Eighth Mississippi State - UAB Conference on Differential Equations and Computational Simulations. Electron. J. Diff. Eqns., Conference 19 (2010), pp. 221-233.

Nonlinear stochastic heat equations with cubic nonlinearities and additive Q-regular noise in R^1

Henri Schurz

Semilinear stochastic heat equations perturbed by cubic-type nonlinearities and additive space-time noise with homogeneous boundary conditions are discussed in R^1. The space-time noise is supposed to be Gaussian in time and possesses a Fourier expansion in space along the eigenfunctions of underlying Lapace operators. We follow the concept of approximate strong (classical) Fourier solutions. The existence of unique continuous L^2-bounded solutions is proved. Furthermore, we present a procedure for its numerical approximation based on nonstandard methods (linear-implicit) and justify their stability and consistency. The behavior of related total energy functional turns out to be crucial in the presented analysis.

Published September 25, 2010.
Math Subject Classifications: 34F05, 35R60, 37H10, 37L55, 60H10, 60H15, 65C30.
Key Words: Semilinear stochastic heat equations; cubic nonlinearities; additive noise; homogeneous boundary conditions; approximate strong solution; Fourier expansion; SPDE; existence; uniqueness; energy; Lyapunov functionals; numerical methods; consistency; stability.

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Henri Schurz
Department of Mathematics
Southern Illinois University, Carbondale (SIUC)
Carbondale, IL 62901-4408, USA
email: hschurz@math.siu.edu

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