A diffusion problem of Kirchhoff type involving the nonlocal fractional p -Laplacian

In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional $p$-Laplacian $\left\{ \begin{array}{*{35}{l}} {{\partial }_{t}}u+M([u]_{s, p}^{p}\text{)}(-\Delta)_{p}^{s}u=f(x, t) & \text{in }\Omega \times {{\mathbb{R}}^{+}}, {{\partial }_{t}}u=\partial u/\partial t, \\ u(x, 0)={{u}_{0}}(x) & \text{in }\Omega, \\ u=0\ & \text{in }{{\mathbb{R}}^{N}}\backslash \Omega, \\\end{array}\text{ }\ \ \right.$ where $[u]_{s, p}$ is the Gagliardo $p$-seminorm of $u$, $Ω\subset \mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partialΩ$, $1


(Communicated by Manuel del Pino)
Abstract. In this paper, we study an anomalous diffusion model of Kirchhoff type driven by a nonlocal integro-differential operator. As a particular case, we are concerned with the following initial-boundary value problem involving the fractional p-Laplacian in Ω, where [u]s,p is the Gagliardo p-seminorm of u, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂Ω, 1 < p < N/s, with 0 < s < 1, the main Kirchhoff function M : R + 0 → R + is a continuous and nondecreasing function, (−∆) s p is the fractional p-Laplacian, u 0 is in L 2 (Ω) and f ∈ L 2 loc (R + 0 ; L 2 (Ω)). Under some appropriate conditions, the well-posedness of solutions for the problem above is studied by employing the sub-differential approach. Finally, the largetime behavior and extinction of solutions are also investigated.
1. Introduction. In this paper, we study the fractional Kirchhoff type parabolic problem in Ω, where ∂ t u = ∂u/∂t, Ω ⊂ R N is a bounded domain with Lipschitz boundary ∂Ω and ϕ X s,p 0 (Ω) = Q |ϕ(x) − ϕ(y)| p K(x − y)dxdy is the norm on the main fractional Sobolev space X s,p 0 (Ω), which we introduce in Section 2 and which is well-defined along any ϕ ∈ C ∞ 0 (R N ). Here and in the follows Q = R 2N \ (CΩ × CΩ) and CΩ = R N \ Ω. (1. 3) The elliptic part of problem (1.1) is represented by L K , which is a nonlocal integrodifferential operator, defined pointwise for each x ∈ R N by |ϕ(x) − ϕ(y)| p−2 [ϕ(x) − ϕ(y)] |x − y| N +ps dy along any ϕ ∈ C ∞ 0 (R N ). We refer to [8,16,17,23,24,28,32,34,35,38,39,40] and the references therein for further details on the fractional Laplacian operator, and to [18] for further details on the fractional Laplacian and on the fractional Sobolev space W s,p (R N ). Throughout the paper, without further mentioning, we always assume for simplicity that N > ps and that K satisfies (1.4). It is worth to note that the case N = 2s, that is when p = 2, N = 1 and s = 1/2, it was recently studied in [19]. As far as we know even the case N = 2s is completely new for (1.1).
Throughout the paper M : R + 0 → R + denotes the main Kirchhoff function, assumed to be continuous and nondecreasing, and u 0 the initial value of class L 2 (Ω). The interest in studying problems like (1.1) relies not only on mathematical purposes, but also on their significance in real models, as explained by Caffarelli in [11,12], Laskin in [26] and Vázquez in [37]. It is worthy pointing out that Applebaum in [7] states that the fractional Laplacian operators of the form (−∆) s , are the infinitesimal generators of stable radially symmetric Lévy processes. Laskin in [27] formulated a fractional Schrödinger equation as a result of expanding the Feynman path integral, from the Brownian-like to the Lévy-like quantum mechanical paths.
Recently, Fiscella and Valdinoci in [23] first proposed a stationary Kirchhoff variational equation, involving the fractional Laplacian, which models the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string. Indeed, the stationary problem (1.1) is a fractional version of a model, the so-called stationary Kirchhoff equation, introduced by Kirchhoff for certain evolution phenomena. We refer e.g. to [9,21,33] for evolution equations of Kirchhoff type, to [32,34,38] for non-degenerate stationary Kirchhoff problems, to [8,14,30,35,39] for the degenerate case and the references therein.
