On the existence and absence of global solutions of the first Darboux problem for nonlinear wave equations (Q946124)

The first Darboux problem to the equation \(u_{tt}-u_{xx}+\lambda u\left| u\right| ^{\alpha }=f(x,t)\) is considered. Existence of a global strong generalized solution of the problem for either \(-1<\alpha<0\) or \(\alpha ,\lambda >0\) is proved. In the case of \(\lambda <0,\alpha >0\) the problem has no global strong generalized solution, but under some assumptions on \(f(x,t)\) local solution exists.