Time fractional diffusion equations: solution concepts, regularity, and long-time behavior

By Rico Zacher. Fractional Differential Equations, 2019

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In this paper we give a survey of results on various analytical aspects of time fractional diffusion equations. We describe the approach via abstract Volterra equations and collect results on strong solutions in the $L_p$ sense. We further discuss the concept of weak solutions for equations with rough coefficients and give an account of recent developments towards a De Giorgi-Nash-Moser theory for such equations. The last part summarizes recent results on the long-time behaviour of solutions, which turns out to be significantly different from that in the heat equation case.

Subfields: Applied Mathematics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/56458931e58dc0060b76d4fd5aee6cd24bb7d39b