Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem

We obtain upper bounds for the decay rate for solutions to the nonlocal problem ∂tu(x, t) = ∫ Rn J(x, y)|u(y, t) – u(x, t)|p–2(u(y, t) – u(x, t))dy with an initial condition u0 ∈ L1(Rn)∩ L∞(Rn) and a fixed p > 2. We assume that the kernel J is symmetric, bounded (and therefore there is no regularizing effect) but with polynomial tails, that is, we assume a lower bounds of the form J(x, y)≥ c1|x – y|–(n+2σ ), for |x – y| > c2 and J(x, y) ≥ c1, for |x – y| ≤ c2. We prove that ‖u(·, t)‖Lq(Rn) ≤ Ct n (p–2)n+2σ (1– 1 q ) for q ≥ 1 and t large. MSC: 35K05; 45P05; 35B40


Introduction.
In this paper we deal with nonlocal Cauchy problems of the form

t))dy
for t ∈ R + and x ∈ R n with n ≥ 2, a fixed p > 2 and an initial condition u(x, 0) = u 0 (x) satisfying u 0 ∈ L 1 (R n )∩L ∞ (R n ). On the kernel J, we will always assume that it is a bounded and symmetric function defined for (x, y) ∈ R n × R n together with the integrability condition J(·, y) ∈ L 1 (R n ) for all y ∈ R n . Under these hypotheses existence and uniqueness of a solution follows from a fixed point argument as in [1].
Nonlocal problems have been recently widely used to model diffusion processes (see [6] and [5] for a general nonlocal vector calculus). Problem (1.1) and its stationary version have been considered recently in connection with real applications, for example to peridynamics or a recent model for elasticity. We quote for instance [2], [11], [12], [13], [14] and the recent book [1].
Our main goal here is to obtain upper bounds for the asymptotic behavior of the solution of (1.1) as t → +∞. It is expected that the diffusive nature of the equation implies that the solution goes to zero when t → +∞.
The main result of this paper reads as follows: Theorem 1.1. Let n ≥ 2, q ∈ [1, +∞) and σ ∈ (0, 1). Let J be a kernel satisfying (1.2). Then the solution of (1.1) associated to an initial condition where the constant C depends on u 0 , q, σ and n.
Let us end the introduction with some comments on the previous bibliography. For the linear case, p = 2, and for smooth kernels J with compact support, it is proven in [8] that the solution u of the equation (1.1) has the decay estimate for any q ∈ [1, ∞). Note that this decay rate is the same as the one that holds for solutions of the classical Heat equation. In the case of an equation in convolution form, that is when J(x, y) = K(x − y) with K a nonnegative radial function, not necessarily compactly supported, it is proven in [3] that the solutions of equations with the form (1.1) have the decay estimate provided the function K has a Fourier transform satisfying the expansionK(ξ) = 1 − A|ξ| 2σ + o(|ξ| 2σ ), where A > 0 is a constant. In this case the decay estimate is analogous to the one for the σ−order fractional heat equation with σ ∈ (0, 1). We also note that the convolution form of the equation allows the use of Fourier analysis to obtain this result. However, the use of Fourier analysis is not helpful here due to fact that our operator is not in convolution form. Despite of this difficulty, energy methods can be applied, see [8], [4]. We borrow ideas and techniques from these references. In particular we use Proposition 3.2 of [4] (whose proof is included here for completeness). However we have to point out that in [4] only the linear case, that is, p = 2, was treated, while here we deal with (1.1) for any p ≥ 2. For examples of kernels with exponential decay bounds we refer to [9] and [10].
The case 1 ≤ p < 2 remains open as well as the corresponding estimate for the L ∞ -norm.

Basic Facts and Preliminaries.
First, we need to introduce fractional Sobolev spaces and its seminorms, we refer to [7] for details. For σ ∈ (0, 1) and r ∈ [1, ∞), W σ,r (R n ) is the fractional Sobolev space of all L r (R n ) functions with finite fractional seminorm [v] σ,r , given by Under these definitions, we have the following fractional Sobolev-type inequality, there exists a constant C > 0 such that for each v ∈ W σ,r (R n ) with σr < n, it holds where s = nr/(n − σr) (see [7]).
First, we consider a positive smooth function ψ : R n → R with the following properties With the aid of this function, we split a function u into two parts. We will denote the "smooth" part of u as v and the remaining as w. We let Sometimes, for simplicity in the notation and where the context is clear, we will write u, v and w as functions depending only of x.
As a first property of this decomposition we have that each L r norm of the functions v and w is controlled by the corresponding norm of u.

