Semilinear Stochastic Hyperbolic Systems in One Dimension

Abstract

The article is concerned with two types of solutions to the Cauchy problems for some semilinear stochastic hyperbolic systems in one space dimension. In the symmetric hyperbolic case, by means of an energy inequality, the existence of unique local and global solutions in a Sobolev space is proved. In addition, for some random regular hyperbolic systems with additive noise, based on the method of characteristics, it is shown that the system has a unique continuous mild solution. Such a solution may exist locally or globally depending on the smoothness of the random coefficients.

Key Words:

• Stochastic hyperbolic systems
• Energy inequalities
• Method of characteristics
• Martingales with spatial parameters

1991 Mathematics Subject Classification:

• 60G
• 62G
• 35R

Acknowledgments