Carleman estimate for a fractional diffusion equation with half order and application

Abstract

We consider a fractional diffusion equation in x ∈ (0, ℓ) where the derivative in time t is of half order in the sense of Caputo and we establish a Carleman estimate. Since the derivatives of non-natural number orders do not satisfy the integration by parts, which is essential for establishing a Carleman estimate, we twice apply the Caputo derivative to convert the original fractional diffusion equation to a system with a usual partial differential operator: . Next we apply the Carleman estimate to prove the conditional stability in a Cauchy problem with data u(0, t), , 0 < t < T.

Keywords:

• fractional diffusion equation
• Carleman estimate
• Cauchy problem
• conditional stability

AMS Subject Classifications:

• 34K37
• 35R30
• 35R25