Fractal Hamilton-Jacobi-KPZ equations
  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. 
Grzegorz Karch and Wojbor A. Woyczynski