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On a method constructing solutions to hyperbolic partial differential equations. (English) Zbl 0909.35083

The author treats an initial-boundary value problem for a system of the type: \[ {\partial^2 u^i\over\partial t^2}- {\partial\over\partial x_\alpha} \Biggl(A^{\alpha\beta}_{ij}(x){\partial u^j\over\partial x_\beta}\Biggr)\quad\text{in }Q= (0, T)\times\Omega,\quad i=1,\dots, n \] with initial conditions \(u(0)= u_0\), \(\partial u/\partial t (0)= v_0\) and boundary conditions \(u= u_0\) on \(\partial\Omega\). Here \(\Omega\) is a bounded domain in \(\mathbb{R}^m\), \(m\geq 2\), \(T>0\) and the coefficients \(A^{\alpha\beta}_{ij}\) are bounded and measurable functions in \(\Omega\).
By Rothe’s approximation he constructs solutions of the problem. There is observed the minimum of a family of variational functionals, and for approximated solutions are established \(L^2\)-estimates of the second derivatives with respect to the time variable. Performing “local estimates”, the author obtains the higher uniform integrability of the spatial gradients of the approximate solutions and hence of the solutions.

MSC:

35L20 Initial-boundary value problems for second-order hyperbolic equations
35A35 Theoretical approximation in context of PDEs
35L55 Higher-order hyperbolic systems
35B45 A priori estimates in context of PDEs
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