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, ut(x, t) = ∫ Ω∪Ω0 J(x− y)|u(y, t)− u(x, t)|(u(y, t)− u(x, t)) dy for (x, t) ∈ Ω × R and u(x, t) = 0 in Ω0 × R. The proof of these estimates is based on bounds for the associated first eigenvalue. Mathematics Subject Classification (2010). 35B40, 35P15, 45K05.


Introduction
In this paper we consider the following evolution problem where Ψ p : R → R is given by Ψ p (s) = |s| p−2 s, p > 1, Ω is a connected and bounded domain, the kernel J is a nonnegative smooth symmetric radial function supported in the unit ball, B(0, 1), which is strictly positive in B(0, 1) (therefore, nonlocal problems governed with the fractional Laplacian are not included in this article) and Ω 0 is a measurable set (with positive measure) included in {x ∈ R N \ Ω : d(x, ∂Ω) < 1}.
Nonlocal evolution problems like (1.1) have been recently used to model diffusion processes in applied mathematics (for example, in population dynamics, phase transitions, elasticity models, etc.), see the survey [11] and the references [4], [5], [9], [10], [13], [16], [17] and [18].This problem can be viewed as a nonlocal analogous to the well known p−Laplacian diffusion problem v t = div(|∇v| p−2 ∇v) , we refer to [3] for a proof of the fact that the local p−Laplacian can be approximated by these kind of problems.Concerning the boundary condition, if Ω 0 = ∅, problem (1.1) is written as Observe that the diffusion takes place only in Ω.Moreover, using the symmetry of J we can integrate the equation to obtain that the total mass of the solution is constant in time.So, in this case, we say that we get a homogeneous Neumann boundary condition.This problem was studied in [2] (see also [1], [6], [7] and [8] for the linear case p = 2).
On the other hand, to get homogeneous Dirichlet boundary condition we take Ω 0 = R N \ Ω.In this case problem (1.1) becomes Observe that in this case the diffusion also takes place in Ω 0 where u is zero.Since J is supported in the unit ball, we also obtain homogeneous Dirichlet boundary condition taking Ω 0 = {x ∈ R N \ Ω : d(x, ∂Ω) < 1}.See [3] and [6] for a study of this problem.
Global well-posedness of this problem for u 0 ∈ L p (Ω) as well as a contraction principle can be found in [3].Moreover, it is proved there, see also the previously mentioned references, that for homogeneous Neumann boundary conditions the solution converge to the mean value of the initial condition while for homogeneous Dirichlet boundary conditions solutions converge to zero.
Our main aim here is to obtain upper bounds for the asymptotic decay of the solutions as t goes to infinity.Our main result is the following.Theorem 1.1.Assume that u 0 ∈ L ∞ (Ω) then 1.For p > 2, we have a polynomial decay, for every 0 ≤ r < ∞ there exists 2. For 1 < p ≤ 2, we have an exponential decay, for every 0 ≤ r < ∞ there exists γ > 0 such that For recent references dealing with decay rates for nonlocal evolution equations we refer to [6], [14] and the book [3].
The proof of our decay estimates is based on the positivity of the following infimum (that we will call form now on the first eigenvalue associated with our problem), In fact, for the linear case the L 2 -norm decay exponentially with a rate given by the first eigenvalue, that is, in (2) γ = λ 1,2 (Ω 0 ), for p = 2 and r = 1.
This constant λ 1,p (Ω 0 ) is the nonlocal analogous to the first eigenvalue for the local p−La-placian operator, ∆ p u = div(|∇u| p−2 ∇u), given by λ1 = inf , where Γ 0 is a nontrivial (in terms of p−capacity) subset of ∂Ω.Remark that, in contrast to what happens with λ1 , it is not known, due to lack of compactness, if the infimum in λ 1,p (Ω 0 ) is attained or not.This fact, the possible nonexistence of eigenfunctions for λ 1,p (Ω 0 ), forces us to work directly with the infimum.
In the linear case with Dirichlet boundary conditions, that is, p = 2 and Ω 0 = {x ∈ R N \ Ω : d(x, ∂Ω) < 1}, the infimum is achieved thanks to the results in [15].In this linear case the first eigenvalue (and its asymptotic behaviour in expanding domains) was studied in [12].
For general 1 ≤ p < ∞, the positivity of λ 1,p (Ω 0 ) was proved in [3] for the Dirichlet case.This positivity result is analogous to the existence of a positive constant for the classical Poincare inequality Ω |v| p ≤ C Ω |∇v| p valid for functions v ∈ W 1,p 0 (Ω).Here we show how to adapt the arguments for the Dirichlet case from [3] to obtain that λ 1,p (Ω 0 ) is strictly positive under mixed boundary conditions.Note that λ 1,p (Ω 0 ) is the best constant in a sort of nonlocal Poincare inequality.
In addition to positivity, we prove further properties of λ 1,p (Ω 0 ).In particular, we show the continuity with respect to Ω 0 and we find some bounds in terms of the measure of Ω 0 which implies that lim |Ω0|→0 λ 1,p (Ω 0 ) = 0 .In fact, we have ).Moreover, there exist constants C 1 , C 2 depending on J, Ω and p but not on Ω 0 such that where The rest of the paper is organized as follows: In Section 2, we prove the properties of λ 1,p (Ω 0 ) stated in Theorem 1.2, in particular we prove a Poincare type inequality which implies the positivity of the first eigenvalue; and, in Section 3, we show the decay bound for the evolution problem.

