On the regularizing effect of nonlinear damping in hyperbolic equations

Global well-posedness in H 2 ( R 3 ) × H 1 ( R 3 ) H^2(\mathbb {R}^3)\times H^1(\mathbb {R}^3) is shown for nonlinear wave equations of the form ◻ u + f ( u ) + g ( u t ) = 0 , \Box u+f(u)+g(u_t)=0, where t ∈ R + . t\in \mathbb {R}_+. The main assumption is that the nonlinear damping g ( u t ) g(u_t) behaves like | u t | m − 1 u t |u_t|^{m-1}u_t with m ≥ 2 m\geq 2 and the defocusing nonlinearity f ( u ) f(u) is like | u | p − 1 u |u|^{p-1}u with p ≥ 2. p\geq 2. The result also applies to certain exponential functions, such as f ( u ) = sinh ⁡ u . 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 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 ) H^1(\mathbb {R}^3)\times L^2(\mathbb {R}^3) is shown for the same equation in the critical case f ( u ) = u 5 f(u)=u^5 and g ( u t ) = | u t | 2 / 3 u t 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 ) ) . L^4(\mathbb {R}_+,L^{12}(\mathbb {R}^{3})).