Existence of weak solutions for an abstract Cauchy problem involving the fractional Laplacian

Existence of weak solutions for an abstract Cauchy problem

Resumen

In this paper, we construct a theory of existence and uniqueness of solutions for an abstract Cauchy problem $ u'(t)+Au(t) = h(t)$, $t\in(0,T )$ and $u(0)=f$, where $A$ is the fractional Laplacian operator acting on a bounded domain with smooth boundary. The construction of the solution is based on the method of temporal discretization.