To explain the motivation of problem (1.1), let us shortly introduce a prototype of nonlocal problem like (1.1) in R + × R N . Indeed, nonlocal evolution equations of the form and its variants, have been recently widely used to model diffusion processes. More precisely, as stated by Fife in [20], if u = u(x, t) is thought of as a density of population at the point x and time t and K(x − y) is thought of as the probability distribution of jumping from location y to location x, then R N u(y, t)K(x − y)dy is the rate at which individuals are arriving at position x from all other places and − R N u(x, t)K(x − y)dy is the rate at which they are leaving location x to travel to all other sites. This consideration, in the absence of external or internal sources, leads immediately to the fact that the density u satisfies problem (1.6). For recent references on nonlocal diffusions, see for example [2,3,4,13,15,31].
If we consider the effects of total population, then problem (1.6) becomes where the coefficient M : R + 0 → R + denotes the possible changes of total population in R N . This means that the behavior of individuals is subject to total population, such as the diffusion process of bacteria. Problem (1.7) is also meaningful, since the way of measurements are usually taken in average sense. In particular, if K(x) = |x| −N −2s and s 1 − , then equation (1.7) reduces to which has been investigated in recent papers. See for instance [25] and the references therein. Moreover, if M ≡ 1, K(x) = |x| −N −ps and s 1 − , then problem (1.7) reduces to where ∆ p u = div(|∇u| p−2 ∇u) and f is a source term not necessarily zero. By using sub-differential calculus, Akagi and Matsuura in [2] exploited the well-posedness and asymptotic behaviors of solutions for equation (1.8) in variable exponent Sobolev spaces, see [3] for further results. Actually, the existence, uniqueness, extinction in finite time, decay and blow-up of solutions for equation (1.9) have been studied extensively in recent years, for example, see [5,6] and the references therein. Motivated by the above works, we focus on the well-posedness and large-time behavior of solutions of (1.1). Main difficulties arise, when dealing with this problem, because of the presence of the Kirchhoff function and of the nonlocal nature of the p-fractional Laplacian. To the best of our knowledge, there are no results on the diffusion problem of Kirchhoff type involving the fractional p-Laplacian. Clearly, the extension from the linear case p = 2 to the semi-linear general case 1 < p < ∞ is also not trivial.
Problems of the above form (1.1) are mathematical models occuring in nonlocal reaction-diffusion theory, non-Newtonian fluid theory, non-Newtonian filtration and turbulent flows of a gas in a porous medium. In the non-Newtonian fluid theory, the quantity p is characteristic of the medium. Media with p > 2 are called dilatant fluids and those with p < 2 are called pseudoplastics. If p = 2, they are Newtonian fluids.
The rest of the paper is organized as follows. In Section 2, we recall some necessary definitions and properties of the fractional Sobolev spaces. In Section 3, we obtain the well-posedness of solutions of problem (1.1) for all p > 1 by employing the subdifferential approach. In Section 4, the asymptotic behavior of solutions of problem (1.1) is studied under some appropriate natural assumptions. Section 5 is devoted to the study of the extinction property of solutions of problem (1.1), without source term (i.e. when f ≡ 0) in the singular case 2N/(N + 2s) ≤ p < 2.
2. Preliminaries. In this section, we first recall some necessary properties of fractional Sobolev spaces and related notations, see also [18] for further details.
Firstly, let us recall that X s,p (Ω) denotes the linear space of Lebesgue measurable functions u : R N → R such that the quantity It is easy to see that bounded and Lipschitz functions belong to X s,p (Ω), thus X s,p (Ω) is not trivial. We refer to [22,38] for further details on the space X s,p (Ω). Moreover, X s,p 0 (Ω) denotes the space of functions u ∈ X s,p (R N ) that vanish a.e. in CΩ = R N \ Ω, endowed with the norm (1.2). It is easy to see that (1.2) is equivalent to (2.1) and that X s,p 0 (Ω) is a uniformly convex Banach space, being p > 1. See [34,35,38] for more details. Indeed, X s,p 0 (Ω) can also be identified with the closure of C ∞ 0 (Ω) in X s,p (Ω), as shown by Fiscella, Servadei and Valdinoci in [22]. It is worthy mentioning that the space X s,p 0 (Ω) is similar to, but different from, the standard fractional Sobolev space W s,p 0 (Ω). The interested reader can refer to [22] for further comments.
