In this paper, global-in-time existence and blow-up results are shown for a reaction-diffusion equation appearing in the theory of aggregation phenomena (including chemotaxis). Properties of the corresponding steady-state problem are also presented. Moreover, the stability around constant equilibria and the non-existence of nonconstant solutions are studied in certain cases.
We consider the following initial boundary value problem:
where and . Here, Ω is an open bounded domain in , and denotes a boundary operator of Neumann or Dirichlet type, i.e.
where γ is the outer unit normal vector of . For the sake of simplicity, we take .
One of the motivations to study such an equation comes from the structure similarities that exist with the parabolic-elliptic Keller–Segel models for chemotaxis, i.e.
where V is the fundamental solution of Poisson equation (or some other given potential in the case of general diffusion aggregation equations). If formally the interaction potential V is replaced by a the Dirac mass , then the above equation is reduced to
where for , , and are i.i.d. Brownian motions.
where is the distribution of the i.i.d. random processes at time t. By Itô’s formula one can obtain the following nonlocal partial differential equation for :
The physical meaning of the unknown u is that of a concentration, therefore one considers only nonnegative solutions corresponding to nonnegative initial data.
Furthermore (like in the case of Keller–Segel system), problem (1.1) with (and the homogeneous Neumann boundary condition) possesses the following entropy structure:
This entropy is a combination of a positive part from the diffusion and a negative one from the aggregation. It needs to be pointed out that here the aggregation phenomenon is much stronger than the one appearing in Keller–Segel systems because of the singular potential that appeared in (1.2)–(1.3).
As for the reaction term, it is considered to be of logistic (mono-stable) type so that (when ) a significant dampening effect is exercised on the density u at those points where u becomes large.
The arrangement of the paper is the following. In Section 2, global existence and uniqueness of classical solutions are obtained for initial data such that parabolicity is expected to hold. The rest of the paper concerns cases in which parabolicity is expected to be lost at some point, so that blowup may happen. Considerations on the possible steady states and their stability as well as direct estimates of blowup are presented. Section 3 is devoted to the study of the steady states. The non-existence of nontrivial steady states is proved via Pohozaev’s type arguments. Furthermore, the linear stability of constant steady states is investigated. Finally, in Section 4, blow-up (in finite time) results are presented. Two different procedures are carried out: Kaplan’s method is used for the problem with Dirichlet boundary condition on one hand, and the concavity method is used for the problem with Neumann boundary condition on the other hand. In the end, we present an annex where blowup is directly observed in a class of explicit solutions linked to Barenblatt profiles, and we draw conclusions in a final section.
In this section, the global existence and uniqueness of a solution is obtained thanks to Leray–Schauder fixed point theorem, under the condition that the initial datum belongs to the parabolic region.
and using the notation , (1.1) can be rewritten as
It can be expected that global existence holds in the case when the parabolicity can be kept in the evolution (that is, when for all time , or equivalently ). At the same time, the logistic term and the expected nonnegativity of u imply that the estimate should also hold. Therefore, a natural sufficient condition for getting global existence for equation (1.1) is , together with . The theorem below states a precise result in this direction:
Let Ω be a smooth bounded domain in . Assume and . Let also , and
with compatibility condition . Then problem (1.1), together with homogeneous Neumann or homogeneous Dirichlet boundary condition has a unique global-in-time classical solution. In addition, it holds that
We first observe that we can take small enough in such a way that and . Then we will prove the existence and uniqueness of a solution u of the problem, which satisfies the estimate
For any fixed , we will use the Leray–Schauder fixed point theorem to prove the existence. Let
We define an operator as follows: for given and , let be the solution (see [8, Chapter V, Theorem 7.4] for the existence and uniqueness of the solution) of the following problem:
In order to build up the map , we have to show that for all .
For , it is obvious that , since in that case u satisfies the heat equation.
For , we first prove that . To this end, let be small and let be the solution of
The solution possesses a uniform in ε estimate , see [8, Chapter V, Theorem 7.4]. With , if , then there exists such that
More precisely, we take here as the last time before the solution takes some negative value. If , we have . If , in the case of the homogeneous Neumann boundary condition, we also have . Then we get (still in the case of Neumann boundary condition)
which is a contradiction. Therefore, .
If (and for all ), in the case of the homogeneous Dirichlet boundary condition, we can prove (see the sequel of the proof) that there exists a sequence satisfying and such that
This limit, together with the fact that is smooth and that the tangential derivative of vanishes because of the homogeneous Dirichlet boundary condition, shows that . Thus we can follow the same argument as above.
