Global existence and decay of small solutions in a viscous half Klein–Gordon equation

We establish global existence and decay of solutions of a viscous half Klein-Gordon equation with a quadratic nonlinearity considering initial data, whose Fourier transform is small in $L^1(\mathbb{R})\cap L^{\mathrm{\infty }}(\mathbb{R})$. Our analysis relies on the observation that nonresonant dispersive effects yield a transformation of the quadratic nonlinearity into a subcritical nonlocal quartic one, which can be controlled by the linear diffusive dynamics through a standard $L^1$-$L^{\mathrm{\infty }}$-argument. This transformation can be realized by applying the normal form method of Shatah or, equivalently, through integration by parts in time in the associated Duhamel formula.