Representation formulas of the solutions to the Cauchy problems for first order systems

Abstract

Representation formulas of the solutions to the Cauchy problems for first order systems of the forms $\partial u/\partial t- \sum_{j=1}^{d} A_j(t) \partial u/ \partial x_j -A_0(t) u=f$ are established. The coefficients $A_j$'s are assumed to be matrix-valued functions of the forms $A_j(t) = \alpha_j(t) I + \beta_j(t) M_j$, where $\alpha_j(t), \beta_j(t)$, $j=1,\ldots,d$, are real-valued continuous functions, the eigenvalues of the matrices $M_j$, $j=1,\ldots,d$, are real, and the commutators $[M_j, M_{\ell}] = 0$ for all $j,\ell =0,1,\ldots,d$. No restrictions on the multiplicities of the characteristic roots are imposed.