General initial data for a class of parabolic equations including the curve shortening problem
Discrete and Continuous Dynamical Systems- Series A
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.
Cauchy problem, Curve shortening problem, Initial trace, Porous medium equations, Widder’s theorem
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.
Department of Mathematical Sciences