Analysis of the sine-Gordon equation with a nonlinear $\delta$-potential

This paper is devoted to the analysis of the following nonlinear wave equation $$u_{tt} - u_{xx} + (1 + q\delta_0(x)) \sin u = 0,$$ where $\delta_0 = \delta_0(x)$ is the Dirac delta function centered at the origin and $q \in \mathbb{R}$ is a constant. Equations of this form arise in the study of propagating solitons in the presence of a localized inhomogeneity. It is proved that the Cauchy problem for this equation is globally well-posed in the energy space $H^1_{\sin} \times L^2$. A complete characterization of stationary waves in the energy space, based on the parameter $q$, is also provided. Finally, a criterion to determine the stability or instability of the stationary waves, which depends upon the sign of the parameter $q$, is established.