Time Regularity for Nonlocal Parabolic Equations

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One of the phenomena that are exclusive to nonlocal parabolic equations is how the boundary data posed in the complement of a given domain might drastically affect the regularity of the solution. Consider the fractional heat equation of order $\sigma\in(0,2)$ \begin{alignat*}{3} u_t &= \Delta^{\sigma/2} u \quad &&\text{ in } \quad &&B_1\times\mathbb R\\ u &= g \quad &&\text{ on } \quad &&(\mathbb R^n \setminus B_1)\times\mathbb R \end{alignat*} If $g$ has a sudden discontinuity in time then it is expected that the nonlocal effect, transmitted into the equation by $\Delta^\sigma$, makes $u_t$ discontinuous in time. A specific example was presented by Chang-Lara and Dávila[1].


References

  1. Chang-Lara, Héctor; Dávila, Gonzalo (2014), "Regularity for solutions of non local parabolic equations", Calc. Var. Partial Differential Equations 49: 139--172, doi:10.1007/s00526-012-0576-2, ISSN 0944-2669, http://dx.doi.org/10.1007/s00526-012-0576-2