scholarly journals Random attractors via pathwise mild solutions for stochastic parabolic evolution equations

2021 ◽  
Author(s):  
Christian Kuehn ◽  
Alexandra Neamţu ◽  
Stefanie Sonner
Keyword(s):  
Stochastic Partial Differential Equations ◽  
Additive Noise ◽  
Evolution Equations ◽  
Differential Operators ◽  
Mild Solutions ◽  
Random Sets ◽  
Exponential Attractors ◽  
Parabolic Evolution Equations ◽  
Finite Fractal Dimension ◽  
Longtime Behavior

AbstractWe investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution problems in Banach spaces with additive noise and prove the existence of random exponential attractors. These are compact random sets of finite fractal dimension that contain the global random attractor and are attracting at an exponential rate. In order to apply the framework of random dynamical systems, we use the concept of pathwise mild solutions.

ATTRACTORS FOR A CAGINALP MODEL WITH A LOGARITHMIC POTENTIAL AND COUPLED DYNAMIC BOUNDARY CONDITIONS

2013 ◽  
Vol 11 (06) ◽  
pp. 1350024 ◽  
Author(s):  
MONICA CONTI ◽  
STEFANIA GATTI ◽  
ALAIN MIRANVILLE
Keyword(s):  
Boundary Conditions ◽  
Phase Field Model ◽  
Logarithmic Potential ◽  
Dynamic Boundary Conditions ◽  
Exponential Attractors ◽  
Dynamic Boundary ◽  
Suitable Definition ◽  
Finite Fractal Dimension ◽  
Definition Of ◽  
Longtime Behavior

We study the longtime behavior of the Caginalp phase-field model with a logarithmic potential and dynamic boundary conditions for both the order parameter and the temperature. Due to the possible lack of distributional solutions, we deal with a suitable definition of solutions based on variational inequalities, for which we prove well-posedness and the existence of global and exponential attractors with finite fractal dimension.

scholarly journals Low-rank approximation of linear parabolic equations by space-time tensor Galerkin methods

2019 ◽  
Vol 53 (2) ◽  
pp. 635-658
Author(s):  
Thomas Boiveau ◽  
Virginie Ehrlacher ◽  
Alexandre Ern ◽  
Anthony Nouy
Keyword(s):  
Evolution Equations ◽  
Differential Operators ◽  
Space Time ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Dual Norm ◽  
Fully Discrete ◽  
Linear Parabolic Equations ◽  
Parabolic Evolution Equations ◽  
Rank Approximation

We devise a space-time tensor method for the low-rank approximation of linear parabolic evolution equations. The proposed method is a Galerkin method, uniformly stable in the discretization parameters, based on a Minimal Residual formulation of the evolution problem in Hilbert–Bochner spaces. The discrete solution is sought in a linear trial space composed of tensors of discrete functions in space and in time and is characterized as the unique minimizer of a discrete functional where the dual norm of the residual is evaluated in a space semi-discrete test space. The resulting global space-time linear system is solved iteratively by a greedy algorithm. Numerical results are presented to illustrate the performance of the proposed method on test cases including non-selfadjoint and time-dependent differential operators in space. The results are also compared to those obtained using a fully discrete Petrov–Galerkin setting to evaluate the dual residual norm.

Solving Becker’s Problem on Periodic Solutions of Parabolic Evolution Equations

2018 ◽  
Vol 18 (2) ◽  
pp. 195-215
Author(s):  
Giovanni Vidossich
Keyword(s):  
Evolution Equation ◽  
Positive Solution ◽  
Open Problem ◽  
Parabolic Equations ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Existence And Multiplicity ◽  
Finite Dimensional ◽  
Parabolic Evolution Equations ◽  
Systems Of Parabolic Equations

Abstract We present existence and multiplicity theorems for periodic mild solutions to parabolic evolution equations. Their peculiarity is a link with the spectrum of the generator of the semigroup rather than with the spectrum of the linearized periodic BVP for the evolution equation. They provide a positive solution to the open problem risen by Becker [3], they extend some results of Castro and Lazer [5] from scalar to systems of parabolic equations, and they are new even for finite-dimensional ODEs.

scholarly journals Antiperiodic Problems for Nonautonomous Parabolic Evolution Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
R. N. Wang ◽  
Y. Zhou
Keyword(s):  
Evolution Equation ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Linear Part ◽  
Neutral Type ◽  
Linear Operators ◽  
Parabolic Evolution Equation ◽  
Parabolic Evolution Equations ◽  
Semilinear Parabolic ◽  
Densely Defined

