Eighth Mississippi State  UAB Conference on Differential Equations
and Computational Simulations.
Electron. J. Diff. Eqns., Conference 19 (2010), pp. 221233.
Nonlinear stochastic heat equations with cubic nonlinearities
and additive Qregular noise in R^1
Henri Schurz
Abstract:
Semilinear stochastic heat equations perturbed by cubictype
nonlinearities and additive spacetime noise with homogeneous
boundary conditions are discussed in R^1.
The spacetime 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^2bounded
solutions is proved. Furthermore, we present a procedure for
its numerical approximation based on nonstandard methods
(linearimplicit) 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 629014408, USA
email: hschurz@math.siu.edu

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