Decay estimates for nonlinear nonlocal diffusion problems in the whole space

In this paper, we obtain bounds for the decay rate in the L r (ℝ d )-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, . We consider a kernel of the form K ( x , y ) = ψ( y − a ( x )) + ψ( x − a ( y )), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a ( x ) = Ax . To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form . The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ℝ d : Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ 1, p 1/ p as p →∞.