Fractional diffusion withNeumann boundary conditions:the logistic equation

Fractional diffusion with
Neumann boundary conditions:
the logistic equation

Eugenio Montefusco, Benedetta Pellacci & Gianmaria Verzini
July 17, 2019
Abstract

Motivated by experimental studies on the anomalous diffusion of biological populations, we introduce a nonlocal differential operator which can be interpreted as the spectral square root of the Laplacian in bounded domains with Neumann homogeneous boundary conditions. Moreover, we study related linear and nonlinear problems exploiting a local realization of such operator as performed in [7] for Dirichlet homogeneous data. In particular we tackle a class of nonautonomous nonlinearities of logistic type, proving some existence and uniqueness results for positive solutions by means of variational methods and bifurcation theory.

1 Introduction

Nonlocal operators, and notably fractional ones, are a classical topic in harmonic analysis and operator theory, and they are recently becoming impressively popular because of their connection with many real-world phenomena, from physics [20, 14, 21] to mathematical nonlinear analysis [1, 24], from finance [4, 13] to ecology [6, 23, 17, 5]. A typical example in this context is provided by Lévy flights in ecology: optimal search theory predicts that predators should adopt search strategies based on long jumps –frequently called Lévy flights– where prey is sparse and distributed unpredictably, Brownian motion being more efficient only for locating abundant prey (see [25, 29, 17]). As the dynamic of a population dispersing via random walk is well described by a local operator –typically the Laplacian– Lévy diffusion processes are generated by fractional powers of the Laplacian for in all . These operators in can be defined equivalently in different ways, all of them enlightening their nonlocal nature, but, as shown in [8] and [9], they admit also local realizations: the fractional Laplacian of a given function corresponds to the Dirichlet to Neumann map of a suitable extension of to . On the contrary, on bounded domains, different not equivalent definitions are available (see e.g. [15, 3, 7] and references therein). This variety reflects the different ways in which the boundary conditions can be understood in the definition of the nonlocal operator. In particular, we wish to mention the recent paper by Cabré and Tan [7], where the operator on a bounded domain and associated to homogenous Dirichlet boundary conditions is defined by Fourier series, using a basis of corresponding eigenfunctions of . Their point of view allows to recover also in the case of a bounded domain the aforementioned local realization: indeed, interpreting as a part of the boundary of the cylinder , the Dirichlet spectral square root of the Laplacian coincides with the Dirichlet to Neumann map for functions which are harmonic in the cylinder and zero on its lateral surface. These arguments can be extended also to different powers of , see [12]. On the other hand, in population dynamic, Neumann boundary data are as natural as Dirichlet ones, as they represent a boundary acting as a perfect barrier for the population. The aim of this paper is then to provide a first contribution in the study of the spectral square root of the Laplacian with Neumann boundary conditions.
Inspired by [7], our first goal is to provide a formulation of the problem

(1.1)

where is a bounded domain in , , and can be thought, for instance, as an function. To this aim, let us denote with an orthonormal basis in formed by eigenfunctions associated to eigenvalues of the Laplace operator subjected to homogenous Neumann boundary conditions, that is

(1.2)

We can define the operator by

(1.3)

The first series in (1.3) starts from since the first eigenvalue and the corresponding eigenfunction in (1.2) are given by . This simple difference with the Laplacian subjected to homogeneous Dirichlet boundary conditions has considerable effects. First of all, this implies that , as the usual Neumann Laplacian, has a nontrivial kernel made of the constant functions, then it is not an invertible operator and (1.1) cannot be solved without imposing additional conditions on the datum ; on the other hand, given any defined on , its harmonic extension on having zero normal derivative on the lateral surface needs not to belong to any Sobolev space, as constant functions show. These features has to be taken into account when establishing the functional framework where to set the variational formulation of (1.1). In this direction, we will first provide a proper interpretation of (1.1), and a corresponding local realization, in the zero mean setting. To this aim, let us introduce the space of functions defined in the cylinder

An easy application of the Poincaré-Wirtinger inequality shows that we can choose as a norm of the norm of the gradient of (see Proposition 2.2 and Lemma 2.3). It comes out that, when the datum has zero mean, a possible solution of (1.1) is the trace of a function belonging to . The corresponding space of traces can be equivalently defined in different ways, since Proposition 2.4 shows that

In proving this result, one obtains that every has an harmonic extension given by

(1.4)

and which is also the unique weak solution of the problem

(1.5)

Thus, given we can find a unique solving (1.5), for which it is well defined the functional acting on as

where is any extension of . Since this functional is actually an element of the dual of , it is well defined the operator between and its dual. Thus, restricting the study to the zero mean function spaces, and taking into account equations (1.3) and (1.4), we have that conincides with , but it is invertible: for every in the dual space of there exists a unique such that , and this function is the trace on of the unique solution of the problem (see Lemma 2.14)

(1.6)