Lemma 2.1.
Let v and w be given by (2.4). For each r ∈ (1, +∞), there exists Proof. We start with v. Denoting r ′ = r/(r − 1) the Hölder conjugate of r and using the definition of v, we have ∫ The inequality for w easily follows immediately from the triangular inequality in L r . Now we state a key result to get the desired estimate on the decay rate.
The constant C depends on ψ, β, r and n.
Proof. For the estimate concerning w, we have ∫ Applying Holder's inequality, we get ∫ Now we deal with the term with v. We split the fractional seminorm as |v(x) − v(y)| r |x − y| n+2β dxdy =: I ext + I int and look at these integrals separately. For I ext , using the definition of v we have Now, we can look at the measure µ(dz) = ψ(z)dz as a probability measure (because of (2.3)) and since the function t → |t| r is convex in R, we can apply Jensen's inequality on the dz−integral in right-hand side of the last expression to obtain which, after an application of Fubini's Theorem, gives Then, applying the changex = x − z,ỹ = y − z in the dxdy integral and using (2.3), we conclude Using this last expression, we obtain from the assumption (1.2) that Now we deal with I int . In this case, using the definition of v, we can write

Note that by using (2.3), we have for all
Thus, using this equality into (2.7), we get However, note that if |x − z| ≥ 2 in the dz integral, since |x − y| < 1 necessarily |y − z| > 1. Then, due to the fact that ψ is supported in the unit ball, the contribution of the integrand when |x − z| ≥ 2 is null in the dz integral. Taking this into account, applying Hölder's inequality into the dz−integral, we have By Fubini's Theorem we can write Using the regularity of ψ, we have and since r > 2β, we conclude that the last integral is convergent, obtaining which leads us to the following estimate for I int From this, it is easy to get which, by the use of (1.2), let us conclude that This last estimate together with (2.6) concludes the proof.

Proof of Theorem 1.1
As mentioned in the introduction, existence and uniqueness of solutions to problem (1.1) follows as in [1]. In fact, the symmetry, boundedness and integrability assumptions over J, allows us to perform a fixed point argument to obtain the following result whose proof is omitted.  1). This solution satisfies ||u(·, t)|| L 1 (R n ) ≤ ||u 0 || L 1 (R n ) and ||u(·, t)|| L ∞ (R n ) ≤ ||u 0 || L ∞ (R n ) for all t ≥ 0. Now, let us introduce the main idea behind the energy methods. To clarify the exposition, let us perform these computations in the local case and next see how we can adapt them to our nonlocal problem with the help of Proposition 2.2. Let us describe briefly how the energy method can be applied to obtain decay estimates for local problems. Let us begin with the simpler case of the estimate for solutions to the p−Lapacian evolution equation in L 2 -norm. Let u be a solution to If we multiply the equation by u and integrate in R n , we obtain If we use interpolation and that ∥u(·, t)∥ L 1 (R n ) ≤ C(u 0 ) for any t > 0, we have with α determined by .
follows. To obtain a decay bound for ∥u(·, t)∥ L q (R n ) we can use the same idea multiplying by u q−1 at the beginning.
Now we are ready to proceed with the proof of our main result.
Proof of Theorem 1.1. The symmetry assumption on J allows us to mimic this idea and use an energy approach in order to get Theorem 1.1. Roughly speaking, this assumption allows us to "integrate by parts" equation (1.1). For q = 1 the proof is finished by Theorem 3.1. For q > 1 we multiply the equation by q|u| q−2 u and integrate, obtaining the identity where we omitted the dependence on t of the function u for simplicity. Now we recall the following inequality (whose proof is straightforward): let q > 1 and a, b ̸ = 0. Then, there exists a constant C depending only on q, such that Hence, using this inequality into (3.1), we conclude Note that we get that the L q -norm of u is decreasing in t. At this point we would like to use Sobolev's inequality, that is not available due to the lack of regularizing effect of our nonlocal operator. Instead we will use Proposition 2.2 that involves a good control of the smooth part v (but we have to take care of the rough part w).
By the definition of v and w in (2.4), we have ) .
Concerning w we can also interpolate and obtain Note that we are using that p > 2 here. Now we use that to get (3.5) ||w|| q L q (R n ) ≤ C||w|| qγ L p+q−2 (R n ) , with C depending on u 0 , q, σ and n.