Note that in both (2.1) and (1.2) the integrals can be extended to the whole spaces R N and R 2N , since u = 0 a.e. in CΩ. By Lemma 2.3 of [38], the Banach space X s,p 0 (Ω) = (X s,p 0 (Ω), · X s,p 0 (Ω) ) is continuously embedded in L r (R N ) for any r ∈ [1, p * s ], where the critical fractional Sobolev exponent is defined by Throughout the paper, the letters C, C i , i = 1, 2, · · · , denote positive constants which vary from line to line, but are independent of the terms which take part in any limit process. For the reader's convenience, we recall some related useful definitions and notations. The Banach space L p (0, T ; X) consists of all strongly measurable functions u : [0, T ] → X, endowed with the norm where T ∈ R + is a given number and X is a reflexive Banach space. Obviously, L p (0, T ; X) is a reflexive Banach space. It follows from [36, Theorem 1.5] that the dual space of L p (0, T ; X) can be identified with L p 0, T ; X , where p = p/(p − 1) and X is the dual space of X. The other spaces can be understood similarly.
3. Well-posedness of solutions. In this section, we prove the well-posedness of (weak) solutions of problem (1.1). To this aim, we require the following assumption (M) M : R + 0 → R + 0 is a continuous and nondecreasing function and there exists a constant m 0 > 0 such that A prototype for Kirchhoff function M is given by Hence the problems we treat in this paper are non-degenerate.
Without further mentioning, throughout the paper we always assume that (M) holds and put for all t ∈ R + 0 . Here and henceforth, C w ([0, T ]; X) denotes the space of weakly continuous functions on [0, T ] into a normed space X and f ∈ L 2 loc (R + 0 ; L 2 (Ω)). Definition 3.1. A function u ∈ C(R + 0 ; L 2 (Ω)) is said to be a solution for problem (1.1), if the following conditions are satisfied (i) u ∈ W 1,2 loc (R + 0 ; L 2 (Ω)) ∩ C ω (R + 0 ; X s,p 0 (Ω)) and L K u ∈ L 2 loc (R + 0 ; L 2 (Ω)), (ii) u(·, 0) = u 0 a.e. in Ω, (iii) For all ϕ ∈ X s,p 0 (Ω), the following equality holds   Proof. It follows from [22,Theorem 6] , and assume, without loss of generality, that t 1 < t 2 . Then setting t = λt 1 + (1 − λ)t 2 , in order to prove that By the Lagrange mean theorem, there exist ξ, η ∈ R + , with t 1 < ξ < t < η < t 2 , such that . It follows from these facts that Then claim (3.3) follows at once, since M is nondecreasing in R + 0 and ξ < η. Hence, being M nondecreasing in R + 0 , then (3.3) implies that Therefore, I is convex in H 0 , as stated. Next, we show that I is lower semi-continuous in H 0 . Let µ ∈ R + 0 be fixed and set Let (u n ) n be a sequence in [I ≤ µ] such that u n → u strongly in H 0 . Clearly, I(u n ) ≤ µ for all n yields u n p X s,p Hence ( u n X s,p 0 (Ω) ) n is bounded and, up to a subsequence, still denoted by (u n ) n , then u n u weakly in X s,p 0 (Ω), since X s,p 0 (Ω) is reflexive. Let I 0 denote the restriction of I to X s,p 0 (Ω). Then I 0 is of class C 1 X s,p 0 (Ω) and weakly lower semi-continuous in X s,p 0 (Ω), see [38] for similar arguments. Moreover, I 0 is convex. Therefore we have lim inf n→∞ I 0 (u n ) ≥ I 0 (u) = I(u), which, together with I 0 (u n ) = I(u n ) ≤ µ, implies that u ∈ [I ≤ µ]. Hence [I ≤ µ] is closed in H 0 , and so I is lower semi-continuous in H 0 .
Moreover, one can show the following lemma, see [38] for similar arguments.
Lemma 3.2. The restriction I 0 of I to X s,p 0 (Ω) is of class C 1 X s,p 0 (Ω) , and the Fréchet derivative dI 0 (u) of I 0 at u ∈ X s,p 0 (Ω) coincides with M u p X s,p 0 (Ω) L K u valuated at u| R N \Ω = 0 in the sense of distribution, that is, for all v ∈ X s,p 0 (Ω). The sub-differential operator ∂I : H 0 → H 0 of I is given by for any u ∈ D(I), where naturally (·, ·) H0 = (·, ·) L 2 (Ω) and the effective domain The abstract evolution equation (3.4) was well studied, mainly by Brézis in [10, Chap. III], see also [36]. Thus Lemma 3.1 yields at once the next result.
