell type boundary conditions

In this paper, we study a basic generation problem concerning the second order differential operator in the space C[0,1] of complex continuous functions equipped with Feller-Wentzell type boundary conditions, which originates from the work of Feller [W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2) 55 (1952) 468-519]. We prove successfully that the operator, under suitable assumptions, generates a strongly continuous cosine function on C[0,1] (or on a subspace of C[0,1]), by means of an operator matrix analysis combined with perturbation, approximation, and similarity techniques.     

Positive solutions of three-point boundary value problems for systems of nonlinear second order ordinary differential equations

In this paper, we study the three-point boundary value problems for systems of nonlinear second order ordinary differential equations of the form

Under some conditions, we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones.     

Infinitely many bounded solutions for the <Emphasis Type="Italic">p</Emphasis>-Laplacian with nonlinear boundary conditions

By applying a method due to Saint Raymond, we prove the existence of infinitely many weak solutions for a quasilinear elliptic partial differential equation, involving the p-Laplacian operator, coupled with a nonlinear boundary condition. Our main assumption is a suitable oscillatory behaviour of the nonlinearity either at infinity or at zero.     

Interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations

We present new interior regularity criteria for suitable weak solutions of the magnetohydrodynamic equations in dimension three: a suitable weak solution is regular near an interior point z if the scaled -norm of the velocity with 1?3/p+2/q?2, 1?q?∞ is sufficiently small near z and if the scaled -norm of the magnetic field with 1?3/l+2/m?2, 1?m?∞ is bounded near z. Similar results are also obtained for the vorticity and for the gradient of the vorticity. Furthermore, with the aid of the regularity criteria, we exhibit some regularity conditions involving pressure for weak solutions of the magnetohydrodynamic equations.     

THE NONLOCAL INITIAL PROBLEMS OF A SEMILINEAR EVOLUTION EQUATION

The purpose of this paper is to investigate the existence of solutions to a nonlocal Cauchy problem for an evolution equation. The methods used here include the semigroup methods in proper spaces and Schauder's theorem. And the results are applied to a system of nonlinear partial differential equations with nonlinear boundary conditions.     

Stationary solutions of SPDEs and infinite horizon BDSDEs with non-Lipschitz coefficients

We prove a general theorem that the -valued solution of an infinite horizon backward doubly stochastic differential equation, if exists, gives the stationary solution of the corresponding stochastic partial differential equation. We prove the existence and uniqueness of the -valued solutions for backward doubly stochastic differential equations on finite and infinite horizon with linear growth without assuming Lipschitz conditions, but under the monotonicity condition. Therefore the solution of finite horizon problem gives the solution of the initial value problem of the corresponding stochastic partial differential equations, and the solution of the infinite horizon problem gives the stationary solution of the SPDEs according to our general result.