Surfaces arising in amorphous thin-film-growth are often described by certain classes of stochastic PDEs. In this paper we address the question of existence of a solution and statistical quantities (e.g. mean interface width or correlation functions). Moreover, we discuss the approximations of such statistical quantities by the spectral Galerkin method. This is an important question, as the numerical computation of statistical quantities plays a key role in the verification of the models.
We consider the stochastic Burgers equationwith periodic boundary conditions, where t ≥ 0, r ∈ [0, 1], and η is some spacetime white noise. A certain Markov jump process is constructed to approximate a solution of this equation.
We establish the existence of time-periodic solutions of semi-linear wave equations on the unit sphere in R 3 . The problem has been studied previously in (1982, V. Benci and Fortunato, Ann. Mat. Pura Appl. 132, 215 242) using variational techniques. Our results here are much sharper: We employ delicate methods of bifurcation theory (1979, H. Kielho fer, J. Math. Anal. Appl. 68, 408 420; 1987, H. Kielho fer and P. Ko tzner, J. Appl. Math. Phys. 38, 201 212) combined with well-known group-theoretic ideas to find branches of small-amplitude solutions. Precise spatio-temporal patterns of solutions are uncovered as a by-product of the analysis. Moreover, in certain cases we prove the existence of global solution branches (in the sense of P. Rabinowitz).
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