Abstract and Applied AnalysisVolume 2011 (2011), Article ID 629692, 21 pagesdoi:10.1155/2011/629692

Research Article

Properness and Topological Degree for Nonlocal Reaction-Diffusion Operators

N. Apreutesei¹ and V. Volpert²

¹Department of Mathematics, “Gh. Asachi” Technical University of Iasi, Bd. Carol. I, 700506 Iasi, Romania²Institut Camille Jordan, UMR 5208 CNRS, Université Lyon 1, 69622 Villeurbanne, France

Received 5 December 2010; Accepted 25 February 2011

Academic Editor: Nobuyuki Kenmochi

Copyright © 2011 N. Apreutesei and V. Volpert. N. Apreutesei and . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

The paper is devoted to integro-differential operators, which correspond to nonlocal reaction-diffusion equations considered on the whole axis. Their Fredholm property and properness will be proved. This will allow one to define the topological degree.

1. Introduction

Consider the semilinear parabolic equation where Here is a bounded function, not necessarily continuous, on . The support of the function is supposed to be bounded, . We will also assume that . Conditions on the function will be specified below.

Integro-differential equations of this type arise in population dynamics (see [1, 2] and references therein). They are referred to as nonlocal reaction-diffusion equations. A travelling wave solution of (1.1) is a solution of this equation of the particular form . It satisfies the equation The constant is the wave speed. It is unknown and should be found together with the function . There are numerous works devoted to the existence [3–7], stability and nonlinear dynamics [1, 1, 2, 8–16] of travelling wave solutions of some particular cases of (1.1). Properties of travelling waves are determined by the properties of the integro-differential operator in the left-hand side of (1.3). In this paper we will study the Fredholm property of this operator and its properness. We will use them to define the topological degree and will discuss some applications.

Let , , the usual Holder spaces endowed with the norms We are interested in the solutions of equation (1.3) with the limits at , where the values are such that . We are looking for the solutions of (1.3) under the form , where , such that for and for . Thus (1.3) becomes Denote by the operator in the left-hand side of (1.5), that is ,

Suppose that is differentiable with respect to both variables. The linearization of about a function is the operator , where and are the derivatives of with respect to the first and to the second variable, respectively.

For the linearized operator , we introduce the limiting operators. Since for , there exist the limits , it follows that as and the limiting operators are given by where

We will now recall the main definitions and results concerning the essential spectrum and Fredholm property for linear operators and the properness of nonlinear operators.

1.1. Essential Spectrum and Fredholm Property

Let us recall that a linear operator acting from a Banach space into another Banach space is called a Fredholm operator if its kernel has a finite dimension, its image is closed, and the codimension of the image is also finite. The last two conditions are equivalent to the following solvability condition: the equation is solvable if and only if for a finite number of functionals from the dual space .

Suppose that . By definition, the essential spectrum of the operator is the set of all complex for which the operator , where is the identity operator, does not satisfy the Fredholm property. The essential spectrum of general elliptic boundary value problems in unbounded domains can be determined in terms of limiting operators [17]. For the integro-differential operators under consideration, since they have constant coefficients at infinity, the essential spectrum can be found explicitly. It is proved [5, 6] that the operator is normally solvable with a finite-dimensional kernel if and only if the equations do not have nonzero bounded solutions. Applying the Fourier transform to the last equations, we obtain where is the Fourier transform of the function . Thus, the operator is normally solvable with a finite-dimensional kernel if and only if the curves on the complex plane do not pass through the origin. Under some additional conditions, it can be also shown that the codimension of the operator is finite, that is, it satisfies the Fredholm property, and its index can be found.

A nonlinear operator is called Fredholm if the linearized operator satisfies this property. In what follows we will use the Fredholm property in some weighted spaces (see below).

1.2. Properness and Topological Degree

An operator is called proper on closed bounded sets if the intersection of the inverse image of a compact set with any closed bounded set in is compact. For the sake of brevity, we will call such operators proper. It is an important property because it implies that the set of solution of the operator equation is compact.

