Blow-up for coupled systems of semilinear damped wave equations on the Heisenberg group

The main purpose of this paper is to study behaviors of solutions to the Cauchy problem for coupled systems of semilinear damped wave equations with power nonlinearities on the Heisenberg group, where the Fujita critical exponent is determined. By means of the Bihari inequality as well as the Gagliardo-Nirenberg-type inequality and the contraction mapping principle, the local well-posedness of the Cauchy problem is successfully proved. Combining energy estimates and decay estimates together with a contradiction argument, the global existence of solutions is also demonstrated. At the same time, the blow-up results and upper bound estimation of the solutions is derived by using the test function technique. Our main new contribution is the derivation of a fundamental inequality utilized in weighted energy estimates, as well as the selection of exponents in the scaling of two convex functions for constructing test functions.