DCDS
Global exponential κ-dissipative semigroups and exponential attraction
Jin Zhang Peter E. Kloeden Meihua Yang Chengkui Zhong
Discrete & Continuous Dynamical Systems - A 2017, 37(6): 3487-3502 doi: 10.3934/dcds.2017148

Globally exponential $κ-$dissipativity, a new concept of dissipativity for semigroups, is introduced. It provides a more general criterion for the exponential attraction of some evolutionary systems. Assuming that a semigroup $\{S(t)\}_{t≥q 0}$ has a bounded absorbing set, then $\{S(t)\}_{t≥q 0}$ is globally exponentially $κ-$dissipative if and only if there exists a compact set $\mathcal{A}^*$ that is positive invariant and attracts any bounded subset exponentially. The set $\mathcal{A}^*$ need not be finite dimensional. This result is illustrated with an application to a damped semilinear wave equation on a bounded domain.

keywords: Exponential dissipativity global attractor measures of noncompactness
DCDS-B
Long-time behavior of stochastic reaction-diffusion equation with dynamical boundary condition
Lu Yang Meihua Yang
Discrete & Continuous Dynamical Systems - B 2017, 22(7): 2627-2650 doi: 10.3934/dcdsb.2017102

In this paper, we study the dynamic behavior of a stochastic reaction-diffusion equation with dynamical boundary condition, where the nonlinear terms $f$ and $h$ satisfy the polynomial growth condition of arbitrary order. Some higher-order integrability of the difference of the solutions near the initial time, and the continuous dependence result with respect to initial data in $H^1(\mathcal{O})× H^{\frac 1 2}(Γ)$ were established. As a direct application, we can obtain the existence of pullback random attractor $A$ in the spaces $L^{p}(\mathcal{O})× L^{p}(Γ)$ and $H^1(\mathcal{O})× H^{\frac 1 2}(Γ)$ immediately.

keywords: Stochastic reaction-diffusion equation dynamical boundary condition higher-order integrability continuity pullback random attractor
DCDS-B
Long term dynamics of second order-in-time stochastic evolution equations with state-dependent delay
Igor Chueshov Peter E. Kloeden Meihua Yang
Discrete & Continuous Dynamical Systems - B 2018, 23(3): 991-1009 doi: 10.3934/dcdsb.2018139

The well-posedness and asymptotic dynamics of second-order-in-time stochastic evolution equations with state-dependent delay is investigated. This class covers several important stochastic PDE models arising in the theory of nonlinear plates with additive noise. We first prove well-posedness in a certain space of functions which are $C^1$ in time. The solutions constructed generate a random dynamical system in a $C^1$-type space over the delay time interval. Our main result shows that this random dynamical system possesses compact global and exponential attractors of finite fractal dimension. To obtain this result we adapt the recently developed method of quasi-stability estimates to the random setting.

keywords: State-dependent delay stochastic wave equation pullback random attractor exponential attractor
DCDS-B
Pullback attractors for non-autonomous quasi-linear parabolic equations with dynamical boundary conditions
Lu Yang Meihua Yang Peter E. Kloeden
Discrete & Continuous Dynamical Systems - B 2012, 17(7): 2635-2651 doi: 10.3934/dcdsb.2012.17.2635
The existence of a unique minimal pullback attractor is established for the evolutionary process associated with a non-autonomous quasi-linear parabolic equations with a dynamical boundary condition in $L^{r_1}(\Omega)\times L^{r_1}(\Gamma)$ under that assumption that the external forcing term satisfies a weak integrability condition, where $r_1$ $>$ $2$ is determined by the order of the nonlinearity.
keywords: quasi-linear parabolic equation p-Laplacian dy- namical boundary condition. Pullback attractor
CPAA
Reaction-diffusion equations with a switched--off reaction zone
Peter E. Kloeden Thomas Lorenz Meihua Yang
Communications on Pure & Applied Analysis 2014, 13(5): 1907-1933 doi: 10.3934/cpaa.2014.13.1907
Reaction-diffusion equations are considered on a bounded domain $\Omega$ in $\mathbb{R}^d$ with a reaction term that is switched off at a point in space when the solution first exceeds a specified threshold and thereafter remains switched off at that point, which leads to a discontinuous reaction term with delay. This problem is formulated as a parabolic partial differential inclusion with delay. The reaction-free region forms what could be called dead core in a biological sense rather than that used elsewhere in the literature for parabolic PDEs. The existence of solutions in $L^2(\Omega)$ is established firstly for initial data in $L^{\infty}(\Omega)$ and in $W_0^{1,2}(\Omega)$ by different methods, with $d$ $=$ $2$ or $3$ in the first case and $d$ $\geq$ $2$ in the second. Solutions here are interpreted in the sense of integral or strong solutions of nonhomogeneous linear parabolic equations in $L^2(\Omega)$ that are generalised to selectors of the corresponding nonhomogeneous linear parabolic differential inclusions and are shown to be equivalent under the assumptions used in the paper.
keywords: Reaction-diffusion equation dead core existence of solutions. memory inclusion equations discontinuous right-hand sides

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