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.
Citation
Masaki Tajiri. Tomio Umeda. "Representation formulas of the solutions to the Cauchy problems for first order systems." Osaka J. Math. 44 (1) 197 - 205, March 2007.
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