Fractal Hamilton-Jacobi-KPZ equations

Nonlinear and nonlocal evolution equations of the form $u_t=\mathcal{L} u \pm\vert\nabla u\vert^q$, where $\mathcal{L}$ is a pseudodifferential operator representing the infinitesimal generator of a Lévy 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 $\mathbb{R}^N$, and supplemented with nonnegative, bounded, and sufficiently regular initial conditions.