#### Title

General initial data for a class of parabolic equations including the curve shortening problem

#### Publication Title

Discrete and Continuous Dynamical Systems- Series A

#### ISSN

10780947

#### E-ISSN

15535231

#### Document Type

Article

#### Abstract

The Cauchy problem for a class of non-uniformly parabolic equations including (4) is studied for initial data with less regularity. When m ∈ (1, 2], it is shown that there exists a smooth solution for t > 0 when the initial data belongs to Lploc, p > 1. When m > 2, the same results holds when the initial data belongs to Wloc1,p, p ≥ m − 1. An example is displayed to show that a smooth solution may not exist when the initial data is merely in Lploc, p > 1. Solvability of the weak solution is also studied.

#### First Page

2963

#### Last Page

2986

#### Publication Date

3-1-2020

#### DOI

10.3934/dcds.2020157

#### Keywords

Cauchy problem, Curve shortening problem, Initial trace, Porous medium equations, Widder’s theorem

#### Recommended Citation

Chou, Kai Seng and Kwong, Ying Chuen, "General initial data for a class of parabolic equations including the curve shortening problem" (2020). *NIU Bibliography*. 564.

https://huskiecommons.lib.niu.edu/niubib/564

#### Department

Department of Mathematical Sciences