Caraballo, Tomás; Langa, José A.; Robinson, James C. A stochastic pitchfork bifurcation in a reaction-diffusion equation. (English) Zbl 0996.60070 Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457, No. 2013, 2041-2061 (2001). Let \(D\subseteq \mathbb R^{m}\), \(m\leq 5\), be an open bounded set with a regular boundary and let \(W\) be a one-dimensional Wiener process. A stochastic reaction-diffusion equation (in a Stratonovich form) \[ du = (\Delta u + \beta u - u^3) dt + \sigma u \circ dW \quad \text{in }D,\qquad u=0 \quad \text{on }\partial D,\tag{1} \] is considered. In their previous work [Discrete Contin. Dyn. Syst. 6, No. 4, 875-892 (2000)] the authors proved that there exists a random attractor \(\mathcal A(\omega)\) for (1) and found an upper bound to its Hausdorff dimension. Namely, let \(0<\lambda_1\leq\lambda_2\leq\ldots\) be the eigenvalues of the Dirichlet Laplacian \(-\Delta\) on \(D\), then \(d_{\text H}(\mathcal A(\omega)) < d\) provided that \(\beta d < \sum^{d}_{j=1} \lambda_{j}\), which yields \(d_{\text H} (\mathcal A(\omega)) \leq C\beta^{m/2}\). In the paper under review, it is shown that an analogous estimate holds also from below: if \(\lambda_{n}<\beta<\lambda_{n+1}\), then \(d_{\text H}(\mathcal A(\omega)) \geq n\), consequently, \(d_{\text H}(\mathcal A(\omega)) \geq c\beta^{m/2}\) for a \(c>0\). In the one-dimensional case \(m=1\) this result is used to show that (1) undergoes a stochastic pitchfork bifurcation at \(\beta = \lambda_1\). Reviewer: Jan Seidler (Praha) Cited in 55 Documents MSC: 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 37H20 Bifurcation theory for random and stochastic dynamical systems Keywords:random attractors; stochastic pitchfork bifurcation; stochastic Chafee-Infante equation; Hausdorff dimension PDFBibTeX XMLCite \textit{T. Caraballo} et al., Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 457, No. 2013, 2041--2061 (2001; Zbl 0996.60070) Full Text: DOI