A singular perturbed problem for semilinear wave equations with small parameter
The author considers the Cauchy problem: \(\delta u_{tt}-\Delta u+u_t= F(u,\nabla u)\), \(t=0:u= \varepsilon u_0(x)\), \(u_t=\varepsilon u_1(x)\), where \(\delta\) and \(\varepsilon\) are small positive parameters, \(\nabla\) is the gradient operator and \(\Delta u=\sum{\partial^2u \over\partial \chi^2_i}\), and \(F\) is a smooth function satisfying given conditions. The author uses the global iteration method to obtain a lower bound independent of \(\delta\) on the life span of the classical solutions to the above system, and compares the solution of the perturbed system to that of the unperturbed system i.e. whether \(\delta=0\) or \(\delta>0\). The author proves that with the exception for a small boundary layer, the solution of the system when \(\delta>0\) converges to the solution of the limit case, i.e. when \(\delta=0\). It is also shown, in a rigorous way, that the perturbation term \(\delta u_n\) has an appreciable effect only for a short time.
scientific article; zbMATH DE number 1492591