Well-posedness results for a class of semi-linear super-diffusive equations

Abstract

In this paper we investigate the following fractional order in time Cauchy problemThe fractional in time derivative is taken in the classical Caputo sense. In the scientific literature such equations are sometimes dubbed as fractional-in time wave equations or super-diffusive equations. We obtain results on existence and regularity of local and global weak solutions assuming that A is a nonnegative self-adjoint operator with compact resolvent in a Hilbert space and with a nonlinearity f ∈ C 1 (R) that satisfies suitable growth conditions. Further theorems on the existence of strong solutions are also given in this general context.

Local existence and nonexistence for fractional in time weakly coupled reaction-diffusion systems

“…where A is a differential operator which generates a strongly continuous semigroup on L 2 ðXÞ. Detailed results can be found in [1,3,8,12]. Let 1 p\1 and q 1 ; q 2 2 ½1; 1 be given constants.…”

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

“…For more explicit relation between H s () and H s (), one can consult Alvarez et al, 24 where it has been established that…”

“…For more explicit relation between and , one can consult Alvarez et al, 24 where it has been established that with Here, denotes the space of infinitely differentiable functions with compact support in . The space is concerned with the particular case s = 1/2.…”

Analysis of nonlinear fractional diffusion equations with a Riemann-liouville derivative

“…In [30], Metzler-Klafter showed that non-Markovian diffusion processes with a memory can be modeled by a time fractional diffusion equation. These time fractional equations have become a primary component in the fields of partial differential equations and has attracted much attention; see for example [27,18,3,13,21,15,19,30].…”