We study the dynamics for the focusing nonlinear Klein Gordon equation with a positive, singular, radial potential and initial data in energy space. More precisely, we deal with utt - Δu + m2u = |x|-a|u|p-1u with 0 < a < 2. In dimension d ≥ 3, we establish the existence and uniqueness of the ground state solution that enables us to define a threshold size for the initial data that separates global existence and blow-up. We find a critical exponent depending on a. We establish a global existence result for subcritical exponents and subcritical energy data. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary sets.
ASJC Scopus subject areas
- Applied Mathematics