  In this paper, we establish Gevrey class regularity of solutions to a class
of dissipative equations with an analytic nonlinearity in the whole space. This
generalizes the results of Ferrari and Titi in the periodic space case with
initial data in $L^2-$based Sobolev spaces to the $L^p$ setting and in the
whole space. Our generalization also includes considering rougher initial data,
in negative Sobolev spaces in some cases including the Navier-Stokes and the
subcritical quasi-geostrophic equations, and allowing the dissipation operator
to be a fractional Laplacian. Moreover, we derive global (in time) estimates in
Gevrey norms which yields decay of higher order derivatives which are optimal.
Applications include (temporal) decay of solutions in higher Sobolev norms for
a large class of equations including the Navier-Stokes equations, the
subcritical quasi-geostrophic equations, nonlinear heat equations with
fractional dissipation, a variant of the Burgers' equation with a cubic or
higher order nonlinearity, and the generalized Cahn-Hilliard equation. The
decay results for the last three cases seem to be new while our approach
provides an alternate proof for the recently obtained $L^p\, (1<p<2)$ decay
result for the Navier-Stokes equations by Bae, Biswas and Tadmor. These
applications follow from our global Gevrey regularity result for initial data
in \emph{critical spaces} with low regularity.
