Invariant sets for systems of partial differential equations I. Parabolic equations

1977 ◽  
Vol 67 (1) ◽  
pp. 41-52 ◽  
Author(s):  
Ray Redheffer ◽  
Wolfgang Walter
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Parabolic Equations ◽  
Invariant Sets ◽  
Partial Differential

Exponential global attractors for semigroups in metric spaces with applications to differential equations

2011 ◽  
Vol 31 (6) ◽  
pp. 1641-1667 ◽  
Author(s):  
ALEXANDRE N. CARVALHO ◽  
JAN W. CHOLEWA
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Global Attractor ◽  
Metric Space ◽  
Metric Spaces ◽  
Global Attractors ◽  
Invariant Sets ◽  
Bounded Subset ◽  
Partial Differential ◽  
Unstable Manifolds

AbstractIn this article semigroups in a general metric space V, which have pointwise exponentially attracting local unstable manifolds of compact invariant sets, are considered. We show that under a suitable set of assumptions these semigroups possess strong exponential dissipative properties. In particular, there exists a compact global attractor which exponentially attracts each bounded subset of V. Applications of abstract results to ordinary and partial differential equations are given.

Similarity and Generalized Finite-Difference Solutions of Parabolic Partial Differential Equations

1972 ◽  
Vol 39 (2) ◽  
pp. 584-590
Author(s):  
A. M. Clausing
Keyword(s):  
Partial Differential Equations ◽  
Finite Difference ◽  
Differential Equations ◽  
Parabolic Equations ◽  
Finite Difference Methods ◽  
Parabolic Type ◽  
Computational Effort ◽  
Partial Differential ◽  
Considerable Saving ◽  
Boundary Layer Equations

Techniques are presented for obtaining generalized finite-difference solutions to partial differential equations of the parabolic type. It is shown that the advantages of similarity in the solution of similar problems are generally not lost if the solution to the original partial differential equations is effected in the physical plane by finite-difference methods. The analysis results in a considerable saving in computational effort in the solution of both similar and nonsimilar problems. Several examples, including both the heat-conduction equation and the boundary-layer equations, are given. The analysis also provides a practical means of estimating the accuracy of finite-difference solutions to parabolic equations.

On smooth approximation for random attractor of stochastic partial differential equations with multiplicative noise

2018 ◽  
Vol 18 (05) ◽  
pp. 1850040 ◽  
Author(s):  
Hongbo Fu ◽  
Xianming Liu ◽  
Jicheng Liu ◽  
Xiangjun Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Random Attractor ◽  
Invariant Sets ◽  
Stochastic Pdes ◽  
Polygonal Approximation ◽  
Smooth Approximation ◽  
Partial Differential ◽  
Random Partial Differential Equations

Wong–Zakai type approximation for stochastic partial differential equations (abbreviate as PDEs) is well studied. Besides the polygonal approximation, a type of smooth noise approximation is considered. After showing the existence of random attractor for a class of random partial differential equations defined on the entire space [Formula: see text], when random color noises tend to white noise, the solutions and invariant sets between original stochastic PDEs and random PDEs are compared. Some continuity results of random attractor in random dynamical systems are indicated.

scholarly journals Extended backward stochastic Volterra integral equations, Quasilinear parabolic equations, and Feynman–Kac formula

2020 ◽  
Vol 21 (01) ◽  
pp. 2150004
Author(s):  
Hanxiao Wang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Equations ◽  
Parabolic Equations ◽  
Natural Extension ◽  
Volterra Integral Equations ◽  
Degenerate Parabolic Equations ◽  
Partial Differential ◽  
Non Local ◽  
Quasilinear Parabolic

This paper is concerned with the relationship between backward stochastic Volterra integral equations (BSVIEs, for short) and a kind of non-local quasilinear (and possibly degenerate) parabolic equations. As a natural extension of BSVIEs, the extended BSVIEs (EBSVIEs, for short) are introduced and investigated. Under some mild conditions, the well-posedness of EBSVIEs is established and some regularity results of the adapted solution to EBSVIEs are obtained via Malliavin calculus. Then it is shown that a given function expressed in terms of the adapted solution to EBSVIEs uniquely solves a certain system of non-local parabolic equations, which generalizes the famous nonlinear Feynman–Kac formula in Pardoux–Peng [Backward stochastic differential equations and quasilinear parabolic partial differential equations, in Stochastic Partial Differential Equations and Their Applications (Springer, 1992), pp. 200–217].

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