## Abstract

We shall study L^{2} energy conserved solutions to the heat equa- tion. We shall first establish the global existence, uniqueness and regularity of solutions to such nonlocal heat flows. We then extend the method to a fam- ily of singularly perturbed systems of nonlocal parabolic equations. The main goal is to show that solutions to these perturbed systems converges strongly to some suitable weak-solutions of the limiting constrained nonlocal heat flows of maps into a singular space. It is then possible to study further properties of such suitable weak solutions and the corresponding free boundary problem, which will be discussed in a forthcoming article.

Original language | English (US) |
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Pages (from-to) | 49-64 |

Number of pages | 16 |

Journal | Discrete and Continuous Dynamical Systems- Series A |

Volume | 37 |

Issue number | 8 |

State | Published - Aug 2017 |

## Keywords

- Global existence
- Nonlocal heat equation
- Singularly perturbed parabolic equations
- Suitable weak solutions

## ASJC Scopus subject areas

- Analysis
- Discrete Mathematics and Combinatorics
- Applied Mathematics