The first eigenvalue
In this section we study some properties of λ 1,p (Ω 0 ) in terms of Ω 0 .Since, Lemma 2.1.Assume that a sequence of sets Proof.We only have to remark that uniformly for y ∈ Ω.Therefore, for any u with u L p (Ω) = 1, we get as n → ∞.Now, we look for lower and upper estimates of λ 1,p (Ω 0 ).To do that, we need the following Poincare type inequality that shows that λ 1,p (Ω 0 ) is strictly positive.Lemma 2.2.There exists a positive constant λ such that Proof.Following [3] we cover the domain Ω∪Ω 0 with a finite family of disjoint sets, B j j = 0, 1, • • • , L and define Then, iterating this inequality and using that u = 0 in B 0 we get that where, Therefore, adding in j, we have the Poincare type inequality Now, we construct the family B j .To do that, we define the sets Observe that the function g(δ) = |Ω 0 ∩ Γ δ | is continuous and g(0) = |Ω 0 | and g(1) = 0, then we can fix a δ such that, Now, we define the two first sets of the family, for some a given bellow.Observe that by definition we can take α 1 = a.
In order to see that B 1 has a positive measure, we note that Thus, taking a = H(δ)/4C, we get In order to localize a piece of B 1 with positive measure we take a finite family of balls of radius ρ = 1/16 which cover Θ δ/2 .We denote this family as Then, there exists E i0 such that

Now, we define
Since the distance between x ∈ B 2 and y ∈ E i0 is small than 1/4 we have that J(s) ≥ CH(δ) . Thus, Moreover, the measure of B 2 is bounded from bellow independent of δ.
The rest of the sets is defined as and it is easy to see that there exists a positive constant K = K(Ω) such that, Summing up, we get that there exists three constants such that (2.3) Remark 2.3.In the particular case when J is the characteristic function, that is J(s) = χ B(0,1) (s), we get that for x ∈ Ω 0 ∩ Γ δ where Λ δ is spherical cap of height δ.Then, Lemma 2.4.The first eigenvalue satisfies Proof.The lower estimate is a consequence of the Poincare inequality (Lemma 2.2) and (2.3), while the upper estimate is obtained choosing as a test function This result and Remark 2.3 give us the following corollary.
Corollary 2.5.If J(s) = χ B(0,1) (s) we have that 1. for a general Ω 0 we have that there exists two positive constants such that < δ} for some small δ > 0, we get Proof.In the general case, we only observe that J is integrable, then as we wanted to prove.

Estimates for the decay of the associated evolution problem
In this section we study the asymptotic behavior of the solution of the problem (1.1).
Proof of Theorem 1.1.We want to obtain the decay rate in L r+1 (Ω).To this end, we multiply the equation by u r , with 0 < r < ∞ (to obtain a decay in L 1 (Ω) we multiply by the sign of u, we leave the details to the reader), and integrate to obtain Notice that for p = 2, αp = r + 1, then we have exponential decay of the L r norm, u L r+1 (Ω) ≤ u 0 L r+1 (Ω) e −C(p,r)λ1,p(Ω0)t .
For the case p > 2 we can use Jensen inequality to obtain .