Random Attractors for Degenerate Stochastic Partial Differential Equations

Abstract

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate $p$-Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate $p$-Laplace equations we prove that the deterministic, $\infty$-dimensional attractor collapses to a single random point if enough noise is added.

Appendix: Existence of Unique Solutions to Monotone SPDE

We recall the variational approach to monotone SPDE (cf. [34, 42, 43]). Let $V \subseteq H \subseteq V^*$ be a Gelfand triple, $(\Omega ,\mathcal{G },\{\mathcal{G }_t\}_{t \in [0,T]},\mathbb{P })$ be a normal filtered probability space and $W$ be a cylindrical Wiener process on some separable Hilbert space $U$. Further assume that

$$\begin{aligned} A: [0,T] \times V \times \Omega \rightarrow V^*,\ B: [0,T] \times V \times \Omega \rightarrow L_2(U,H) \end{aligned}$$

are $\mathcal{G }_t$-progressively measurable. We extend $A, B$ by $0$ to all of $[0,T] \times H \times \Omega$.

Definition 6.1

A continuous, $H$-valued, $\mathcal{G }_t$-adapted process $\{X_t\}_{t \in [0,T]}$ is called a solution to

$$\begin{aligned} dX_t = A(t,X_t)dt + B(t,X_t)dW_t \end{aligned}$$

(6.1)

if $X \in L^{\alpha }([0,T]\times \Omega ;V)$ and $\mathbb{P }$-a.s.

$$\begin{aligned} X_t = X_0 + \int _0^t A(r,X_r)dr + \int _0^t B(r,X_r)dW_r,\quad \forall t \in [0,T]. \end{aligned}$$

We assume that there are $\alpha > 1, c,C > 0$ and $f \in L^1([0,T]\times \Omega )$ positive and $\mathcal{G }_t$-adapted such that

• (H1): (Hemicontinuity) The map $s \mapsto {}_{V^*}\langle A(t,v_1+sv_2),v\rangle _V$ is continuous on $\mathbb{R }$,

• (H2): (Monotonicity)

  $$\begin{aligned} 2 {}_{V^*}\langle A(t,v_1)-A(t,v_2),v_1-v_2\rangle _V + \Vert B(t,v_1)-B(t,v_2)\Vert _{L_2(U,H)}^2 \le C\Vert v_1-v_2\Vert _H^2, \end{aligned}$$

• (H3): (Coercivity)

  $$\begin{aligned} 2 {}_{V^*}\langle A(t,v),v\rangle _V + \Vert B(t,v)\Vert _{L_2(U,H)}^2 + c\Vert v\Vert _V^\alpha \le f_t + C\Vert v\Vert _H^2, \end{aligned}$$

• (H4): (Growth)

  $$\begin{aligned} \left\Vert A(t,v) \right\Vert_{V^*}^\frac{\alpha }{\alpha -1} \le C\Vert v\Vert _V^\alpha +f_t, \end{aligned}$$

for all $v_1, v_2, v \in V$ and $(t,\omega ) \in [0,T]\times \Omega$.

Theorem 6.2

([43], Theorem 4.2.4) Assume $(H1)$–$(H4)$. Then, for every $X_0 \!\in \! L^2(\Omega ,\mathcal{G }_0;H)$ (5.15) has a unique solution $X$ satisfying

$$\begin{aligned} \mathbb{E }\left(\sup _{t \in [0,T]}\Vert X_t\Vert _H^2 + \int _0^T \Vert X_r\Vert _V^\alpha dr \right) < \infty . \end{aligned}$$