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

Henri Schurz

doi:10.3934/proc.2013.2013.673

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2013, Volume 2013: 673-684. Doi: 10.3934/proc.2013.2013.673 open access

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

Henri Schurz^1,

1.
Southern Illinois University, Department of Mathematics, MC 4408, 1245 Lincoln Drive, Carbondale, IL 62901-7316

Received: August 31, 2012

Revised: February 28, 2013

Published: October 2013

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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.

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Mathematics Subject Classification: Primary: 34F05, 37H10, 60H10, 60H30; Secondary: 65C30.

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