Solution to the Cauchy problem for quasi-hyperbolic systems by vector-valued convolution

Abstract

We solve the initial value problem for quasi-hyperbolic systems of partial differential equations with distributional initial data by vector-valued convolution. We show that every initial value problem of a quasi-hyperbolic system of partial differential equations with temperate initial data has a distributional solution in the sense of L. Hörmander which can be represented as a convolution of the fundamental matrix with the initial data. Moreover, we show that for systems that are correct in the sense of Petrovsky, the solution of the initial value problem is a differentiable distribution-valued function and the solution converges to the given initial data if converges to zero. The special case of the distributional Cauchy problem for the heat equation is considered by R. Dautray and J. L. Lions and by Z. Szmydt and, more recently, by M. A. Chaudhry and M. H. Kazi and by W. Kierat and K. Skornik. The use of the theory of vector-valued distributions allows for a generalization to systems of partial differential equations which are correct in the sense of Petrovsky or, yet more general, quasi-hyperbolic systems.

Keywords:

• quasi-hyperbolic systems
• distributional initial value problem
• correctness in the sense of Petrovsky

Acknowledgements

We thank our colleague Peter Wagner for valuable hints.