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Nonlinear stochastic wave equation with Colombeau generalized stochastic processes. (English) Zbl 1024.60027

The authors use the theory of Colombeau-type generalized function spaces to overcome the multiplication problem in the stuy of the nonlinear Klein-Gordon equation with an additive and/or multiplicative stochastic process in one and three dimensions. The nonlinearity is given by a smooth function, polynomially bounded together with all its derivatives. In all cases, the solutions are obtained and proved to be unique. The normalizations of the initial data and stochastic processes appear to be crucial points for the existence of solutions of the above equations.
Reviewer: L.Vazquez (Madrid)

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35Q72 Other PDE from mechanics (MSC2000)
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[1] DOI: 10.1080/17442509608834039 · Zbl 0887.60069 · doi:10.1080/17442509608834039
[2] DOI: 10.1137/0523049 · Zbl 0757.35068 · doi:10.1137/0523049
[3] Gelfand I. M., Dokl. Akad. Nauk SSSR 102 pp 1065– (1955)
[4] Leandre R., Birkhäuser Prog. Probab. 31 pp 285– (1992)
[5] DOI: 10.1080/10652469808819152 · Zbl 0912.60072 · doi:10.1080/10652469808819152
[6] Oberguggenberger M., Birkhäuser 199 pp 319–
[7] DOI: 10.1007/s004400050257 · Zbl 04555164 · doi:10.1007/s004400050257
[8] Russo F., Kluwer 199 pp 329–
[9] DOI: 10.1090/S0273-0979-1992-00225-2 · Zbl 0767.35045 · doi:10.1090/S0273-0979-1992-00225-2
[10] Urbanik K., Studia Math. 16 pp 268– (1958)
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