Lower bound of the time of existence for the nonlinear Klein-Gordon equation in one space dimension
Let \(u\) be a solution to the semilinear Klein-Gordon equation in one space dimension, \[ \partial^2_tu-\partial^2_x u+u=F(u, \partial_tu, \partial_xu, \partial_t \partial_xu, \partial^2_xu) \] where \(F\) is a quadratic nonlinearity and Cauchy data \(u(t=0)=\varepsilon u_0\), \(\partial_t u(t=0)= \varepsilon u\), are small in \(C_0^\infty\). The aim of the paper is to obtain an explicit lower bound of the existence time of the solution \((T)\): \[ \lim\inf_{\varepsilon\to 0}\varepsilon^2\log T_\varepsilon \geq A. \] This is an improvement of the known result for \(A=0\). The expression for \(A\) is directly computed from the Cauchy data and the given nonlinearity. The author first constructs the asymptotic solution and then derives the approximate solution. An interesting constraint on the nonlinearity (called null-condition) is obtained for which \(A\) is \(\infty\).
scientific article; zbMATH DE number 1339269
scientific article