Well-posedness and regularity of the Cauchy problem for nonlinear   fractional in time and space equations
  The purpose is to study the Cauchy problem for non-linear in time and space pseudo-differential equations. These include the fractional in time versions of HJB equations governing the controlled scaled CTRW. As a preliminary step which is of independent interest we analyse the corresponding linear equation proving its well-posedness and smoothing properties. 
V. Kolokoltsov, M. Veretennikova
math.AP