Abstract
We consider nonlinear diffusion equations of the form ∂tu = Δφ(u) in RN with N ≥ 2. When φ(s) ≡ s, this is just the heat equation. Let Ω be a domain in RN, where ∂Ω is bounded and ∂Ω=∂RNΩ̄. We consider the initial-boundary value problem, where the initial value equals zero and the boundary value equals 1, and the Cauchy problem where the initial data is the characteristic function of the set Ωc=RNΩ. We settle the boundary regularity issue for the characterization of the sphere as a stationary level surface of the solution u:, no regularity assumption is needed for ∂Ω.
Original language | English |
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Pages (from-to) | 2023-2032 |
Number of pages | 10 |
Journal | Mathematical Methods in the Applied Sciences |
Volume | 36 |
Issue number | 15 |
DOIs | |
Publication status | Published - 2013 Oct |
Keywords
- Cauchy problem
- heat equation
- initial behavior
- initial-boundary value problem
- nonlinear diffusion
- sphere
- stationary level surface
ASJC Scopus subject areas
- Mathematics(all)
- Engineering(all)