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Global existence of classical solutions to systems of nonlinear wave equations with different speeds of propagation. (English) Zbl 0988.35106
The Cauchy problem is studied for the standard wave equation in \(\mathbb{R}^3\) with the nonlinear term depending explicitly on the amplitude as well as on the first and second derivatives. The main results on global solutions of the Cauchy problem are based on the two kinds of weighted \(L^\infty\) norms estimates for the linear wave equations. The results of Kato and Majda on local solutions of quasilinear symmetric hyperbolic systems are applied to the Cauchy problem. Finally, certain \(L^\infty\)-\(L^2\) estimates for the local solutions and two energy estimates are used to establish the theorem about the global solutions.

35L55 Higher-order hyperbolic systems
35L30 Initial value problems for higher-order hyperbolic equations
35L70 Second-order nonlinear hyperbolic equations