Talk:Open problems and Time Regularity for Nonlocal Parabolic Equations: Difference between pages
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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<ref name="MR3148110"/>. | |||
== References == | |||
{{reflist|refs= | |||
<ref name="MR3148110">{{Citation | last1=Chang-Lara | first1= Héctor | last2=Dávila | first2= Gonzalo | title=Regularity for solutions of non local parabolic equations | url=http://dx.doi.org/10.1007/s00526-012-0576-2 | journal=Calc. Var. Partial Differential Equations | issn=0944-2669 | year=2014 | volume=49 | pages=139--172 | doi=10.1007/s00526-012-0576-2}}</ref> | |||
}} | |||
Revision as of 11:09, 8 July 2016
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
- ↑ 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