On the regularizing effect of nonlinear damping in hyperbolic equations

Abstract:

Global well-posedness in $H^2(\mathbb {R}^3)\times H^1(\mathbb {R}^3)$ is shown for nonlinear wave equations of the form $\Box u+f(u)+g(u_t)=0,$ where $t\in \mathbb {R}_+.$ The main assumption is that the nonlinear damping $g(u_t)$ behaves like $|u_t|^{m-1}u_t$ with $m\geq 2$ and the defocusing nonlinearity $f(u)$ is like $|u|^{p-1}u$ with $p\geq 2.$ 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 $H^1(\mathbb {R}^3)\times L^2(\mathbb {R}^3)$ 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 $L^4(\mathbb {R}_+,L^{12}(\mathbb {R}^{3})).$