  Nonlinear and nonlinear evolution equations of the form $u_t=\L u \pm|\nabla
u|^q$, where $\L$ is a pseudodifferential operator representing the
infinitesimal generator of a L\'evy stochastic process, have been derived as
models for growing interfaces in the case when the continuous Brownian
diffusion surface transport is augmented by a random hopping mechanism. The
goal of this paper is to study properties of solutions to this equation
resulting from the interplay between the strengths of the "diffusive" linear
and "hyperbolic" nonlinear terms, posed in the whole space $\bbfR^N$, and
supplemented with nonnegative, bounded, and sufficiently regular initial
conditions.
