On Cauchy—Dirichlet Problem for Parabolic Quasilinear SPDEs

We study the Cauchy–Dirichlet problem for a second-order quasilinear parabolic stochastic differential equation (SPDE) in a domain with a zero order noise term driven by a cylindrical Brownian motion. Considering its solution as a function with values in a probability space and using the methods of deterministic partial differential equations, we establish the existence and uniqueness of a strong solution in Hölder classes with weights.     

Global solutions of abstract quasilinear parabolic equations

Local and global existence and uniqueness for strict solutions of abstract quasilinear parabolic equations are studied. Applications to quasilinear parabolic partial differential equations are also given.     

Large Deviations for Quasilinear Parabolic Stochastic Partial Differential Equations

Potential Analysis - In this paper, we establish the Freidlin-Wentzell’s large deviations for quasilinear parabolic stochastic partial differential equations with multiplicative noise, which...     

Construction of solutions to quasilinear partial differential equations

We propose a method for constructing solutions to a class of quasilinear parabolic partial differential equations (PDEs) basing on a new property of these equations. The method applies to quasilinear hyperbolic and elliptic equations as well. The results of this article broaden the class of exact solutions to the quasilinear equations, in particular, to the nonlinear heat equations, the equations of chemical kinetics and mathematical biology.     

带Robin边界条件的拟线性抛物方程解的整体存在性与爆破

主要研究了一类带Robin边界条件的拟线性抛物方程解的整体存在性与爆破问题,利用微分不等式技术,获得了方程的解发生爆破时的爆破时间的下界.然后给出了方程解整体存在的充分条件,最后得到了方程的解发生爆破时发生爆破时间的上界.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the uniform norm of solutions of quasilinear SPDE's

In this paper we prove L^p estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE.     

On The Coupled Cahn-hilliard Equations

This paper is concerned with a system of nonlinear partial differential equations, in short, the coupled Cahn-Hilliard equations, which consists of a fourth order quasilinear parabolic equation and a second order quasilinear parabolic equation. This system was recently derived by Penrose and Fife and also by Alt and Pawlow to describe the non-isothermal phase separation of a two-component system. The global existence and uniqueness of classical solutions is proved. The results about the asymptotic behavior, as time goes to infinity, of solution and about the existence and multiplicity of solutions to the corresponding stationary problem, which is a nonlinear boundary value problem involving nonlocal term and constraints, are also obtained.     

Macroscopic limits for stochastic partial differential equations of McKean–Vlasov type

A class of quasilinear stochastic partial differential equations (SPDEs), driven by spatially correlated Brownian noise, is shown to become macroscopic (i.e., deterministic), as the length of the correlations tends to 0. The limit is the solution of a quasilinear partial differential equation. The quasilinear SPDEs are obtained as a continuum limit from the empirical distribution of a large number of stochastic ordinary differential equations (SODEs), coupled though a mean-field interaction and driven by correlated Brownian noise. The limit theorems are obtained by application of a general result on the convergence of exchangeable systems of processes. We also compare our approach to SODEs with the one introduced by Kunita.     

Homeomorphism of solutions to backward SDEs and applications

In this paper we study the homeomorphic properties of the solutions to one dimensional backward stochastic differential equations under suitable assumptions, where the terminal values depend on a real parameter. Then, we apply them to the solutions for a class of second order quasilinear parabolic partial differential equations.     

Exponential mixing properties of stochastic PDEs through asymptotic coupling

 We consider parabolic stochastic partial differential equations driven by white noise in time. We prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions. These conditions amount essentially to the fact that the equation transmits the noise to all its determining modes. Several examples are investigated, including some where the noise does not act on every determining mode directly. Received: 20 September 2001 / Revised version: 21 April 2002 / Published online: 10 September 2002     

On nonnegative solutions of elliptic and parabolic equations

A theorem on the nonexistence of a nonnegative nontrivial generalized solution inR ⁿ is proved for general quasilinear second-order degenerate elliptic equations. Analogous results are obtained for a large class of systems of partial differential equations, second-order parabolic and inverse parabolic equations, which are nonlinear and may be degenerate.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 16, pp. 114–136, 1992.     

关于拟线性混合型边界问题的概率表示

关于某些抛物型和椭圆型偏微分方程的混合边界问题的解被表示为一类联系于Ito正向反射边界随机微分方程的反向随机微分方程的解.     

