REGULAR DYNAMICS AND BOX-COUNTING DIMENSION FOR A RANDOM REACTION-DIFFUSION EQUATION ON UNBOUNDED DOMAINS

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 $ L^{2}( {\mathbb{R}}^N) $ into $ L^{p}( {\mathbb{R}}^N) \cap H^{1}( {\mathbb{R}}^N) $ is Hölder continuous for arbitrary space dimension $ N\geq 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 $ L^{2}( {\mathbb{R}}^N) $-pullback attractor is exactly a pullback attractor in $ L^{p}( {\mathbb{R}}^N) \cap H^{1}( {\mathbb{R}}^N) $, and furthermore it is attracting in $ L^{\delta}( {\mathbb{R}}^N) $ for any $ \delta\geq2 $, 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 $ L^{p}( {\mathbb{R}}^N) \cap H^{1}( {\mathbb{R}}^N) $, and two comparison formulas with $ L^2 $-dimension are derived, which are a straightforward consequence of Hölder continuity of the systems