This work focuses on the antiperiodic problem of nonautonomous semilinear parabolic evolution equation in the formu′(t)=A(t)u(t)+f(t,u(t)),t∈R,u(t+T)=-u(t),t∈R, where(At)t∈R(possibly unbounded), depending on time, is a family of closed and densely defined linear operators on a Banach spaceX. Upon making some suitable assumptions such as the Acquistapace and Terreni conditions and exponential dichotomy on(At)t∈R, we obtain the existence results of antiperiodic mild solutions to such problem. The antiperiodic problem of nonautonomous semilinear parabolic evolution equation of neutral type is also considered. As sample of application, these results are applied to, at the end of the paper, an antiperiodic problem for partial differential equation, whose operators in the linear part generate an evolution family of exponential stability.

THE MONOTONICITY INEQUALITY FOR LINEAR STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

2009 ◽  
Vol 12 (04) ◽  
pp. 575-591 ◽  
Author(s):  
L. GAWARECKI ◽  
V. MANDREKAR ◽  
B. RAJEEV
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Stochastic Partial Differential Equations ◽  
Evolution Equations ◽  
Differential Operators ◽  
Probabilistic Representation ◽  
Partial Differential ◽  
Finite Dimensional ◽  
Monotonicity Inequality ◽  
Elliptic Differential Operators

We prove the monotonicity inequality for differential operators A and L that occur as coefficients in linear stochastic partial differential equations associated with finite-dimensional Itô processes. We characterize the solutions of such equations. A probabilistic representation is obtained for solutions to a class of evolution equations associated with time dependent, possibly degenerate, second-order elliptic differential operators.

Existence results of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay

2014 ◽  
Vol 17 (4) ◽  
Author(s):  
Shengli Xie
Keyword(s):  
Differential Equations ◽  
Differential Evolution ◽  
Existence Theorem ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Evolution Equations ◽  
Infinite Delay ◽  
Mild Solutions ◽  
Existence Results ◽  
Order Case

AbstractIn this paper we prove the existence and uniqueness of mild solutions for impulsive fractional integro-differential evolution equations with infinite delay in Banach spaces. We generalize the existence theorem for integer order differential equations to the fractional order case. The results obtained here improve and generalize many known results.

scholarly journals On Almost Automorphic Mild Solutions for Nonautonomous Stochastic Evolution Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
Jing Cui ◽  
Litan Yan
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Point Theorem ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Banach Fixed Point Theorem ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Almost Automorphic ◽  
Composition Theorem

We consider a class of nonautonomous stochastic evolution equations in real separable Hilbert spaces. We establish a new composition theorem for square-mean almost automorphic functions under non-Lipschitz conditions. We apply this new composition theorem as well as intermediate space techniques, Krasnoselskii fixed point theorem, and Banach fixed point theorem to investigate the existence of square-mean almost automorphic mild solutions. Some known results are generalized and improved.

Fractional evolution equations with nonlocal conditions in partially ordered Banach space

2018 ◽  
Vol 34 (3) ◽  
pp. 379-390
Author(s):  
HEMANT KUMAR NASHINE ◽  
◽  
HE YANG ◽  
RAVI P. AGARWAL ◽  
◽  
...  
Keyword(s):  
Fixed Point ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Nonlocal Conditions ◽  
Coupled Fixed Point ◽  
Positive Cone ◽  
Bounded Linear ◽  
Partially Ordered ◽  
C0 Semigroup ◽  
Uniformly Bounded

In the present work, we discuss the existence of mild solutions for the initial value problem of fractional evolution equation of the form where C Dσ t denotes the Caputo fractional derivative of order σ ∈ (0, 1), −A : D(A) ⊂ X → X generates a positive C0-semigroup T(t)(t ≥ 0) of uniformly bounded linear operator in X, b > 0 is a constant, f is a given functions. For this, we use the concept of measure of noncompactness in partially ordered Banach spaces whose positive cone K is normal, and establish some basic fixed point results under the said concepts. In addition, we relaxed the conditions of boundedness, closedness and convexity of the set at the expense that the operator is monotone and bounded. We also supply some new coupled fixed point results via MNC. To justify the result, we prove an illustrative example that rational of the abstract results for fractional parabolic equations.

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