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

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.