ON THE EXISTENCE THEORY OF A TIME-SPACE FRACTIONAL KLEIN–GORDON–SCHRÖDINGER SYSTEM

Abstract

We analyze a nonlinear Klein–Gordon–Schrödinger system in ${ℝ}^n\times {ℝ}^{+}$, $n\ge 1$, in a space-time fractional setting, considering the time fractional variation in the Caputo sense, and a fractional spatial dispersion. Assuming general polynomial nonlinearities, we prove the existence of local and global mild solutions, as well as the asymptotic stability of global mild solutions, with initial data in a large class of singular spaces, namely, the weak $L^p$ spaces. We also derive the existence of local and global solutions in the same framework for the nonlinear time-space fractional Klein–Gordon equation.