BISWAS, A. and LORINCZI, J., 2019. Maximum principles for time-fractional Cauchy problems with spatially non-local components. Fractional Calculus and Applied Analysis, In Press.
We show a strong maximum principle and an Alexandrov-Bakelman-Pucci estimate for the weak solutions of a Cauchy problem featuring Caputo time-derivatives and non-local operators in space variables given in terms of Bernstein functions of the Laplacian. To achieve this, first we propose a suitable meaning of a weak solution, show their existence and uniqueness, and establish a probabilistic representation in terms of time-changed Brownian motion. As an application, we also discuss an inverse source problem.
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This research of AB was supported in part by an INSPIRE faculty fellowship and a DST-SERB