Abstract: By using the test function method, combining properties of the Heisenberg Laplace operator and K function, and utilizing relevant differential inequality techniques, we studied the rupture problem of the initial value problem solutions for  a class of wave equations (systems) with convolution terms and weak damping terms in n-dimensional space. The results show that there is no global weak solution to the problem when the initial value satisfies certain assumptions. This obtained result gives the influence of parameters in the convolution term on rupture of the solution, and provided a reference for the study of existence of solutions to wave equations (systems) with weak damping and convolution type nonlinear terms in high dimensional spaces.

Key words: weakly damped wave equations, Heisenberg group, test function method, rupture