It appears that elliptic (or ordinary differential) operators are not generally proper when considered in Hölder or Sobolev spaces in unbounded domains. We illustrate this situation with a simple example. Consider the equation where . It can be verified that this equation has a positive solution , which converges to zero at infinity. This convergence is exponential. So the solution belongs to Hölder and to Sobolev spaces. Along with the function , any shifted function , is also a solution. Hence there is a family of solutions, and the set of solutions is not compact. Similar examples can be constructed for the integro-differential equation.

In order to obtain proper operators, we introduce weighted spaces with a growing at infinity polynomial weight function . The norm in this space is given by the equality Let us return to the previous example. The family of functions is not uniformly bounded in the weighted space. If we take any bounded closed set in the function space, it can contain the solutions only for a compact set of the values of . Therefore the set of solutions is compact in any bounded closed set. This example shows the role of weighted spaces for the properness of the operators.

Properness of general nonlinear elliptic problems in unbounded domains and in weighted spaces is proved in [18]. In this work, we will prove properness of the integro-differential operators. After that, using the construction of the topological degree for Fredholm and proper operators with the zero index [18], we will define the degree for the integro-differential operators. We will finish this paper with some applications of these methods to travelling waves solutions.

2. Properness in Weighted Spaces

In this section we study the properness of the semilinear operator .

Definition 2.1. If , are Banach spaces, an operator is called proper if for any compact set and any bounded closed set , the intersection is a compact set in .

Remark 2.2. The operator may not be proper from to (see the comments related to (1.11) from the introduction).

We will show in the sequel that is proper in some weighted spaces associated to and .

Let be the function given by . Denote , endowed with the usual norms and . We will work in the weighted Holder spaces and , which are and , respectively, with the norms and .

We begin with the following estimate for the integral term .

Lemma 2.3. Suppose that and is a function such that on , is bounded and . Then for some constant .

Proof. If , then . First we write Since is bounded for and , we have , for some positive constant .For every , , denoteIf , we have . If , thenSince and , then . In this case, the boundedness of and of , implies that for some . Thus the desired estimate holds and the lemma is proved.

We study the operator acting from into . In order to introduce a topological degree (in Section 4), we prove the properness of in the more general case when the coefficient and function depend also on a parameter . Let , be the operator defined through

We note that the linearization of about a function is while its limiting operators are given by

Assume that the following hypotheses are satisfied For any , the function and its derivatives with respect to and satisfy the Lipschitz condition: there exists such that for any . Similarly for and : , and the derivatives of are Lipschitz continuous in , that is, there exists a constant such that , for all from any bounded set in . (Condition NS) For any , the limiting equations do not have nonzero solutions in .

Lemma 2.4. Suppose that conditions hold. If and in , then

Proof. We have the equality Condition leads to the estimate of the first difference In view of hypothesis in , the above inequality allows us to conclude that the weighted norm converges to zero.In order to estimate the second difference, we begin with the following representation: Similarly, Therefore, ( denotes the first line in the right-hand side, : the second, : the third). The expressions and converge to zero in the weighted norm of , due to the Lipschitz condition with respect to of the derivatives of (see ). The expression is a function with a finite support. It also converges to zero in the weighted norm as . This concludes the proof.

We can now prove the properness of the -dependent operator . Denote by and similarly and .

Theorem 2.5. If , under assumptions , the operator from (2.6) is proper with respect to on .

