Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D

Pages: 673 - 684, Issue special, November 2013

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Henri Schurz - Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316, United States (email)

Abstract: Semilinear heat equations on rectangular domains in $\mathbb{R}^2$ (conduction through plates) with cubic-type nonlinearities and perturbed by an additive Q-regular space-time white noise are considered analytically. These models as 2nd order SPDEs (stochastic partial differential equations) with non-random Dirichlet-type boundary conditions describe the temperature- or substance-distribution on rectangular domains as met in engineering and biochemistry. We discuss their analysis by the eigenfunction approach allowing us to truncate the infinite-dimensional stochastic systems (i.e. the SDEs of Fourier coefficients related to semilinear SPDEs), to control its energy, existence, uniqueness, continuity and stability. The functional of expected energy is estimated at time $t$ in terms of system-parameters.

Keywords:  Semilinear heat equations, SPDEs, cubic nonlinearity, Q-regular space-time noise, Gaussian noise, Wiener process, Lyapunov functions, total energy functional, Fourier solutions, approximating Fourier coefficients, conservation laws, trace formula.
Mathematics Subject Classification:  Primary: 34F05, 37H10, 60H10, 60H30; Secondary: 65C30.

Received: September 2012;      Revised: March 2013;      Published: November 2013.