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2013, 2013(special): 673-684. doi: 10.3934/proc.2013.2013.673

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  September 2012 Revised  March 2013 Published  November 2013

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: Gaussian noise, Q-regular space-time noise, Semilinear heat equations, Wiener process, SPDEs, cubic nonlinearity, trace formula., Fourier solutions, Lyapunov functions, total energy functional, conservation laws, approximating Fourier coefficients.
Mathematics Subject Classification: Primary: 34F05, 37H10, 60H10, 60H30; Secondary: 65C3.
Citation: Henri Schurz. Stochastic heat equations with cubic nonlinearity and additive space-time noise in 2D. Conference Publications, 2013, 2013 (special) : 673-684. doi: 10.3934/proc.2013.2013.673
References:
[1]

E. Allen, "Modeling with Stochastic Differential Equations," Springer-Verlag, New York, 2007.

[2]

L. Arnold, "Stochastic Differential Equations," John Wiley & Sons, Inc., New York, 1974

[3]

A. Bensoussan and R. Temam, Équations aux dérivées partielles stochastiques non linéaires. I. (in French), Israel J. Math., 11 (1972) 95-129.

[4]

A. Bensoussan, Some existence results for stochastic partial differential equations, in Stochastic partial differential equations and applications, (Trento, 1990), p. 37-53, Pitman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992.

[5]

P.L. Chow, "Stochastic Partial Differential Equations," Chapman & Hall/CRC, Boca Raton, FL, 2007.

[6]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, Cambridge, 1992.

[7]

G. Da Prato and J. Zabzcyk, "Ergodicity for Infinite Dimensional Systems," Cambridge University Press, Cambridge, 1996.

[8]

L.C. Evans, "Partial Differential Equations," AMS, Providence, 2010.

[9]

T.C. Gard, "Introduction to Stochastic Differential Equations," Marcel Dekker, Basel, 1988.

[10]

W. Grecksch and C. Tudor, "Stochastic Evolution Equations. A Hilbert space approach," Akademie-Verlag, Berlin, 1995.

[11]

A.L. Hodgkin and W.A.H. Rushton, The electrical constants of a crustacean nerve fibre, Proc. Roy. Soc. London. B 133 (1946) 444-479.

[12]

R.Z. Khasminskiĭ, "Stochastic Stability of Differential Equations," Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.

[13]

C. Koch, "Biophysics of Computation: Information Processing in Single Neurons," Oxford U. Press, Oxford, 1999.

[14]

C. Koch and I. Segev, "Methods in Neuronal Modeling: From Ions to Networks (2-nd edition)," MIT Press, Cambridge, MA, 1998.

[15]

E. Pardoux, Équations aux dérivées partielles stochastiques non linéaires monotones, PhD. Thesis, U. Paris XI, 1975.

[16]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3 (1979), no. 2, 127-167.

[17]

B.L. Rozovskii, "Stochastic Evolution Systems," Kluwer, Dordrecht, 1990.

[18]

H. Schurz, "Stability, Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations and Applications'', Logos-Verlag, Berlin, 1997.

[19]

H. Schurz, Nonlinear stochastic wave equations in $\mathbbR^1$ with power-law nonlinearity and additive space-time noise, Contemp. Math., 440 (2007), 223-242.

[20]

H. Schurz, Existence and uniqueness of solutions of semilinear stochastic infinite-dimensional differential systems with H-regular noise, J. Math. Anal. Appl., 332 (1) (2007), 334-345.

[21]

H. Schurz, Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), no. 2, 353-363.

[22]

H. Schurz, Nonlinear stochastic heat equations with cubic nonlinearities and additive Q-regular noise in $\mathbbR^1$, Electron. J. Differ. Equ. Conf., 19 (2010), 221-233.

[23]

A.N. Shiryaev, "Probability," Springer-Verlag, Berlin, 1996.

[24]

G.J. Stuart and B. Sakmann, Active propagation of somatic action potentials into neocortical pyramidal cell dendrites, Nature 367 (1994) 69-72.

[25]

H.C. Tuckwell and J.B. Walsh, Random currents through nerve membranes. I. Uniform poisson or white noise current in one-dimensional cables, Biol. Cybern., 49 (1983), no. 2, 99-110.

[26]

C. Tudor, On stochastic evolution equations driven by continuous semimartingales, Stochastics 23 (1988), no. 2, 179-195.

[27]

J.B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math., 1180, Springer, Berlin-New York, 1986, 265-439.

