Orbital stability of standing waves for semilinear wave equations with indefinite energy
The paper deals with the orbital stability of standing waves for the nonlinear Klein-Gordon equation \(u_{tt}-\Delta u+m(x)u +g(x,\left| u\right| )u=0\) in \(\mathbb{R}^{+}\times \Omega ,\) where \(\Omega \) denotes \(\mathbb{R}^{n}\) or a bounded domain or a compact manifold. The orbital stability is determined by the convexity of the lowest energy \(d(\omega)\) of the standing waves with the frequency \(\omega .\) The arguments rely on the conservation of energy and charge and the construction of suitable invariant manifolds of solution flows. Existence of standing waves for some special equations, e.g. the case \(m(x)\equiv 1, g=\left| u\right| ^{p-1}\) is obtained as the applications of the results.
scientific article; zbMATH DE number 5289203
scientific article