Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem
Boundary Value Problems volume 2014, Article number: 109 (2014)
We obtain upper bounds for the decay rate for solutions to the nonlocal problem with an initial condition and a fixed . 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 , for and , for . We prove that for and t large.
MSC: 35K05, 45P05, 35B40.
In this paper we deal with nonlocal Cauchy problems of the form
for and with , a fixed and an initial condition satisfying . As regards the kernel J, we will always assume that it is a bounded and symmetric function defined for together with the integrability condition for all . Under these hypotheses existence and uniqueness of a solution follow from a fixed point argument as in .
Nonlocal problems have been recently widely used to model diffusion processes (see  and  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 [4–8] and the recent book .
Our main goal here is to obtain upper bounds for the asymptotic behavior of the solution of (1.1) as . It is expected that the diffusive nature of the equation implies that the solution goes to zero when .
To obtain our results the key assumptions are the following lower bounds for J:
for certain constants , and . For simplicity we will assume .
The main result of this paper reads as follows.
Theorem 1.1Let, and. LetJbe a kernel satisfying (1.2). Then the solution of (1.1) associated to an initial conditiondecays inwith the upper bound
where the constantCdepends on, q, σ, andn.
Let us end the introduction with some comments on the previous bibliography. For the linear case, , and for smooth kernels J with compact support, it is proven in  that the solution u of (1.1) has the decay estimate
for any . 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 with K a nonnegative radial function, not necessarily compactly supported, it is proven in  that the solutions of equations with the form (1.1) have the decay estimate
provided the function K has a Fourier transform satisfying the expansion for , where is a constant. In this case the decay estimate is analogous to the one for the σ-order fractional heat equation, , with . 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. In spite of this difficulty, energy methods can be applied; see [9, 11]. We also mention the recent reference , where similar estimates can be found for nonlocal equations with unbounded kernels. We borrow ideas and techniques from these references. In particular we use Proposition 3.2 of  (whose proof is included here for completeness). However, we have to point out that in  and  only the linear case, that is, , was treated, while here we deal with (1.1) for any . For examples of kernels with exponential decay bounds we refer to  and . Finally, we point out that our results are also valid for unbounded kernels (in the spirit of ) since only lower bounds are assumed in (1.2).
The case remains open as well as the corresponding estimate for the -norm.
2 Basic facts and preliminaries
First, we need to introduce fractional Sobolev spaces and their seminorms, we refer to  for details. For and , is the fractional Sobolev space of all functions with finite fractional seminorm , given by
Under these definitions, we have the following fractional Sobolev-type inequality: there exists a constant such that, for each with , we have
where (see ).
First, we consider a positive smooth function 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 find that each norm of the functions v and w is controlled by the corresponding norm of u.
Lemma 2.1Letvandwbe given by (2.4). For each, we have
Proof We start with v. Denoting 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 . □
Now we state a key result to get the desired estimate on the decay rate.
Proposition 2.2Letand letbe a kernel satisfying (1.2), ψsatisfying (2.3), , and. Then there exists a constantsuch that, for allandv, wdefined in (2.4), we have
The constantCdepends onψ, β, r, andn.
Proof For the estimate concerning w, we have
Applying Hölder’s inequality, we get
Since ψ is supported in , we have for all , and, since J verifies for , there exists a constant C depending only on such that . Then
Now we deal with the term with v. We split the fractional seminorm as
and look at these integrals separately. For , using the definition of v we have
Now, we can think of the measure as a probability measure (because of (2.3)), and since the function 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 change , in the integral and using (2.3), we conclude
Using this last expression, we obtain from the assumption (1.2) that
Now we deal with . 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 with (2.7), we get
However, note that if in the dz integral, since necessarily . Then, due to the fact that ψ is supported in the unit ball, the contribution of the integrand when 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 , we conclude that the last integral is convergent, obtaining
which leads to the following estimate for :
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. □
3 Proof of Theorem 1.1
As mentioned in the introduction, existence and uniqueness of solutions to problem (1.1) follows as in . In fact, the symmetry, boundedness, and integrability assumptions over J allow us to perform a fixed point argument to obtain the following result, whose proof is omitted.
Theorem 3.1Let, then there exists a unique solutionof equation (1.1). This solution satisfiesandfor all.
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 -norm. Let u be a solution to
If we multiply the equation by u and integrate in , we obtain
Now we use Sobolev’s inequality
with to obtain
If we use interpolation and that for any , we have
with α determined by
Hence we get
from where the decay estimate
follows. To obtain a decay bound for we can use the same idea multiplying by 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’ (1.1). For the proof is finished by Theorem 3.1. For we multiply the equation by 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 and . Then there exists a constant C depending only on q, such that
Hence, using this inequality with (3.1), we conclude
Note that we find that the -norm of u is decreasing in t. At this point we would like to use Sobolev’s inequality, which is not available due to the lack of regularizing effect of our nonlocal operator. Instead we will use Proposition 2.2, which involves a good control of the smooth part v (but we have to take care of the rough part w).
Let us fix (the case will be tackled at the end of this proof using interpolation). By the definition of v and w in (2.4), we have
Now we note that v belongs to for all p. Hence, we can interpolate, obtaining
where θ is given by
Recalling and the Sobolev-type inequality (2.2), we obtain
where and the constant C depends on , q, σ, and n.
Concerning w we can also interpolate and obtain
with γ given by
Note that we are using that here. Now we use
with C depending on , q, σ, and n.
From (3.3), (3.4), and (3.5) we obtain
Now we use Proposition 2.2, with and , to obtain
and we conclude that
with . Since (recall that the -norm of the solution decreases) and , we have
Then, from (3.2), we obtain
from which it follows that
as we wanted to show.
Now we deal with the case . We can interpolate, obtaining
As we have already proved that
for . □
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This work was partially supported by MEC MTM2010-18128 and MTM2011-27998 (Spain).
The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
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Esteve, C., Rossi, J.D. & San Antolin, A. Upper bounds for the decay rate in a nonlocal p-Laplacian evolution problem. Bound Value Probl 2014, 109 (2014). https://doi.org/10.1186/1687-2770-2014-109