Global existence of solutions of the time fractional Cahn–Hilliard equation in $${\mathbb {R}}^3$$

Abstract

Cauchy problem for the Caputo-type time fractional Cahn–Hilliard equation in $${\mathbb {R}}^3$$ is examined. The local existence and uniqueness of mild solutions and strong solutions are obtained for the initial data $$u_0$$ satisfying $$u_0-{\bar{u}}\in L^\infty ({\mathbb {R}}^3)\cap L^1({\mathbb {R}}^3)$$ , where $${\bar{u}}$$ is an equilibrium constant. The local solutions are extended globally if $$u_0-{\bar{u}}$$ is small in $$L^1({\mathbb {R}}^3)$$ . These results are consistent with those of the traditional Cahn–Hilliard equation such as the property of mass conservation. However, extra difficulties arise in dealing with the singularity of Mittag-Leffler operators and non-Markovian property in the Caputo-type time fractional problem.