Critical Exponents for Differential Inequalities with Riemann-Liouville and Caputo Fractional Derivatives

Abstract

We find the critical exponents for global in time solutions to differential inequalities with power nonlinearities, supplemented by an initial data condition. The operator for which the differential inequality is studied contains a Caputo or Riemann-Liouville time derivative of fractional order and a sum of homogeneous spatial partial differential operators. In the special case of a fractional diffusive equation, the obtained critical exponents are sharp. In particular, global existence of small data solutions to the fractional diffusive equation with Caputo and Riemann-Liouville time derivative of order in (0, 1) and in (1, 2), holds for supercritical powers. The existence result for the superdiffusive case (α ∈ (1, 2)), which interpolates a semilinear heat equation and a semilinear wave equation, was recently obtained in the general setting by the author and his collaborators. We use a simple representation of Mittag-Leffler functions to show that global existence of small data solutions for supercritical powers also holds for to the subdiffusive equation with Caputo and Riemann-Liouville time derivative (α ∈ (0, 1)).