Abstract
The authors show the existence and uniqueness of solution for a class of stochastic wave equations with memory. The decay estimate of the energy function of the solution is obtained as well.
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Alabau-Boussouira, F., Cannarsa, P. and Sforza, D., Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254, 2008, 1342–1372.
Barbu, V., Da Prato, G. and Tubaro, L., Stochastic wave equations with dissipative damping, Stochastic Process. Appl., 117, 2007, 1001–1013.
Bo, L. J., Tang, D. and Wang, Y. J., Explosive solutions of stochastic wave equations with damping on Rd, J. Diff. Eqs., 244(1), 2008, 170–187.
Cannarsa, P. and Sforza, D., An existence result for semilinear equations in viscoelasticity: the case of regular kernels, Mathematical Models and Methods for Smart Materials, Series on Advances in Mathematics for Applied Sciences, Vol. 62, B. Fabrizio, B. Lazzari and A. Morro (eds.), World Scientific Publishing, Singapore, 2002, 343–354.
Cardon-Weber, C., Cahn-Hilliard stochastic equation: existence of the solution and of its density, Bernoulli, 7(5), 2001, 777–816.
Cavalcanti, M. and Oquendo, H., Frictional versus viscoelastic damping in a semilinear wave equation, SIAM J. Control Optim., 42(4), 2003, 1310–1324.
Chow, P.-L., Stochastic wave equations with polynomial nonlinearity, Ann. Appl. Probab., 12(1), 2002, 361–381.
Chow, P.-L., Asymptotics of solutions to semilinear stochastic wave equations, Ann. Appl. Probab., 16(2), 2006, 757–789.
Chow, P.-L., Asymptotic solutions of a nonlinear stochastic beam equation, Discrete Contin. Dyn. Syst. Ser. B., 6(4), 2006, 735–749.
Dafermos, C. M., An abstract Volterra equation with application to linear viscoelasticity, J. Diff. Eqs., 7, 1970, 554–589.
Da Prato, G. and Zabczyk, J., Stochastic Equations in Infinite Dimensions, Cambridge University Press, Cambridge, 1992.
Prüss, J., Evolutionary Intergral Equations and Applications, Monographs in Mathematics, Vol. 87, Birkhäuser, Basel, 1993.
Rivera, J. E. M., Asymptotic behaviour in linear viscoelasticity, Quart. Apll. Math., 52, 1994, 629–648.
Rivera, J. E. M. and Salvatierra, A. P., Asymptotic behaviour of the energy in partially viscoelastic materials, Quart. Appl. Math., 59(3), 2001, 557–578.
Walsh, J. B., An introduction to stochastic partial differential equations, École d’Été de Probabilités de Saint Flour XIV—1984, Lecture Notes in Mathematics, 1180, Springer-Verlag, Berlin, 1986, 265–439.
Zuazua, E., Exponetial decay for the semilinear wave equation with locally ditributed damping, Comm. Part. Diff. Eqs., 15(2), 1990, 205–235.
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Project supported by the National Natural Science Foundation of China (No. 10871103).
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Wei, T., Jiang, Y. Stochastic wave equations with memory. Chin. Ann. Math. Ser. B 31, 329–342 (2010). https://doi.org/10.1007/s11401-009-0170-x
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DOI: https://doi.org/10.1007/s11401-009-0170-x