Proof. Consider a convergent sequence , say in . Let be a solution in of the equation , such that We prove that one can choose a convergent in subsequence of the sequence . Without loss of generality we may assume that as . Equation can be written as Multiplying the equation by and denoting , , we derive that Indeed, since and , by (2.21) one easily obtains (2.22).The sequence is uniformly bounded in : Then it is locally convergent on a subsequence. More exactly, for every bounded interval of , there is a subsequence (denoted again ) converging in to a limiting function . By a diagonalization process we can prolong to such that . Since , , we can easily see that .Let be the limit that corresponds to . Then in and .We now want to pass to the limit as in (2.21) and (2.22). To this end observe that implies that Since is continuous from to (see ) and in , we derive that Passing to the limit as , uniformly on bounded intervals of in (2.21) and (2.22), one obtains thatSubtracting (2.27) from (2.22) and denoting , one findsRecall that as in . We show that in . Suppose that it is not the case. Then, without any loss of generality, we can chose a sequence such that . This means that . Let Then, Writing (2.28) in , one obtains We will pass to the limit as in (2.31). First we note that by (2.29) and (2.23), there exists such that as in . Next, it is obvious that while condition in leads to . Inequality (2.23) implies a similar estimate for , so and are bounded in . We also have for and for and Denote by , , and the three terms in the right-hand side. Hypothesis for infers that Next, (2.29) leads to for some . By (2.20) we obtain , , hence By the change of variable , it follows that uniformly on bounded intervals of . Hypothesis shows that On the other hand, for some , where . For , , with the aid of (2.29), we arrive at in . As above, since , , uniformly on bounded intervals of , we deduce that Now we may pass to the limit in (2.31). With the aid of (2.33)–(2.41) and , one arrives at which contradicts . Therefore we have proved that in .Now we have to show that in . To this end, we write (2.28) in the form , where is the linear part from the left-hand side and Using Lemma 2.1 from [6] for the linear operator , we can write We make use of Lemma 2.4, hypothesis for , and of the convergence in , to deduce that in . Since in , we conclude that in . The theorem is proved.

3. Fredholm Property in Weighted Spaces

Consider the operator , and its limiting operators

Recall here condition NS for , that is, hypothesis : for each , the limiting equations do not have nonzero solutions.

We prove now the Fredholm property of as an operator acting between the above weighted Holder spaces.

Theorem 3.1. If condition NS is satisfied, then the operator (acting between weighted spaces) is normally solvable with a finite-dimensional kernel.

Proof. Like in Theorem 2.2 from [6], we can prove that from to is normally solvable with a finite-dimensional kernel. To verify the property in the weighted spaces, we use Lemma 2.24 in [18]: if is normally solvable with a finite-dimensional kernel and the operator , is compact, then is normally solvable with a finite-dimensional kernel.Let be a sequence such that . We prove the existence of a subsequence of which converges in . Consider the sequence . Since , one can find a subsequence, denoted again , which converges locally in to a function , which can be prolonged to by a diagonalization process. We have , and in (in , for every bounded interval ).Let be such that . Then Observe that and in . Now we can write But Since , where as , is uniformly bounded and as locally with respect to , it follows that as , uniformly with respect to on . Similarly, , are uniformly bounded, , as locally and , as , so the first two terms from (3.4) tend to zero uniformly with respect to , as . This implies that as in . Therefore, with the aid of the local convergence in , we conclude that as in . The theorem is proved.

We prove now the Fredholm property for , under an additional hypothesis. To this end, let be the identity operator on .

Condition NS[]For each , the limiting equations associated to the operator do not have nonzero solutions in , for any .

We recall an auxiliary result from [6] which will be employed below.

Lemma 3.2. Consider the operators defined by , and the homotopy , , . Then there exists large enough such that the limiting equations do not have nonzero solutions for any .

Theorem 3.3. If Condition NS is satisfied, then , regarded as an operator from to , has the Fredholm property and its index is zero.

Proof. We put , and , , . Condition NS for implies Condition NS for . Then, Theorem 3.1 ensures that , regarded from to , is normally solvable with a finite-dimensional kernel. For operator , we have , , hence is a Fredholm operator and its index is .By Lemma 3.2 applied for , there exists large enough such that Condition NS holds for all , . In view of Theorem 3.1, it follows that the operators are normally solvable with a finite-dimensional kernel. In other words, the homotopy gives a continuous deformation from the operator to the operator , in the class of the normally solvable operators with finite-dimensional kernels. Such deformation preserves the Fredholm property and the index. Since the index of is zero, we derive that the index of all is zero. In particular, for and , one arrives at the conclusion that has the Fredholm property and its index is zero. This completes the proof.

