In this paper we investigate the long-time behavior of stochastic reaction-diffusion equations of the type du = (Au + f (u))dt + σ(u)dW (t), where A is an elliptic operator, f and σ are nonlinear maps and W is an infinite dimensional nuclear Wiener process. The emphasis is on unbounded domains. Under the assumption that the nonlinear function f possesses certain dissipative properties, this equation is known to have a solution with an expectation value which is uniformly bounded in time. Together with some compactness property, the existence of such a solution implies the existence of an invariant measure which is an important step in establishing the ergodic behavior of the underlying physical system. In this paper we expand the existing classes of nonlinear functions f and σ and elliptic operators A for which the invariant measure exists, in particular, in unbounded domains. We also show the uniqueness of the invariant measure for an equation defined on the upper half space if A is the Shrödinger-type operator A = 1 ρ (divρ∇u) where ρ = e −|x| 2 is the Gaussian weight.

The bidomain equations driven by stochastic forces and subject to nonlinearities of FitzHugh-Nagumo or Allen-Cahn type are considered for the first time. It is shown that this set of equations admits a global weak solution as well as a stationary solution, which generates a uniquely determined invariant measure.

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We consider minimizers of a Ginzburg-Landau energy with a discontinuous and rapidly oscillating pinning term, subject to a Dirichlet boundary condition of degree d > 0. The pinning term models an unbounded number of small impurities in the domain. We prove that for strongly type II superconductor with impurities, minimizers have exactly d isolated zeros (vortices). These vortices are of degree 1 and pinned by the impurities. As in the standard case studied by Bethuel, Brezis and Hélein, the macroscopic location of vortices is governed by vortex/vortex and vortex/ boundary repelling effects. In some special cases we prove that their macroscopic location tends to minimize the renormalized energy of Bethuel-Brezis-Hélein. In addition, impurities affect the microscopic location of vortices. Our technics allows us to work with impurities having different size. In this situation we prove that vortices are pinned by the largest impurities.

In this paper, we study limiting behavior of the invariant measures for reaction–diffusion equations in the whole space [Formula: see text] with regular and singular perturbations. In the regular case, we show the convergence of the unique stationary solution of [Formula: see text] to a stationary solution of the limiting equation [Formula: see text]. We also consider the asymptotic behavior of the stationary solution under the perturbations of spectrum. Finally, for the singular perturbation of homogenization type, we show the weak convergence of invariant measure to its homogenized limit.

We study the minimization problem for simplified Ginzburg-Landau functional in doubly connected domain. This minimization problem is a subject to "semi-stiff" boundary conditions: |u| = 1 and prescribed degrees p and q on the outer and inner boundaries respectively. Following the work of Berlyand and Rybalko [J. Eur. Math. Soc. 12 (2010), 1497-1531], we additionally prescribe the degree in the bulk (approximate bulk degree) to be d. The work [J. Eur. Math. Soc. 12 (2010), 1497-1531] established the sufficient conditions on the existence of Ginzburg-Landau minimizers, given in terms of p, q and d. The present work complements the result of [J. Eur. Math. Soc. 12 (2010), 1497-1531] by providing the necessary conditions for the existence of nontrivial (nonconstant) minimizers.The solutions of (2) with isolated zeros (vortices) are of special importance since they model the observable physical states during phase transitions in superconductors.Minimization problems for Ginzburg-Landau type functionals have been extensively studied by a variety of authors. The pioneering work on modeling Ginzburg-Landau vortices is the work of Bethuel, Brezis and Hélein [8]. In this work the authors suggested to consider a simplified Ginzburg-Landau model (1), in which the physical source of vortices, the external magnetic field, is modeled via a Dirichlet boundary condition with a positive degree on the boundary. The analysis of full Ginzburg-Landau functional, with induced and applied magnetic fields, was later performed by Sandier and Serfaty in [14].

Given a bounded doubly connected domain G ⊂ R 2 , we consider a minimization problem for the Ginzburg-Landau energy functional when the order parameter is constrained to take S 1 -values on ∂G and have degrees zero and one on the inner and outer connected components of ∂G, correspondingly. We show that minimizers always exist for 0 < λ < 1 and never exist for λ 1, where λ is the coupling constant ( √ λ/2 is the Ginzburg-Landau parameter). When λ → 1 − 0 minimizers develop vortices located near the boundary, this results in the limiting currents with δ-like singularities on the boundary. We identify the limiting positions of vortices (that correspond to the singularities of the limiting currents) by deriving tight upper and lower energy bounds. The key ingredient of our approach is the study of various terms in the Bogomol'nyi's representation of the energy functional.