Random exponential attractor for a stochastic reaction-diffusion equation in \(L^{2p}(D)\)

In this paper, we establish some sufficient conditions for the existence of a random exponential attractor for a random dynamical system in a Banach space. As an application, we consider a stochastic reaction-diffusion equation with multiplicative noise. We show that the random dynamical system $\varphi (t,\omega )$ $\phi(t,\omega)$ generated by this stochastic reaction-diffusion equation is uniformly Fréchet differentiable on a positively invariant random set in $L^{2p}(D)$ $L^{2p}(D)$ and satisfies the conditions of the abstract result, then we obtain the existence of a random exponential attractor in $L^{2p}(D)$ $L^{2p}(D)$ , where p is the growth of the nonlinearity satisfying $1<p\le 3$ $1< p\leq 3$ .