Classical solvability of a parabolic system of two nonlinear equations

We study the complete regularity of solutions of a nondiagonal parabolic system of quasilinear second-order differential equations in divergence form assuming that the coefficients are sufficiently slowly varying functions of their arguments and the off-diagonal terms in the coefficient matrix are sufficiently small. To this end, we use a method based on the successive approximation to the solution by smooth functions with the use of Schauder estimates at each step.     

On nonlinear systems consisting of different types of differential equations

We consider a system consisting of a quasilinear parabolic equation and a first order ordinary differential equation where both equations contain functional dependence on the unknown functions. Then we consider a system which consists of a quasilinear parabolic partial differential equation, a first order ordinary differential equation and an elliptic partial differential equation. These systems were motivated by models describing diffusion and transport in porous media with variable porosity. Supported by the Hungarian NFSR under grant OTKA T 049819.     

On the existence of optimal controls for quasilinear parabolic partial differential equations

A class of systems governed by quasilinear parabolic partial differential equations with first boundary conditions is considered. Existence of solutions for this class of systems and theira priori estimates are established. Further, a theorem on the existence of optimal controls for the corresponding control problem is obtained. Its proof is based on Filippov's implicit functions lemma. The control restraint setU is taken as a measurable multifunction.The authors wish to thank Professor L. Cesari for his most valuable comments and suggestions. In fact, a condition assumed in the original version of this paper was substantially relaxed by him. For details, see Remark 4.1.     

The motion of a particle on a light viscoelastic bar: asymptotic analysis of its quasilinear parabolic–hyperbolic equation

We study the large longitudinal motion of a nonlinearly viscoelastic bar with one end fixed and the other end attached to a heavy particle. This problem is a precise continuum-mechanical analog of the basic discrete mechanical problem of the motion of a particle on a (massless) spring. This motion is governed by an initial-boundary-value problem for a class of third-order quasilinear parabolic–hyperbolic partial differential equations subject to a nonstandard boundary condition, which is the equation of motion of the particle. The ratio of the mass of the bar to that of the tip mass is taken to be a small parameter ε. We prove that this problem has a unique globally defined solution that admits a valid asymptotic expansion, including an initial-layer expansion, in powers of ε for ε near 0. The validity of the expansion gives a precise meaning to the solution of the reduced problem, obtained by setting ε=0, which curiously is seldom governed by the expected ordinary differential equation. The fundamental constitutive hypothesis that the tension be a uniformly monotone function of the strain rate plays a critical role in a delicate proof that each term of the initial-layer expansion decays exponentially in time. These results depend on new decay estimates for the solution of quasilinear parabolic equations.     

THE UNCONDITIONAL CONVERGENT DIFFERENCE METHODS WITH INTRINSIC PARALLELISM FOR QUASILINEAR PARABOLIC SYSTEMS WITH TWO DIMENSIONS

In the present work we are going to solve the boundary value problem for the quasilinear parabolic systems of partial differential equations with two space dimensions by the finite difference method with intrinsic parallelism.Some fundamental behaviors of general finite difference schemes with intrinsic parallelism for the mentioned problems are studied.By the method of a priori estimation of the discrete solutions of the nonlinear difference systems,and the interpolation formulas of the various norms of the discrete functions and the fixed-point technique in finite dimensional Euclidean space,the existennce of the discrete vector solutions of the nonliear difference system with intrinsic parallelism are proved .Moreover the convergence of the discrete vector solutions of these difference schemes to the unique generalizd solution of the original quasilinear parabolic problem is proved.     

完全耦合的正倒向随机微分方程及相应的偏微分方程系统

本文利用完全耦合的正倒向随机微分方程,对一类耦合了一个代数方程的二阶拟线性抛物型偏微分方程系统,给出概率表示。在适当的假设下,得到这类偏微分方程系统粘性解的存在唯一性结果。     

Stochastic Evolution Equations Driven by a Fractional White Noise

Abstract

In this paper we study stochastic evolution equations driven by a fractional white noise with arbitrary Hurst parameter in infinite dimension. We establish the existence and uniqueness of a mild solution for a nonlinear equation with multiplicative noise under Lipschitz condition by using a fixed point argument in an appropriate inductive limit space. In the linear case with additive noise, a strong solution is obtained. Those results are applied to stochastic parabolic partial differential equations perturbed by a fractional white noise.     

Concavity maximum principle for viscosity solutions of singular equations

We prove a concavity maximum principle for the viscosity solutions of certain fully nonlinear and singular elliptic and parabolic partial differential equations. Our results parallel and extend those obtained by Korevaar and Kennington for classical solutions of quasilinear equations. Applications are given in the case of the singular infinity Laplace operator.