A nonlocal \(p\)-Laplacian evolution equation with nonhomogeneous Dirichlet boundary conditions. (English) Zbl 1183.35034
The authors study the existence, the uniqueness and a contraction property for solutions of the equation
\[
-{\partial u(x,t)\over\partial t}= \int_{\mathbb{R}^N} J(x- y)|u(x,t)- u(y,t)|^{p-2} (u(x,t)- u(y,t))\,dy,
\]
where \((x,t)\in\Omega\times (0,T)\) and \(J\) is a compactly supported suitable function. The function \(u(x,t)\) is prescribed when \(x\in \mathbb{R}^N\setminus\Omega\).
Some convergence properties are studied. It is indicated that the case \(p\to\infty\) has applications to the formation of sandpiles.
Some convergence properties are studied. It is indicated that the case \(p\to\infty\) has applications to the formation of sandpiles.
Reviewer: Peter Lindqvist (Trondheim)