The critical exponent(s) for the semilinear fractional diffusive equation

TL;DR

This paper identifies two distinct critical exponents determining the global existence of small data solutions for a semilinear fractional diffusive equation with Caputo derivative, highlighting the impact of fractional order.

Contribution

It establishes the existence of two different critical exponents for the equation, depending on the initial data conditions, which is a novel insight for fractional diffusive equations.

Findings

Two critical exponents for global solutions are identified.

The second critical exponent appears when the second initial data is zero.

Linear estimates are derived to support the contraction argument.

Abstract

In this paper we show that there exist two different critical exponents for global small data solutions to the semilinear fractional diffusive equation with Caputo fractional derivative in time. The second critical exponent appears if the second data is assumed to be zero. This peculiarity is related to the fact that the order of the equation is fractional. To prove our result, we first derive Lr-Lq linear estimates for the solution to the inhomogeneous linear Cauchy problem and then we apply a contraction argument.