In order to show the limit (2.3), we consider a sequence that converges to and ( ) one of the minimal points of such that (note that if several points satisfy (2.2), one at least can be selected in such a way that the construction above makes sense). Then because of the continuity of . We get therefore
Then the mean value theorem implies that there exists a sequence such that . Therefore,
where and the last line comes again from the homogeneous Dirichlet boundary condition. As a consequence, since we are working with bounded classical solutions, the limit vanishes.
On the other hand, satisfies the following linear problem:
where all the coefficients are uniformly bounded in ε. Therefore by the maximum principle, we have
By taking the limit in , we obtain that in .
Next we prove that in . Suppose that there exists such that
Then we have , and moreover if we consider the Dirichlet boundary condition, so that
In the case of the Neumann boundary conditions, might appear on the boundary, but in this case, we still have , therefore the above argument also works. This implies , which is a contradiction with the assumption . Therefore .
Thus the map is well defined. Due to the compact embedding from to , we know that is a compact operator.
Next we show that the map is continuous in w and σ. For all and , let be a sequence such that as , and let be a sequence such that . Let , the Schauder estimates show that uniformly in j. Notice that satisfies the following linear problem:
Using Schauder’s theory for linear parabolic equations, we get the estimate
Hence, is continuous in w and σ.
Furthermore, it is obvious that . Additionally, for any fixed point of , the uniform estimates for quasilinear parabolic equation show ([8, Chapter V, Theorem 7.2]) that there exists a constant M depending only on , c, such that
Therefore, by Leray–Schauder’s fixed point theorem, there exists a fixed point to the map , i.e. u is a solution of the following problem:
which is equivalent to equation (1.1).
The uniqueness of classical solutions follows directly from comparison principles. ∎
In this section, two results concerning stationary states are given. One of them shows that nontrivial nonnegative solutions do not exist for equation (1.1) with Dirichlet boundary condition. The other one has to do with the linear instability of constant steady states to equation (1.1) with homogeneous Neumann boundary condition.
The steady states corresponding to (1.1) satisfy the equation
Let , let Ω be a star shaped domain of with respect to the origin and suppose that are functions defined on such that
Then the problem
does not have any nontrivial (that is, different from ) classical solution.
This last condition is satisfied in particular when c and d are negative and or . Note also that since , the homogeneous Dirichlet boundary condition on implies that on , so that the first part of Theorem 2 can be applied.
By testing (3.3) with , we get
First, notice that
Integrating over Ω and applying the divergence lemma to the left-hand side, we get
so that using (3.4), we obtain
We first compute
where . Integrating by parts, we get
For the second term, using problem (3.3), we get
For the last term in (3.5), we compute
Using the Dirichlet boundary condition, we see that , so that on , we have
Thus, the above relation becomes
Since Ω is star shaped, there exists such that
and relation (3.9) yields
Therefore, a sufficient condition for the non-existence of (nontrivial, nonnegative, classical) solutions is
Next, if we set , , then
Using (3.10), we get the sufficient condition of non-existence of nontrivial solutions to the corresponding steady-state problem, which consists in finding such that
As stated in the theorem, this happens when for example, and . ∎
We obtain (3.10) from (3.9) by neglecting the first boundary term (since Ω is starshaped). Now we keep this first boundary integral (the second boundary integral in (3.9) is 0 because of the boundary conditions) and compute
where we have used the geometry of the domain, Cauchy–Schwarz inequality, , the divergence lemma and the problem itself. Then relation (3.9) yields
Therefore, we can get a more precise description for the non-existence of solutions, since now we need to check the less stringent inequality
We present here a computation related to the linear stability of steady states for the Neumann boundary condition. We denote by the solution of the eigenvalue problem for the Laplacian with homogeneous Neumann boundary condition, with for and .
We assume that . Then the equilibrium for equation (1.1) with homogeneous Neumann boundary condition is asymptotically linearly stable if and only if .
Indeed, we set , so that the problem is transformed into
By projecting the equation onto the k-th eigenspace (and by using the notation ), we obtain
The condition for linear asymptotic stability of the steady state is therefore, for all ,
whence the result.
In this section, we present blow-up results (for different boundary conditions). Namely, we show that the solution to equation (1.1) blows up, under appropriate conditions, for both Dirichlet and Neumann boundary conditions, by using two different classical methods, i.e. Kaplan’s and concavity method.
