Decay estimates for a nonlocal $p-$Laplacian evolution problemwith mixed boundary conditions

Raúl Ferreira; Julio D. Rossi

doi:10.3934/dcds.2015.35.1469

Article Contents

2015, Volume 35, Issue 4: 1469-1478. Doi: 10.3934/dcds.2015.35.1469

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Decay estimates for a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions

Raúl Ferreira^1, and
Julio D. Rossi^2,

1.
Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid
2.
Departamento de Análisis Matemático, Universidad de Alicante, Ap 99, 03080, Alicante

Received: March 2013

Revised: October 2013

Published: May 2017

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Abstract

In this paper we prove decay estimates for solutions to a nonlocal $p-$Laplacian evolution problem with mixed boundary conditions, that is, $$u_t (x,t)=\int_{\Omega\cup\Omega_0} J(x-y) |u(y,t)-u(x,t)|^{p-2}(u(y,t)-u(x,t))\, dy$$ for $(x,t)\in \Omega\times \mathbb{R}^+$ and $u(x,t)=0$ in $ \Omega_0\times \mathbb{R}^+$. The proof of these estimates is based on bounds for the associated first eigenvalue.

Keywords:

Nonlocal diffusion,
eigenvalues.

Mathematics Subject Classification: 35B40, 35P15, 45K05.

Citation:

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References

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