L1-L1 estimates for a doubly dissipative semilinear wave equation

In this paper, we derive energy estimates and L1- L1 estimates, for the solution to the Cauchy problem for the doubly dissipative wave equation (Formula Presented.).The solution is influenced both by the diffusion phenomenon created by the damping term ut, and by the smoothing effect brought by the damping term - Δ ut. Thanks to these two effects, we are able to obtain linear estimates which may be effectively applied to find global solutions in any space dimension n≥ 1 , to the problems with power nonlinearities | u| p, | ut| p and | ∇ u| p, in the supercritical cases, by only assuming small data in the energy space, and with L1 regularity. We also derive optimal energy estimates and L1- L1 estimates, for the solution to the semilinear problems.