Fractional semilinear damped wave equation on the Heisenberg group

Title: Fractional semilinear damped wave equation on the Heisenberg group

Abstract:  In this talk, we will address the Cauchy problem for the semilinear damped wave equation for the fractional  sub-Laplacian on the Heisenberg group with power-type nonlinearity. With the presence of a positive damping term and nonnegative mass term, we derive decay estimates for the solution of the homogeneous linear fractional damped wave equation on the Heisenberg group, for its time derivative, and for its space derivatives. We also discuss how these estimates can be improved when we consider additional regularity for the Cauchy data in the absence of the mass term.  Also, in the absence of a mass term, we prove the global well-posedness for a certain range of p. However,  in the presence of the mass term, the global (in time) well-posedness for small data holds for an improved range of p.  Finally, as an application of the linear decay estimates,  we investigate well-posedness for the Cauchy problem for a weakly coupled system with two semilinear fractional damped wave equations with a positive mass term on the Heisenberg group. This is a joint work with Dr. Aparajita Dasgupta and Dr. Shyam Swarup Mondal.