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

for $t\in {\mathbb{R}}_{+}$ and $x\in {\mathbb{R}}^n$ with $n\ge 2$, a fixed $p>2$ and an initial condition $u(x,0)=u_0(x)$ satisfying $u_0\in L^1({\mathbb{R}}^n)\cap L^{\mathrm{\infty }}({\mathbb{R}}^n)$. As regards the kernel J, we will always assume that it is a bounded and symmetric function defined for $(x,y)\in {\mathbb{R}}^n\times {\mathbb{R}}^n$ together with the integrability condition $J(\cdot ,y)\in L^1({\mathbb{R}}^n)$ for all $y\in {\mathbb{R}}^n$. Under these hypotheses existence and uniqueness of a solution follow from a fixed point argument as in [1].

Nonlocal problems have been recently widely used to model diffusion processes (see [2] and [3] 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 [1].

Our main goal here is to obtain upper bounds for the asymptotic behavior of the solution of (1.1) as $t\to +\mathrm{\infty }$. It is expected that the diffusive nature of the equation implies that the solution goes to zero when $t\to +\mathrm{\infty }$.

for certain constants $c_1,c_2>0$, and $\sigma \in (0,1)$. For simplicity we will assume $c_2=1$.

The main result of this paper reads as follows.

where the constantCdepends on$u_0$, q, σ, andn.

provided the function K has a Fourier transform satisfying the expansion $\hat{K}(\xi )=1-A{|\xi |}^{2\sigma }+o({|\xi |}^{2\sigma })$ for $\xi \sim 0$, where $A>0$ is a constant. In this case the decay estimate is analogous to the one for the σ-order fractional heat equation, $v_t=-{(-\mathrm{\Delta })}^{\sigma }v$, with $\sigma \in (0,1)$. 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. In spite of this difficulty, energy methods can be applied; see [9, 11]. We also mention the recent reference [12], 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 [11] (whose proof is included here for completeness). However, we have to point out that in [12] and [11] only the linear case, that is, $p=2$, was treated, while here we deal with (1.1) for any $p\ge 2$. For examples of kernels with exponential decay bounds we refer to [13] and [14]. Finally, we point out that our results are also valid for unbounded kernels (in the spirit of [12]) since only lower bounds are assumed in (1.2).

The case $1\le p<2$ remains open as well as the corresponding estimate for the $L^{\mathrm{\infty }}$-norm.

where $s=nr/(n-\sigma r)$ (see [15]).

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 $L^r$ norm of the functions v and w is controlled by the corresponding norm of u.

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 constantCdepends onψ, β, r, andn.

where $r^{\prime }=r/(r-1)$.

This last estimate together with (2.6) concludes the proof. □

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 allow us to perform a fixed point argument to obtain the following result, whose proof is omitted.

Theorem 3.1Let$u_0\in L^1({\mathbb{R}}^n)\cap L^{\mathrm{\infty }}({\mathbb{R}}^n)$, then there exists a unique solution$u\in C([0,+\mathrm{\infty }),L^1({\mathbb{R}}^n)\cap L^{\mathrm{\infty }}({\mathbb{R}}^n))$of equation (1.1). This solution satisfies${\parallel u(\cdot ,t)\parallel }_{L^1({\mathbb{R}}^n)}\le {\parallel u_0\parallel }_{L^1({\mathbb{R}}^n)}$and${\parallel u(\cdot ,t)\parallel }_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}\le {\parallel u_0\parallel }_{L^{\mathrm{\infty }}({\mathbb{R}}^n)}$for all$t\ge 0$.

follows. To obtain a decay bound for ${\parallel u(\cdot ,t)\parallel }_{L^q({\mathbb{R}}^n)}$ 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.

where we omitted the dependence on t of the function u for simplicity.

Note that we find that the $L^q$-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).

where $\tilde{\sigma }=2\sigma {(p+q-2)}^{-1}$ and the constant C depends on $u_0$, q, σ, and n.

with C depending on $u_0$, q, σ, and n.

as we wanted to show.

for $1<q<2$. □