4. Topological Degree

In this section we apply the topological degree construction for Fredholm and proper operators with the zero index constructed in [18] to the integro-differential operators.

Definitions 4. Recall in the beginning the definition of the topological degree. Consider two Banach spaces , , a class of operators acting from to and a class of homotopies Let be an open bounded set and such that , , where is the boundary of . Suppose that for such a pair , there exists an integer with the following properties.(i)Homotopy invariance. If and , for , , then (ii)Additivity. If , is the closure of and are open sets, such that and for all , then (iii)Normalization. There exists a bounded linear operator with a bounded inverse defined on all such that, for every bounded set with , The integer is called a topological degree.

4.1. Degree for Fredholm and Proper Operators

We now recall a general result concerning the existence of a topological degree which was proved in [18, 19].

Let and be Banach spaces, algebraically and topologically and let be an open bounded set.

Denote by the imbedding operator, , and by a class of bounded linear operators satisfying the following conditions:(a)the operator is Fredholm for all ,(b)for every operator , there is such that has a uniformly bounded inverse for all .

Denote by the class where is the Fréchet derivative of the operator .

Finally, one introduces the class of homotopies given by Here the properness of is understood in both variables and .

Theorem 4.1 (see [18]). For every and every open set , with there exists a topological degree .

Remark 4.2. Condition (b) can be weakened. Let and be two Banach spaces such that , where the inclusion is understood in the algebraic and topological sense. In the case of the Hölder space , this can be the space with an integer nonnegative . We can also consider some integral spaces [17]. Instead of (b) above we can impose the following condition [20]:for every operator , there is such that has a uniformly bounded inverse for all .

4.2. Degree for the Integrodifferential Operators

Now, let and be the weighted spaces introduced in the previous section, with , . We will apply Theorem 4.1 for the integro-differential operator of the form (1.6), where function , , for , for and and its derivatives with respect to and are Lipschitz continuous in ;the limiting equations do not have nonzero solutions in , .

Under these hypotheses, Theorem 2.5 assures that operator is proper. Moreover, its Fréchet derivative is from (1.7) and it is a Fredholm operator with the index zero (Theorem 3.3).

Consider the class of operators defined through (1.6), such that are satisfied. Consider also the class of homotopies , , of the form (2.6), satisfying andfor every , the equations do not have nonzero solutions in , . By Theorem 2.5 and Theorem 3.3, we infer that operators are Fréchet differentiable, proper with respect to and their Fréchet derivatives verify condition (a) above. Condition follows from the lemma in the appendix. Hence has the form (4.6). Applying Theorem 4.1 for the class of operators and the class of homotopies , we are led to the following result.

Theorem 4.3. Suppose that functions and satisfy conditions and . Then a topological degree exists for the class of operators and the class of homotopies.

5. Applications to Travelling Waves

In this section we will discuss some applications of the Fredholm property, properness and topological degree to study travelling wave solutions of (1.1). Let us begin with the classification of the nonlinearities. Denote We obtain this function from if we formally replace the kernel of the integral by the -function. The corresponding reaction-diffusion equation is called bistable if , monostable if one of these derivatives is positive and another one negative and, finally, unstable if . As it is well-known, it can have travelling wave solutions, that is solutions, which satisfy the problem Let be a solution of (5.3) with some . The operator linearized about this solution, has the essential spectrum given by two parabolas: Therefore the operator satisfies the Fredholm property if and only if . If this condition is satisfied, then the index of the operator is well defined. In the bistable case it equals 0, in the monostable case 1, in the unstable case 0 [9].

