Global existence for nonlinear Klein-Gordon equations in infinite homogeneous waveguides in two dimensions
This paper deals with the Cauchy problem for the semilinear 2D Klein-Gordon equation \[ (\partial^2_t- \Delta_x- \Delta_g+ m^2)u= f(y, u,\partial_t u,\partial_x u,\nabla_g u)\tag{1} \] in infinite homogeneous waveguides \(\mathbb{R}^2_x\times\mathbb{M}\), \(m= \text{const}> 0\), where \(\mathbb{M}= (M,g)\) is a Zoll manifold, i.e. compact Riemannian manifold all of whose geodesics are periodic with the same period, \(\Delta_g\) is the Laplace-Beltrami operator on \(\mathbb{M}\) and \(f\) is a homogeneous polynomial of second degree. The initial data are ``sufficiently small'' and belong to \(C^\infty_0(\mathbb{R}^2\times \mathbb{M})\). Then it is proved that for almost all values of \(m\) the Cauchy problem for (1) admits a global smooth solution \(u\in C^\infty(\mathbb{R}_+\times \mathbb{R}^2\times\mathbb{M})\).
nonlinear Klein-Gordon equation in two dimensions
scientific article; zbMATH DE number 5149543
scientific article