A problem of the Riemann-Hilbert type for a nonregular system of partial differential equations (Q1380245)

Let \(A(\xi)\), \(\xi\in \mathbb{R}\), be a polynomial matrix with constant coefficients. Let \(\rho (\xi)\) be the number of roots of the characteristic equation \(\text{det} (\lambda E_n-A(\xi)) =0\) \((E_n\) is the unit matrix) that belong to the domain \(\text{Re} \lambda\leq 0\). The main conditions for the matrix \(A(\xi)\) are as follows: \(\rho (\xi) =m\), \(\xi<0\); \(\rho(\xi) =k\), \(\xi >0\); for definiteness \(m>k\). The author considers the problem \[ {\partial u\over \partial t}= A(i \partial/ \partial x)u\quad \text{in } \mathbb{R}\times \mathbb{R}^+, \quad B(i\partial/ \partial x)u(x,0) =f(x,0), \quad \text{Re} \bigl(C (i\partial/ \partial x)u(x,0) \bigr)=g(x), \] where \(B(\xi)\) and \(C(\xi)\) are polynomial matrices of dimension \(k\times n\) and \((m-k) \times n\) respectively. The author proves a solvability theorem.