On the solution behavior of a nonlinear time-fractional Klein–Gordon equation: Theoretical study and numerical validation

Abstract

This work is devoted to the behavior analysis of solutions to a time-fractional Klein–Gordon equation in a multidimensional bounded domain. To get a more flexible representation, the diffusive formalism is adopted to deal with well-posedness issues. In case of negative initial energy, we discuss some conditions related to the solution blow-up in finite time. Prior to the blow-up phenomenon occurrence, we show that the solution component u holds under some conditions an exponential growth in Lp+1 norm. The validity of the theoretical findings is illustrated through the development of a finite difference approach for the one dimension problem, where the convergence of the proposed numerical method is also addressed.