The problem under consideration in this subsection is
Let be the solution to the eigenvalue problem
where μ is the first eigenvalue and Ω is a connected bounded domain. Then and ϕ is strictly positive and bounded in Ω. For convenience, we also impose the normalization condition . The main result in this subsection is:
Assume that Ω is a bounded smooth domain of and let satisfy
When , the above blow-up condition on the initial data can be roughly translated as , which is coherent with our global existence result, and with the assumption of Theorem 7. Note also that the homogeneous Dirichlet boundary condition on could be replaced in the theorem above by the less stringent condition on .
where we have used problem (4.2). Next we recall that
After applying Jensen’s inequality, we remind that , we get
from which the blowup of the solution can be obtained. Namely, by using the change of variables
we can obtain
or (as long as , remembering that )
When , we see that cannot remain true for
so that blowup occurs before time . When , a similar computation shows that a blowup also occurs, under the extra assumption . ∎
As has been stated in the beginning of Section 2, after the transformation , the equation can be rewritten into
In this subsection, we consider the following more general equation with homogeneous boundary condition,
and after giving a result about the blowup for the above general problem, we explain how (and under which conditions) it applies to problem (2.1). We refer the interested reader to [7, 3, 4, 10]. The main result of this subsection is the following:
Suppose that , and that h is a continuous real function such that for all , one has , where We assume that is a smooth nonnegative solution to problem (4.3) on such that
Then there exists (depending only on m, b, h and ) such that .
In other words, a blowup occurs before . More precisely,
Note that the function h is not assumed to be nonnegative. Actually, Theorem 7 still holds when h is negative, or when it changes sign.
Note that a significant limitation of this result is related to the assumption that pointwise. Indeed, this estimate is propagated at the formal level by the equation only in very special cases in which are linked by some equality, like when .
The proof is given by contradiction argument. Assume that the solution is global, and define
The idea of the concavity method is to find an and a such that is a concave function on . Then, from the concavity property of written at time in a differential way, we get
Using this inequality together with the fact that for all , we obtain an upper bound for the blow-up time (that is, the first time such that )
To prove this concavity property, we compute
from which it can be deduced that a sufficient condition for to be concave (on ) is that
In fact, we start by computing the derivative of the functional
and its second derivative
Next, we test (4.3) with , and get
We now test (4.3) with , and get
where we recall that
From (4.10), we can also deduce that
If the initial energy is strictly positive, namely
At this point, we use the assumption on to obtain
and conclude that
From the above inequality, we also get that is strictly increasing. Furthermore,
We now prove that there exists and such that
Therefore, we finally obtain
As observed at the beginning of the proof, the above inequality implies that we cannot extend the solution for all times, since (4.5) holds at a some point . ∎
In this annex, we provide a few explicit computations concerning problem (1.1) in the case when :
and we still consider only the nonnegative solutions.
We recall that it is equivalent to studying the following problem:
with if and only if .
This equation is a (reverse in time) porous medium equation, for which explicit solutions of Barenblatt type  can be computed (on a given time interval for the first type given below):
It is possible to take linear combinations of those solutions and still get solutions, though the equation is nonlinear, as long as the support of those solutions remain separate (more precisely, when each two solutions have support with empty intersections during the time of existence of the solutions).
For an initial datum
with , , ( ), the function defined by
is a solution to equation (5.1) on the time interval for
(if , ) provided that
Condition (5.2) can be rewritten without any direct reference to the time t in the following way:
If , then
If , then
If , then
Note also that for all and as soon as for all , .
The explicit solutions defined above feature in an explicit way the properties of blowup discussed previously. The value (or ) plays a decisive role in the existence or not of a blowup, as can be guessed from the study of the parabolicity regions of the equation.
Finally, we propose a figure illustrating the computations above. In Figure 1, a solution is drawn, with one positive bump and two negative ones, with the specific feature that when the branches of the bump coincide and connect. For this solution, we drew three different time instances.
This paper is a first attempt to tackle problems of the form
whose main characteristic is the fact that the quantity inside the Laplacian does not a priori have a fixed sign, so that global-in-time existence of solutions does not always hold. We proved the global existence and uniqueness of classical solutions for initial data and parameters such that the problem is of parabolic type. The non-existence of nontrivial steady states is studied, and some blow-up results using Kaplan’s method on the one hand, and the concavity method on the other hand, are also presented.
We thank Stephan Knapp for providing us with Figure 1.
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