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On a construction of weak solutions to semilinear dissipative hyperbolic systems with the higher integrable gradients. (English) Zbl 0941.35044

The authors proves existence of a weak solution to the semilinear hyperbolic systems of the type \(\partial^2_tu-\text{div}(A\nabla u)+|\partial_tu|^{\gamma-2}\partial_tu=0\) in \(\Omega\subset\mathbb{R}^m\), \(2<\gamma< {2m\over m-2}\) \((u\) vector-valued), with Dirichlet boundary condition. The gradient of this solution \(u\) is in \(L^p_{\text{loc}}\) for \(p\in[2,2+ \varepsilon(\gamma,m))\). Rothe’s approximative solutions are used in the proof.
Reviewer: A.Doktor (Praha)

MSC:

35L55 Higher-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B99 Qualitative properties of solutions to partial differential equations
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