Uniform estimates for solutions of nonlinear Klein-Gordon equations
The estimates derived in this paper concern the solutions of a nonlinear Klein-Gordon equation \(u_{tt}-\Delta u+mu=g(u)\) in \(\Omega\), \(u=0\) in \(\partial \Omega\), where \(\Omega\) is an open subset of \({\mathbb{R}}^ N\), \(m>0\), and g satisfies some growth conditions. The main result of this paper is that under some superlinearity condition on g (namely if \(xg(x)\geq (2+\epsilon)\int^{x}_{0}g(s)ds)\) then all global solutions are uniformly bounded in \(H^ 1_ 0(\Omega)\). Furthermore, for large time, the bound only depends on the energy of the initial data, that is, on the quantity \[ E(u_ 0,v_ 0)=\int_{\Omega}\{| v_ 0(x)|^ 2+| \nabla u_ 0(x)|^ 2+m/2| u_ 0(x)|^ 2-G(u_ 0(x))\}dx. \] In addition, solutions on the whole line, that is, solutions that exist for \(t\in]-\infty,+\infty [\), are uniformly bounded in terms of the energy of the initial data. As simple consequences we obtain conditions on the initial data under which the solution of (K.G) blows up in a finite time. Similar results were obtained for certain types of ordinary differential equation and for parabolic equations. Here the author demonstrates existence of such phenomena for the K-G equation and conjectures possible extensions.
scientific article; zbMATH DE number 3907001