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A stochastic pitchfork bifurcation in a reaction-diffusion equation. (English) Zbl 0996.60070

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 \(\beta = \lambda_1\).

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
37H20 Bifurcation theory for random and stochastic dynamical systems
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