A WEIGHTED SOBOLEV SPACE THEORY FOR THE DIFFUSION-WAVE EQUATIONS WITH TIME-FRACTIONAL DERIVATIVES ON C( )(1)DOMAINS

Abstract

We introduce a weighted L-p-theory (p > 1) for the time-fractional diffusion-wave equation of the type partial derivative(alpha )(t)u(t, x) = a(ij) (t, x)u(x)i(x)j (t, x) f (t, x), t > 0, x is an element of Omega, where alpha is an element of (0,2), partial derivative(alpha)(t) denotes the Caputo fractional derivative of order alpha, and Omega is a C-1 domain in R-d. We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.