GLOBAL NONEXISTENCE OF SOLUTIONS FOR EQUATIONS WITH A NONLINEAR MEMORY TERM ON THE HEISENBERG GROUP

Abstract

This paper investigates the nonexistence of global solutions for a nonlinear hyperbolic equation with a memory term posed on the Heisenberg
group. Using the modified test function method, combined with Youngs
inequality and tools from fractional calculus, we establish sufficient conditions guaranteeing finite-time blow-up of weak solutions. The analysis
relies on the structure of the Heisenberg group, the properties of the subLaplacian, and the interplay between nonlinear damping, source terms,
and fractional integral operators. A detailed study of auxiliary test functions is developed, allowing precise estimates of all nonlinear and memory contributions. By deriving sharp integral inequalities and identifying
critical exponents, we show that global solutions cannot exist whenever
the nonlinearities exceed certain thresholds connected to the geometry of
the underlying group. Our results improve and generalize several known
blow-up criteria for fractional and ultra-parabolic models. These findings highlight the significant influence of nonlinear memory kernels and
the non-Euclidean geometry of the Heisenberg group on the qualitative
behavior of solutions.