The link between and now becomes transparent since

that is, the image of a function trough is the same of the one yield by acting on the zero mean component of (see Definition 2.12). In this way we have recovered the local realization of as a map Dirichlet-Neumann since

where solves (1.5) with Dirichlet datum instead of . Therefore, if has zero mean, denoting with the unique solution of then the solutions set of (1.1) is given by for .
Since we are interested in ecological applications, as a first study we focus our attention on the logistic equation. More precisely, consider a population dispersing via the above defined anomalous diffusion in a bounded region , with Neumann boundary conditions, growing logistically within the region; then , the population density, solves the diffusive equation

where acts as a diffusion coefficient, the term express the self-limitation of the population and corresponds to the birth rate of the population if self-limitation is ignored. The weight may be positive or negative in different regions, denoting favorable or hostile habitat, respectively. The stationary states of this equation are the solutions of the following nonlinear problem

(1.7)

where . When the diffusion follows the rules of the Brownian motion this model has been introduced in [26] and studied by many authors (see [11] and the references therein). One of the major task in this problem is describing how favorable and unfavorable habitats, represented by the interaction between and , affects the overall suitability of an environment for a given populations [10]. The typical known facts for the stationary problem associated to Brownian motion can be summarized as follows:

Theorem 1.1 ([16, 28]).

i) If the function has negative mean inside and it is positive somewhere, then there exists a positive number such that for every there exists a unique positive solution

(1.8)

and as .
ii) If has nonnegative average, then for every there exists a unique positive solution of (1.8) and as , for expressed by

(1.9)

The number appearing in i) is the first positive eigenvalue with positive eigenfunction of the operator with Neumann boundary condition and with a weight satisfying the hypotheses in i).
In our situation, we have, first of all, to clarify that by a weak positive solution of (1.7) we mean a function , , with and , so that for and is a weak solution of the nonlinear problem

(1.10)

in the sense that

In other words, we impose that the right hand side has zero mean, choosing, in this way, the mean of a solution as . Then we obtain the well posedeness of the problem

since now the right hand side has zero mean, and we obtain in this way the zero part mean of . Moreover, notice that the mean of the function solution of (1.10) with and is exactly the mean of .
Our main existence result is the following

Theorem 1.2.

Let . Then the following conclusion hold:
i) If and there exists such that , then there exists a positive number such that for every there exists a unique positive solution of (1.7), with as .
ii) If , then for every there exists a unique positive solution of (1.7) with for .

As in the standard diffusion case, is the first positive eigenvalue with positive eigenfunction of the problem

which existence is proved in Theorem 3.7. Theorem 1.2 will be obtained via classical bifurcation theory, indeed, in case i), we can show that a smooth cartesian branch of positive solutions bifurcates from the trivial solution , this branch can be continued in all the interval , and contains all the positive solutions of (1.7), that is to say that for every there exists a unique positive solution (see Proposition 3.15, and Theorem3.20). We tackle case ii) first assuming that the mean of is positive. This allows us choose as a bifurcation parameter , the future mean of , instead of , and find a branch bifurcating from the trivial solution , with defined as in (1.9). As in the previous case we can show that this branch is global and contains all the positive solutions (see Proposition 3.16, and Theorem3.20). Finally, we complete the proof of case ii) by approximation in Theorem 3.22.
All the effort made in finding the proper formulation for the linear and the nonlinear problem enables us to prove the existence results for (1.7), which are in accordance with the case of standard diffusion. But, trying to enlighten the differences between the two models, one has to take care of the eigenvalues appearing in Theorems 1.1 and 1.2, that is and . Since such eigenvalues act as a survival threshold in hostile habitat, it is a natural question to wonder which is the lowest one, indeed this indicates whether or not the fractional search strategy is preferable with respect to the brownian one. This appears to be a difficult question, since the eigenvalues depend in a nontrivial way on , and also on the sequence defined in (1.2). At the end of Section 3 we report some simple numerical experiments to hint such complexity.

2 Functional setting

In this section we will introduce the functional spaces where the spectral Laplacian associated to homogeneous Neumann boundary conditions will be defined. Moreover, we will study the main properties of this operator and find the proper conditions under which the inverse operator is well defined. Finally, we will prove summability and regularity properties enjoyed by the solutions of the linear problem.
Throughout the paper is a bounded domain and we will use the notation .
In this plan we will make use of the following projections operators.

Definition 2.1.

Let us define the operators by

for denoting the Lebesgue measure of the domain . and give the average (with respect to ) and the zero-averaged part of a function , respectively. Analogously, for , we write

(2.1)

When no confusion is possible, we drop the subscript in , .

It is standard to prove that, in both cases, and are linear and continuous, and that . Since the integration in the definition of is performed only with respect to the variable, it is natural to interpret the image of a function through the operator as a function of one variable. enjoys the following properties.

Proposition 2.2.

If then . In particular, it is a continuous function up to , and it vanishes as tends to infinity.

Proof.

Since for almost every , we can compute and obtain, by Hölder’s inequality,

As a consequence, , so that it is continuous in and it vanishes as tends to . ∎

Introducing the following functional spaces

(2.2)

it is worth noticing that the former is well defined by Proposition 2.2. Moreover, we can choose as a norm on the quantity

(2.3)

as it is equivalent to the -norm thanks to the following lemma.

Lemma 2.3.

There exists a positive constant such that for every it holds

Proof.

