Energy Estimates for Fractional Evolution Equations

Abstract: This work presents a more broadly applicable version of an energy inequality for weak solutions of evolution equations involving fractional time derivatives. Unlike the classical identity that relates the time derivative of the squared norm to the inner product of the derivative and the function itself, its fractional analogue typically requires a regularity condition that is difficult to verify in practice. We revisit this inequality in the setting of square integrable functions over an open domain and provide a new proof based solely on the assumption that the solution is essentially bounded in time. In our approach, the required regularity emerges naturally within the argument, rather than being imposed beforehand. This refinement expands the scope of applicability of the inequality and offers a rigorous foundation for using the Faedo Galerkin method in problems governed by fractional evolution equations. As an application, we prove existence, uniqueness, and regularity of weak solutions to the fractional heat equation.