Global weak entropy solutions to quasilinear wave equations of Klein-Gordon and sine-Gordon type
The objective of the paper is to prove the existence of global Lipschitz-continuous solutions to the Cauchy problem for the hyperbolic nonlinear partial differential equations of the following form: \[ w_{tt} -(\sigma (w_x))_x + f(w) = 0, \] where \(f\) is a smooth nonlinear function which is subject to the condition \(f(0) = 0\), and \(\sigma\) is another smooth function with a positive first derivative, satisfying an additional condition \(u\sigma'' (u) > 0\) for \(u\neq 0\). The existence is proved for all values of time, \(0 < t < +\infty\), and for the solutions on the whole axis, \(| x| < \infty\). For the proof, the equation is rewritten as a hyperbolic system. Then, an appropriate implicit finite-difference scheme for that system is introduced, which allows the authors to define some approximating solutions. The existence of the latter solution is established, and, finally, it is proved that the solution to the underlying equation also exists as a converging limit of the approximating solutions.
scientific article; zbMATH DE number 1185908