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

Department

Department of Mathematical Sciences

Link to Full Text

Share

COinS