Fractional diffusion with
Neumann boundary conditions:
the logistic equation
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  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.
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  and ,
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 , 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 .
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 , our first goal is to provide a formulation of the problem
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
We can define the operator by
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
and which is also the unique weak solution of the problem
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)
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
where . When the diffusion follows the rules of the Brownian motion this model has been introduced in  and studied by many authors (see  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 . The typical known facts for the stationary problem associated to Brownian motion can be summarized as follows:
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
and as .
ii) If has nonnegative average, then for every there exists a unique positive solution of (1.8) and as , for expressed by
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
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
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
As in the previous case we can show that this branch is global and contains
all the positive solutions (see Proposition 3.16, and
Finally, we complete the proof of case ii) by approximation in Theorem
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.
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
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.
If then . In particular, it is a continuous function up to , and it vanishes as tends to infinity.
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
it is worth noticing that the former is well defined by Proposition 2.2. Moreover, we can choose as a norm on the quantity
as it is equivalent to the -norm thanks to the following lemma.
There exists a positive constant such that for every it holds
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 .
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 .
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
Let us fix such that is finite and take . We have
implying the desired inclusion. On the other hand, let , and let us define
It is a direct check to verify that (see also Lemma 2.10 in ), obtaining that all the equalities in (i) hold.
Let us now show conclusion (ii), starting with proving that there exist constants such that
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.
For every there exists an unique achieving
Moreover, the function is the unique (weak) solution of the problem
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
In a standard way this implies both that is harmonic in and that it satisfies the boundary condition on (in the -sense).
We will refer to the unique solving (2.6) as the Neumann harmonic extension of the function .
As we already noticed,
Furthermore, it is well known that the two 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.
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.
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
Let , and let denote its Neumann harmonic extension. Then the functional is well defined as
where and is any -extension of . Moreover,
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 . ∎
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.
We define the operator as
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.
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.
For every there exists a unique such that
Moreover, the function is the unique (weak) solution of the problem