Stochastic wave equations with polynomial nonlinearity. (English) Zbl 1017.60071

The author studies the stochastic wave equation with polynomial nonlinearity, \[ \partial^2_t u= \nabla^2 u+ f(u)+ \sigma(u)\partial_\tau W(t,x),x\in\mathbb{R}^d,\;t> 0,\quad u(0,x)= g(x),\quad \partial_t u(0,x)= h(x), \] \(d\leq 3\), where \(W(t,.)\) is a Wiener random field and the nonlinear terms \(f(u)\) and \(\sigma(u)\) are assumed to be polynomials in \(u\). The author proves an existence and uniqueness result of local and global solutions in Sobolev space \(H_1\) for a class of these stochastic equations. The proofs are based on an energy inequality for a linear stochastic wave equation. Nevertheless, in order to show that in general the global solution does not exist, the author shows a cubically nonlinear stochastic wave equation with an explosive solution.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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[1] CHOW, P., KOHLER, W. and PAPANICOLAOU, G. (1981). Multiple Scattering and Waves in Random Media. North-Holland, Amsterdam. · Zbl 0464.00013
[2] DA PRATO, G. and ZABCZYK, J. (1992). Stochastic Equations in Inifinite Dimensions. Cambridge Univ. Press. · Zbl 0761.60052
[3] JOHN, F. (1990). Nonlinear Wave Equations, Formation of Singularities. Amer. Math. Soc., Providence, RI. · Zbl 0716.35043
[4] KUNITA, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press. · Zbl 0743.60052
[5] METIVIER, M. and PELLAUMAIL, J. (1980). Stochastic Integration. Academic Press, New York. · Zbl 0426.60059
[6] MIZOHATA, S. (1973). The Theory of Partial Differential Equations. Cambridge Univ. Press. · Zbl 0263.35001
[7] MUELLER, C. (1997). Long time existence for the wave equation with a noise term. Ann. Probab. 25 133-151. · Zbl 0884.60054
[8] PARDOUX, E. (1975). Équations aux dérivees partielles stochastiques non linéaires monotones. These. Univ. Paris XI.
[9] PAZY, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York. · Zbl 0516.47023
[10] REED, M. (1976). Abstract Non-Linear Wave Equations. Lecture Notes in Math. 507. Springer, Berlin. · Zbl 0317.35002
[11] REED, M. and SIMON, B. (1975). Methods of Modern Mathematical Physics II. Academic Press, New York. · Zbl 0459.46001
[12] WALSH, J. (1984). An Introduction to Stochastic Partial Differential Equations. École d’dté de Probabilité de Saint Flour XIV. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · Zbl 0608.60060
[13] WHITHAM, G. (1974). Linear and Nonlinear Waves. Wiley, New York. · Zbl 0373.76001
[14] DETROIT, MICHIGAN 48202 E-MAIL: plchow@math.wayne.edu
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