Energy identity for global solutions of space-time fractional heat equation with critical exponent

We derive the energy identity for global solutions of space-time fractional heat equations with critical exponents. Using the parabolic extension technique, we reformulate the problem as a local parabolic one with Neumann boundary conditions in the $ n+1 $ dimensional upper half space. First, we give a criterion for the nonexistence of global solutions and demonstrate that the $ \omega $-limit set is nonempty. Next, we generalize the concentration compactness principle to the $ \sigma $-extension problem, based on which we do the bubbling analysis. We then prove that as time goes to infinity, the energy converges to the sum of the energy for stationary solution and finitely many positive solutions of the limit equation.