A nonlocal p-Laplacian evolution equation with Neumann boundary conditions
In this paper we study the nonlocal p-Laplacian type diffusion equation,ut (t, x) = under(∫, Ω) J (x - y) | u (t, y) - u (t, x) |p - 2 (u (t, y) - u (t, x)) d y . If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div (| ∇ u |p - 2 ∇ u...
Guardado en:
| Autor principal: | |
|---|---|
| Publicado: |
2008
|
| Materias: | |
| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00217824_v90_n2_p201_Andreu http://hdl.handle.net/20.500.12110/paper_00217824_v90_n2_p201_Andreu |
| Aporte de: |
| Sumario: | In this paper we study the nonlocal p-Laplacian type diffusion equation,ut (t, x) = under(∫, Ω) J (x - y) | u (t, y) - u (t, x) |p - 2 (u (t, y) - u (t, x)) d y . If p > 1, this is the nonlocal analogous problem to the well-known local p-Laplacian evolution equation ut = div (| ∇ u |p - 2 ∇ u) with homogeneous Neumann boundary conditions. We prove existence and uniqueness of a strong solution, and if the kernel J is rescaled in an appropriate way, we show that the solutions to the corresponding nonlocal problems converge strongly in L∞ (0, T ; Lp (Ω)) to the solution of the p-Laplacian with homogeneous Neumann boundary conditions. The extreme case p = 1, that is, the nonlocal analogous to the total variation flow, is also analyzed. Finally, we study the asymptotic behavior of the solutions as t goes to infinity, showing the convergence to the mean value of the initial condition. © 2008 Elsevier Masson SAS. All rights reserved. |
|---|