Existence and uniqueness of solutions of a mixed problem for a symmetric hyperbolic system with discontinuous coefficients
The author considers in the cylindrical space-time domain \(\Omega\) \(\times [0,T]\) the first order linear differential system \((1)\quad (\partial /\partial t)(Au)+\sum^{n}_{i=1}A^ i\partial u/\partial x_ i=f,\) where the \(m\times m\) matrix A is symmetric and coercive and the \(m\times m\) matrices \(A^ i\) are symmetric, together with the initial condition (2) \(u(x,0)=h(x)\), \(x\in \Omega\), and with conservative boundary conditions (3) \({\mathfrak B}u=0\) on \(\partial \Omega \times [0,T]\). - The entries of the matrix A are bounded but may be strongly irregular whereas the entries of the matrices \(A^ i(i=1,...,n)\) are constant. With the aid of well-known results of the general theory of mixed first- order problems with regular coefficients, the author proves existence and uniqueness theorems for the weak \(L^ 2\)-solutions of problem (1)-(2)- (3). - On this ground a uniqueness theorem is also proved for a nonlinear variant of the same problem.
scientific article; zbMATH DE number 3895531
scientific article