In the case of the integro-differential operator the essential spectrum is given by the curves where is the Fourier transform of the function . If we replace by , that is, by the -function, then the spectrum of the integro-differential operator coincides with the spectrum of the reaction-diffusion operator.

We note that

5.1. Fredholm Property

Bistable CaseLet (we recall that ). Suppose that and . Then for all since Hence the essential spectrum is completely in the left-half plane. This allows us to prove properness of the corresponding operators in weighted spaces and to define the topological degree.

Consider now the case where and . The principal difference with the previous case is that the essential spectrum of the integro-differential operator may not be completely in the left-half plane (Figure 1) though this is the case for the reaction-diffusion operator. Depending on the parameters, the essential spectrum can cross the imaginary axis for some pure imaginary values. However the linear operator remains Fredholm since the essential spectrum does not cross the origin; the nonlinear operator remains proper in the corresponding weighted spaces.

Figure 1: Schematic representation of a possible location of the essential spectrum of the operator .

Thus, the bistable case for the reaction-diffusion equation gives rise to two different cases for the integro-differential equation. We will call both of them bistable but will distinguish them when necessary.

Monostable CaseSuppose that and . Then is in the left-half plane for all ; is partially in the right-half plane, . The essential spectrum of the integro-differential operator given by the curves has a similar structure. It does not cross the origin, so that the operator satisfies the Fredholm property. The curve is partially in the right-half plane, . The curve can be completely in the left-half plane or partially in the right-half plane (Figure 1). Similar to the bistable case, there are two subcases in the monostable case.

IndexIn order to find the index of the operator , we consider the operator which depends on the parameter characterizing the width of the support of the function , supp . We recall that . Let , that is the value corresponds to the function in the operator .

Since the essential spectrum of the operator can be determined explicitly, then we can affirm that it converges to the essential spectrum of the operator as . Moreover, converges to in the operator norm. The essential spectrum of the operator does not cross the origin. Therefore it is normally solvable with a finite-dimensional kernel. Hence the index of the operator equals the index of the operator [12]. It is 0 in the bistable case and 1 in the monostable case (cf. [9]).

5.2. Topological Degree and Existence of Solutions

In the bistable case we can define the topological degree for the integro-differential operator and use the Leray-Schauder method to prove existence of solutions. In order to use this method we need to obtain a priori estimates of solutions. In [10], a priori estimates are obtained in the case where Thus, we can now conclude about the existence of waves for this particular form of the nonlinearity. More general functions will be considered in the subsequent works.

5.3. Local Bifurcations and Branches of Solutions

Other conventional applications of the degree are related to local bifurcations and global branches of solutions (see, e.g., [14]). We can now use the corresponding results for the integro-differential operator in the bistable case. Let us emphasize that these results apply in particular for the case where the essential spectrum of the linearized operator crosses the imaginary axis (see above). Therefore the wave persists in this case unless a priori estimates are lost.

Appendix

Sectorial property of an operator implies certain location of its essential spectrum and an estimate of the resolvent. For general elliptic problems in unbounded domains it is proved in [20]. A simple particular case of second-order operators on the axis is considered in [21]. In the lemma below we prove an estimate of the resolvent using this last result.

Lemma A.1. Let , where the coefficients of this operator are sufficiently smooth bounded functions. Then the operator , considered as acting in the same spaces, has a bounded inverse with the norm independent of for , where is sufficiently large.

Proof. Consider the equation We need to obtain the estimate of this equation where is independent of for all sufficiently large. Here and below we denote by the constants independent of , , and .We first prove the estimate Since the operator is sectorial [21], then Estimate (A.4) follows from the last one for sufficiently large.We can write (A.2) in the form We can choose such that the operator in the left-hand side is invertible. Hence This estimate and (A.4) give (A.3). The lemma is proved.

This lemma remains valid for the operators acting in the weighted spaces.

Acknowledgments

This work was partially supported by the LEA Math Mode between CNRS France and Romanian Academy through the joint project “Existence of travelling waves for nonlocal reaction-diffusion equations”.

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