We set and we notice that for any the Poincaré-Wirtinger inequality implies

proving the claim. ∎

The following proposition gives a complete description of the space .

Proposition 2.4.

Let be defined in (2.2). Then the following conclusions hold:

(ii) is an Hilbert space with the norm

equivalent to the usual one in .

Proof.

Since is of class , we have that can be equivalently characterized as , where we write tr. Then Proposition 2.2 provides the inclusion

In order to show the opposite one, consider and consider such that . Notice that and Proposition 2.2 implies that

then we have found belonging to and such that , yielding the first equality in (i). As far as the second equality is concerned, we start by proving the inclusion

Indeed any can be written as , with

then

Let us fix such that is finite and take . We have

implying the desired inclusion. On the other hand, let , and let us define

(2.4)

It is a direct check to verify that (see also Lemma 2.10 in [7]), obtaining that all the equalities in (i) hold.

Let us now show conclusion (ii), starting with proving that there exist constants such that

(2.5)

As

The right hand side inequality holds for ; in order to show the left hand side inequality, let us argue by contradiction and suppose that there exists a sequence , with and . Then is uniformly bounded in and there exists such that converges to weakly in and strongly in (notice that we do not know that the quantity is a norm on ). As a consequence, and

which is an obvious contradiction. As a byproduct of inequalities (2.5) we obtain that is a well defined norm and since is a closed subspace of with respect to the usual norm conclusion (ii) holds. ∎

Carefully reading the proof of the second equality in (i) of the previous proposition, one realizes that for any we can construct a suitable extension which is harmonic and that can be written in terms of a Fourier expansion as shown in (2.4). In the next lemma we provide a variational characterization of such extension.

Lemma 2.5.

For every there exists an unique achieving

Moreover, the function is the unique (weak) solution of the problem

(2.6)

Finally,

(2.7)
Proof.

We observe that the functional to be minimized is simply the square of the norm in , and the set on which we minimize is non empty and weakly closed thanks to the compact embedding of in , for any exponent . The strict convexity of the functional implies the existence and uniqueness of the minimum point.

As usual, the unique minimum point satisfies the boundary condition on (in the -sense) by constraint, and

As a consequence, for every such that , it is possible to choose as a test function in the previous equation. This provides

(2.8)

In a standard way this implies both that is harmonic in and that it satisfies the boundary condition on (in the -sense).

Finally, if is given as in (2.7), then as in (2.7) solves problem (2.6) and the uniqueness of the solution provides the claim. ∎

Definition 2.6.

We will refer to the unique solving (2.6) as the Neumann harmonic extension of the function .

Remark 2.7.

As we already noticed,

Furthermore, it is well known that the two norms

are equivalent. Reasoning as in the proof of Proposition 2.4, and taking into account Lemma 2.5, we obtain that can be equipped with the equivalent norms

where the terms are the Fourier coefficients of . In particular, the harmonic extension of depends on in a linear and continuous way.

In order to introduce and study the dual space of let us first introduce the following space.

Definition 2.8.

Let us define the following subspace of .

where denotes the duality pairing.

The subspace just introduced as a strict connection with the dual space of as well explained in the following proposition.

Proposition 2.9.

It holds

Proof.

We can exploit the splitting in order to obtain

More precisely, on one hand if then, for every ,

on the other hand, if then and

Moreover, both the maps defined above are linear and continuous. This proves that is isomorphic to , for every fixed , and in particular for . ∎

As a first step to arrive to a correct definition of the half Laplacian operator, let us prove the following lemma

Lemma 2.10.

Let , and let denote its Neumann harmonic extension. Then the functional is well defined as

where and is any -extension of . Moreover,

Proof.

The functional is well defined, indeed if and are two extensions of we have that and, arguing as in equation (2.8), yields

Moreover is linear and continuous: indeed, let us choose as an extension of , where is the harmonic extension of ; by Remark 2.7 applied to we have that

As a consequence . Finally, since belongs to , by definition we obtain that

which vanishes because . ∎

Remark 2.11.

If the harmonic extension is more regular (for instance ), then we can employ integration by parts in order to prove that the definition of given above agrees with the usual one.

Thanks to the previous lemmas, we are now in a position to define the fractional operators we work with.

Definition 2.12.

We define the operator as

(2.9)

where is the harmonic extension of according to (2.6). Analogously, we define the operator by

In Definition 2.12 we have introduced the fractional Laplace operator associated to homogeneous Neumann boundary conditions as a Dirichlet to Neumann map. Moreover, thanks to the equivalences of Proposition 2.4, we realize the spectral expression of this operator as explained in the following remark.

Remark 2.13.

Since the harmonic extension operator is linear and continuous by Remark 2.7, we have that both and are linear and continuous. Moreover, if and , we can use equation (2.7) to infer that . This allows to write

In particular, if then

provides the usual Laplace operator associated to homogeneous Neumann boundary conditions on .

We remark that we can think to as acting between and its dual thank to Proposition 2.9. While is neither injective nor surjective, we have that is invertible.

Lemma 2.14.

For every there exists a unique such that

(2.10)

Moreover, the function is the unique (weak) solution of the problem

(2.11)