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

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

Author: Henri Schurz (Southern Illinois Univ., Carbondale, IL, USA)
 
Abstract: 
 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.