On the regularizing effect of nonlinear damping in hyperbolic equations

Abstract

Global well-posedness in H 2 (R 3 ) × H 1 (R 3 ) is shown for nonlinear wave equations of the form u + f (u) + g(u t ) = 0, where t ∈ R + . The main assumption is that the nonlinear damping g(u t ) behaves like |u t | m−1 u t with m ≥ 2 and the defocusing nonlinearity f (u) is like |u| p−1 u with p ≥ 2. The result also applies to certain exponential functions, such as f (u) = sinh u. It is observed that the nonlinear damping gives rise to a new monotone quantity involving the second-order derivatives of u and leading to a priori estimates for initial data of any size.Global well-posedness in H 1 (R 3 ) × L 2 (R 3 ) is shown for the same equation in the critical case f (u) = u 5 and g(u t ) = |u t | 2/3 u t . The main tool is a new estimate for the solution of the nonlinear equation in L 4 (R + , L 12 (R 3 )).