[28]

J.B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Anal., 23 (2005), no. 1, 1-43.

show all references

References:
[1]

E. Allen, "Modeling with Stochastic Differential Equations," Springer-Verlag, New York, 2007.

[2]

L. Arnold, "Stochastic Differential Equations," John Wiley & Sons, Inc., New York, 1974

[3]

A. Bensoussan and R. Temam, Équations aux dérivées partielles stochastiques non linéaires. I. (in French), Israel J. Math., 11 (1972) 95-129.

[4]

A. Bensoussan, Some existence results for stochastic partial differential equations, in Stochastic partial differential equations and applications, (Trento, 1990), p. 37-53, Pitman Res. Notes Math. Ser., 268, Longman Sci. Tech., Harlow, 1992.

[5]

P.L. Chow, "Stochastic Partial Differential Equations," Chapman & Hall/CRC, Boca Raton, FL, 2007.

[6]

G. Da Prato and J. Zabczyk, "Stochastic Equations in Infinite Dimensions," Cambridge University Press, Cambridge, 1992.

[7]

G. Da Prato and J. Zabzcyk, "Ergodicity for Infinite Dimensional Systems," Cambridge University Press, Cambridge, 1996.

[8]

L.C. Evans, "Partial Differential Equations," AMS, Providence, 2010.

[9]

T.C. Gard, "Introduction to Stochastic Differential Equations," Marcel Dekker, Basel, 1988.

[10]

W. Grecksch and C. Tudor, "Stochastic Evolution Equations. A Hilbert space approach," Akademie-Verlag, Berlin, 1995.

[11]

A.L. Hodgkin and W.A.H. Rushton, The electrical constants of a crustacean nerve fibre, Proc. Roy. Soc. London. B 133 (1946) 444-479.

[12]

R.Z. Khasminskiĭ, "Stochastic Stability of Differential Equations," Sijthoff & Noordhoff, Alphen aan den Rijn, 1980.

[13]

C. Koch, "Biophysics of Computation: Information Processing in Single Neurons," Oxford U. Press, Oxford, 1999.

[14]

C. Koch and I. Segev, "Methods in Neuronal Modeling: From Ions to Networks (2-nd edition)," MIT Press, Cambridge, MA, 1998.

[15]

E. Pardoux, Équations aux dérivées partielles stochastiques non linéaires monotones, PhD. Thesis, U. Paris XI, 1975.

[16]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics 3 (1979), no. 2, 127-167.

[17]

B.L. Rozovskii, "Stochastic Evolution Systems," Kluwer, Dordrecht, 1990.

[18]

H. Schurz, "Stability, Stationarity, and Boundedness of Some Implicit Numerical Methods for Stochastic Differential Equations and Applications'', Logos-Verlag, Berlin, 1997.

[19]

H. Schurz, Nonlinear stochastic wave equations in $\mathbbR^1$ with power-law nonlinearity and additive space-time noise, Contemp. Math., 440 (2007), 223-242.

[20]

H. Schurz, Existence and uniqueness of solutions of semilinear stochastic infinite-dimensional differential systems with H-regular noise, J. Math. Anal. Appl., 332 (1) (2007), 334-345.

[21]

H. Schurz, Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise, Discrete Contin. Dyn. Syst. Ser. S, 1 (2008), no. 2, 353-363.

[22]

H. Schurz, Nonlinear stochastic heat equations with cubic nonlinearities and additive Q-regular noise in $\mathbbR^1$, Electron. J. Differ. Equ. Conf., 19 (2010), 221-233.

[23]

A.N. Shiryaev, "Probability," Springer-Verlag, Berlin, 1996.

[24]

G.J. Stuart and B. Sakmann, Active propagation of somatic action potentials into neocortical pyramidal cell dendrites, Nature 367 (1994) 69-72.

[25]

H.C. Tuckwell and J.B. Walsh, Random currents through nerve membranes. I. Uniform poisson or white noise current in one-dimensional cables, Biol. Cybern., 49 (1983), no. 2, 99-110.

[26]

C. Tudor, On stochastic evolution equations driven by continuous semimartingales, Stochastics 23 (1988), no. 2, 179-195.

[27]

J.B. Walsh, An introduction to stochastic partial differential equations, Lecture Notes in Math., 1180, Springer, Berlin-New York, 1986, 265-439.

[28]

J.B. Walsh, Finite element methods for parabolic stochastic PDE's, Potential Anal., 23 (2005), no. 1, 1-43.

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