اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدMultivalued Non-Autonomous Random Dynamical Systems for Wave Equations without Uniqueness
164
0
0.0
(
0
)
تأليف
Bixiang Wang
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three dimensional case. We introduce the concept of weak upper semicontinuity of multivalued functions and use such continuity to prove the measurability of multivalued functions from a metric space to a separable Banach space. By this approach, we show the measurability of pullback attractors of the multivalued random dynamical system of the wave equations regardless of the completeness of the underlying probability space. The asymptotic compactness of solutions is proved by the method of energy equations, and the difficulty caused by the non-compactness of Sobolev embeddings on $R^n$ is overcome by the uniform estimates on the tails of solutions.
قيم البحث
اقرأ أيضاً
The upper semicontinuity of random attractors for non-compact random dynamical systems is proved when the union of all perturbed random attractors is precompact with probability one. This result is applied to the stochastic Reaction-Diffusion with white noise defined on the entire space R^n.
We study pullback attractors of non-autonomous non-compact dynamical systems generated by differential equations with non-autonomous deterministic as well as stochastic forcing terms. We first introduce the concepts of pullback attractors and asympto
tic compactness for such systems. We then prove a sufficient and necessary condition for existence of pullback attractors. We also introduce the concept of complete orbits for this sort of systems and use these special solutions to characterize the structures of pullback attractors. For random systems containing periodic deterministic forcing terms, we show the pullback attractors are also periodic. As an application of the abstract theory, we prove the existence of a unique pullback attractor for Reaction-Diffusion equations on $R^n$ with both deterministic and random external terms. Since Sobolev embeddings are not compact on unbounded domains, the uniform estimates on the tails of solutions are employed to establish the asymptotic compactness of solutions.
We study the long time behavior of solutions of the non-autonomous Reaction-Diffusion equation defined on the entire space R^n when external terms are unbounded in a phase space. The existence of a pullback global attractor for the equation is establ
ished in L^2(R^n) and H^1(R^n), respectively. The pullback asymptotic compactness of solutions is proved by using uniform a priori estimates on the tails of solutions outside bounded domains.
In this article we study the asymptotic behavior of solutions, in sense of global pullback attractors, of the evolution system $$ begin{cases} u_{tt} +etaDelta^2 u+a(t)Deltatheta=f(t,u), & t>tau, xinOmega, theta_t-kappaDelta theta-a(t)Delta u_t=0, &
t>tau, xinOmega, end{cases} $$ subject to boundary conditions $$ u=Delta u=theta=0, t>tau, xinpartialOmega, $$ where $Omega$ is a bounded domain in $mathbb{R}^N$ with $Ngeqslant 2$, which boundary $partialOmega$ is assumed to be a $mathcal{C}^4$-hypersurface, $eta>0$ and $kappa>0$ are constants, $a$ is an Holder continuous function, $f$ is a dissipative nonlinearity locally Lipschitz in the second variable.
We prove the existence and uniqueness of tempered random attractors for stochastic Reaction-Diffusion equations on unbounded domains with multiplicative noise and deterministic non-autonomous forcing. We establish the periodicity of the tempered attr
actors when the stochastic equations are forced by periodic functions. We further prove the upper semicontinuity of these attractors when the intensity of stochastic perturbations approaches zero.
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