Next let us prove the existence of periodic solutions for problem (3.4). In the argument it is required that 2 < p * s , that is that p > 2N/(N + 2s), and in turn p > max{1, 2N/(N + 2s)}. Proof. By Corollary 3.4 of [10], it suffices to check the coercivity of I in H 0 . Since the embedding X s,p 0 (Ω) → L 2 (R N ) is continuous, being p > 2N/(N +2s), we obtain by (M) where C * > 0 comes from (2.2). Hence the functional I is coercive in H 0 , since being p > 1.

Large-time behaviors of solutions.
This section is concerned with the largetime behavior of solutions.

(4.2)
Let u be the unique solution of problem (1.1), with initial datum u 0 ∈ X s,p 0 (Ω). Then there exists u * ∈ X s,p 0 (Ω) such that u(·, t) → u * strongly in L 2 (Ω) as t → ∞, where I is defined in (3.2). Moreover, u * is the unique weak solution of the problem To prove Theorem 4.1, we need the following crucial result.
The proof is thus complete.
Proof of Theorem 4.1. Let (t n ) n be any sequence in R + 0 such that t n → ∞. As shown in the proof of Lemma 4.1, there exists a subsequence (θ n ) n , still denoted by (θ n ) n , such that u(·, θ n ) u * weakly in X s,p 0 (Ω), (4.5) which, together with p > 2N/(N + 2s), yields that u(·, θ n ) → u * strongly in L 2 (Ω). (4.6) Combining these facts with (4.5), we conclude from the demiclosedness of ∂I that which is equivalent to an L 2 -formulation of (4.3).
On the other hand, from the definition of sub-differential, we have as n → ∞. Note also that the weak lower semicontinuity of I in H 0 implies that lim inf n→∞ I(u(·, θ n )) ≥ I(u * ).
On the other hand, (4.6) implies that as n → ∞ By the monotonicity of Eu, we know that Eu(t n ) converges to the same limit as t n → ∞. Consequently, we conclude that as n → ∞ I(u(·, t n )) = Eu(t n ) + (f * , u(·, t n )) L 2 (Ω) + 1 2 This completes the proof.
We note in passing that the restriction p > max{1, 2N/(N + 2s)} is automatic whenever p ≥ 2, being s > 0. Proof. The proof is similar to that of Corollary 3.2, so we leave it to the interested reader.
In order to study the large-time behavior, let us introduce some notation and let us assume from now on that p ≥ 2. Fix α ∈ [p − 1, ∞) and take L M > 0 so small that (4.8) where C > 0 is the number given in Lemma 4.1, m 0 in (M) and C p is a positive constant for which the celebrated inequality holds, being p ≥ 2.
On the Kirchhoff function M let us also assume Condition ( M), when p = 2 and α = 1, has been used to get the uniqueness of solutions of stationary Kirchhoff problems by Ma in [29]. Here, we use condition ( M) to get some useful asymptotic estimates.  (4.8), and that f (·, t) ≡ f * ∈ L 2 (Ω) in R + 0 . Let u = u(x, t) be the unique solution of problem (1.1), with initial datum u 0 ∈ L 2 (Ω), and let u * be the corresponding weak solution of problem (4.3).
(i) If p > 2, then there exists a constant C > 0 such that for all t ≥ 0 .
If p > 2, a weaker assumption on f and f * than that of Theorem 4.3 can be given by assuming that for all t ≥ 0 where p = p/(p − 1) and C 0 > 0 is a given constant. We have the next result.
Assume that f * ∈ L 2 (Ω) and f ∈ L 2 loc (R + 0 ; L 2 (Ω)) satisfy (4.1) and (4.15). Let u = u(x, t) be the unique solution of problem (1.1), with initial data u 0 ∈ L 2 (Ω), and let u * be the corresponding weak solution of (4.3). Then there exists C > 0 such that Proof. A similar argument as that of the proof of Theorem 4.3 gives that , where C * > 0 is the embedding constant given in (2.2). Furthermore, by the Young inequality, (4.9) and (M), we obtain Then from (4.15), (4.18) and the fractional Sobolev embedding X s,p 0 (Ω) → L 2 (Ω) it follows that for all t ≥ 0 where by (4.16) be a solution of the nonlinear ordinary differential equation, with source term, where the constant λ * > 0 has to be determined later. A solution of (4.19) is given by the explicit formula X(t) = α(1 + t) −2/(p−2) , t ∈ R + 0 ,

Extinction of solutions.
For the singular case 1 < p < 2, we prove below that when f ≡ 0 the solution u of (1.1) vanishes at a finite time T * , called extinction time of u. We are planning to investigate these two open problems in a forthcoming paper.