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Mixed problems in several variables. (English) Zbl 0149.06602

Author’s summary: We solve the mixed initial-boundary value problem in the quarter-space \(t\ge 0\), \(x\ge 0\), for a first-order system of differential equations with constant, not necessarily symmetric, matrix coefficients, \[ LU = U_t + AU_x + \sum_{ j=1}^m B_jU_{yj} = 0,\qquad \det A\ne 0. \] The problem is well-posed for smooth initial data if and only if (1) for real \(\xi, \eta\) the matrix \(\xi A + \sum η_jB_j\) has real (not necessarily distinct) eigenvalues, and (2) on \(x=0\) the solution is required to lie in a linear space \(N\) of dimension equal to the number of negative eigenvalues of \(N\), such that, for all real \(\eta\) and for all \(\tau\) with positive real part, \(N\) is free of linear combinations of generalized eigenvectors of \(A^{-1}(\tau + i \sum_{ j=1}^m \eta_jB_j)\) corresponding to eigenvalues with positive real part. The proof is by a Laplace transformation in \(t\), Fourier transformation in \(y\), and then solving an ordinary differential equation in \(x\). Solutions are arbitrarily smooth if the data are sufficiently smooth. Information is obtained about domain of dependence and growth of solution with time.

MSC:

35-XX Partial differential equations
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