Regular dynamics and box-counting dimension for a random reaction-diffusion equation on unbounded domains

Abstract

In this article, we study a random reaction-diffusion equation driven by a Brownian motion with a wide class of nonlinear multiple. First, it is exhibited that the weak solution mapping L2(RN) into Lp(RN) ∩ H1(RN) is Hölder continuous for arbitrary space dimension N ≥ 1, where p > 2 is the growth degree of the nonlinear forcing. The main idea to achieve this is the classic induction technique based on the difference equation of solutions, by using some appropriate multipliers at different stages. Second, the continuity results are applied to investigate the sample-wise regular dynamics. It is showed that the L2(RN)-pullback attractor is exactly a pullback attractor in Lp(RN) ∩ H1(RN), and furthermore it is attracting in Lδ (RN) for any δ ≥ 2, under almost identical conditions on the nonlinearity as in Wang et al [31], whose result is largely developed in this paper. Third, we consider the box-counting dimension of the attractor in Lp(RN)∩H1(RN), and two comparison formulas with L2-dimension are derived, which are a straightforward consequence of Hölder continuity of the systems.