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Abstract

This paper is concerned with a scalar nonlinear convolution equation which appears naturally in the theory of traveling waves for monostable evolution models. First, we prove that each bounded positive solution of the convolution equation should either be asymptotically separated from zero or it should converge (exponentially) to zero. This dichotomy principle is then used to establish a general theorem guaranteeing the uniform persistence and existence of semi-wavefront solutions to the convolution equation. Finally, we apply our abstract results to several well-studied classes of evolution equations with asymmetric non-local and non-monotone response. We show that, contrary to the symmetric case, these equations can possess at the same time the stationary, the expansion and the extinction waves.

Convolution, monostable equation, asymmetric non-local response.

Separation dichotomy and wavefronts ] Separation dichotomy and wavefronts for
a nonlinear convolution equation Carlos Gomez, Humberto Prado and Sergei Trofimchuk] \subjclassPrimary: 34K12, 35K57; Secondary: 92D25.

Carlos Gomez, Humberto Prado Sergei Trofimchuk

1 Instituto de Mátematica y Física, Universidad de Talca

Casilla 747, Talca, Chile

2 Departamento de Matemática, Universidad de Santiago de Chile

Casilla 307, Correo-2, Santiago-Chile


1 Introduction and main results

In this paper, we continue to study the nonlinear scalar convolution equation

(1)

introduced in [1]. Here is a finite measure space, an appropriate kernel is integrable on with while measurable is continuous in for every fixed and there exists . Our goal here is to establish a satisfactory criterion for the existence of semi-wavefronts (i.e. positive, bounded, and vanishing at either or solutions) to (1). Then in Section 5 we will apply this criterion to two non-local and asymmetric monostable evolution equations. In this way, we develop further some ideas from [19]. It should be noted that equation (1) is one of valid general forms for the description of traveling wave profiles. Other similar yet non-equivalent functional equations can be found in [2, 5, 6, 17, 21, 22].

It was shown in [1] that the characteristic function

plays a key role in the investigation of equation (1). In particular, the following holds (see [1, Theorem 2]):

Proposition 1.

Assume . Let be a bounded solution to equation (1). If and for each fixed , then is well defined and has a zero on some non-degenerate interval .

And as we will prove below under the additional mild conditions

(C)

For each there is a measurable such that

(P)

Bounded solution of (1) vanishes at some point only if ,

the conclusion of Proposition 1 remains true even if we replace assumption by a weaker . Moreover, in Theorem 1.2 below we prove the equivalence of these two properties for solutions of equation (1). In view of Theorem 1 and Lemma 3 from [1], this result has the following nice consequence: under a few natural restrictions on , each bounded positive solution with converges exponentially to zero at .

Note that assumption (P) can be easily checked due to

Lemma 1.1.

Assume that there are , and a measurable such that implies (i) if and only if ; (ii) for all . Then implies .

Proof.

Suppose that . Then we have almost everywhere on . Hence, for some , we obtain that for all . Thus , . Similarly, if for some , then for all in an open neighborhood of . In consequence, the set of zeros of continuous is open and closed, and we may conclude that . ∎

We are ready to state our first main result:

Theorem 1.2.

Assume (C), (P) with . Then the following dichotomy holds for each bounded solution of (1): either or . The similar alternative is also valid at .

An easy combination of results from Proposition 1 and Theorem 1.2 leads to

Corollary 1.

If does not have any positive [negative] zero and is a positive bounded solution of (1), then [respectively, ]. As a consequence, equation (1) can not have positive pulse solutions (i.e. solutions satisfying ).

Proof.

Since and is concave on its maximal domain of definition, all real zeros of should be of the same sign (if they exist). ∎

Let denote either or . By Corollary 1, we have the following point-wise persistence property: for each bounded positive solution of Eq. (1) satisfying there is some such that . This fact allowed us to exclude the latter inequality from the definition of semi-wavefronts (cf. with boundary conditions (1.6) in [3]). Now, in order to prove the uniform persistence (this means that the above mentioned can be chosen independent of ) as well as the existence of solutions to equation (1), we will impose additional conditions on its nonlinearity:

(N)

N1. There exists such that increases in for

each fixed and . Consider the monotone function

N2. There exists such that is strictly increasing on , and where .

Set It is clear that and that the graphs of and have similar geometrical shapes. In particular, they share the same critical points.

Figure 1: Nonlinearity under hypotheses (N) and .

If is a constant solution of (1), then because of the relation

Several additional important properties of are listed below:

Lemma 1.3.

Let and (C), (N) hold. Then, for some ,
1. is positive for and there exists ;
2. and ;
3. while for .

Proof.

Let us show, for instance, that . In view of (C), this derivative exists and is equal to . Thus if and only if . Observe that since and we do not exclude the case . ∎

Using the above framework, we can improve conclusions of Theorem 1.2:

Theorem 1.4.

Assume along with all conditions of Theorem 1.2 and take as in Lemma 1.3. Let be a positive bounded solution of equation (1). If then and for some .

Our third result can be considered as a further development of Theorem 6.1 from [5] which was proved for a single-point space and under more restrictive conditions on the nonlinearity :

Theorem 1.5.

Assume (N), that is finite and that for all If is defined and changes its sign on some open interval [respectively, on ], then equation (1) has at least one semi-wavefront, with and [respectively, with ]. Moreover, if equation has exactly two solutions and on , and the point is globally attracting with respect to the map then .

Remark 1.

It is worth noting that the existence of (and consequently of ) is not at all obligatory for the existence of semi-wavefronts. Indeed, suppose that there is a measurable satisfying and consider associated characteristics

We assume also that (N) holds, possesses the second and the third properties of Lemma 1.3, and (this generalizes assumption ). Then all conclusions of Theorem 1.5 remain valid if we replace in its formulation with . See the second part of Section 4 for more details.

The paper is organized as follows. In Section 2, we prove the dichotomy principle. The first part of Section 3 shows how to avoid possible troubles with unbounded solutions of the convolution equation. The second part of the same section presents a short proof of the uniform persistence property. These preliminary results are essential for proving the existence theorem in Section 4. Finally, several applications are considered in the last section of the paper. Associated characteristic equation is analyzed in Appendix.

2 The proof of the dichotomy principle (Theorem 1.2)

.

1. Let be a bounded solution of (1). It is easy to see that is uniformly continuous on . Indeed, setting we find that

where because of the continuity of translation in and the Lebesgue’s dominated convergence theorem.

2. Next we prove an analog of Proposition 1 when and is bounded and positive. We have

Set , then and

(2)

Let [] be characteristic equation for Eq. (1) [Eq. (2), respectively]. We have

and thus . By Proposition 1, has at least one positive root. Therefore has at least one negative zero.

3. Now, let suppose that and . Since and is concave on its maximal domain of definition, all real zeros of should be of the same sign (if they exist). Suppose that does not have any real negative [respectively, positive] root. For a fixed there exists a sequence of intervals , , such that , [respectively, , ] and , . Note that . Indeed, otherwise we can suppose that . By the pre-compactness of in the compact-open topology of , the sequence [respectively, ] of solutions to Eq. (1) contains a subsequence converging to a non-negative bounded function such that , . Since, due to the Lebesgue’s dominated convergence theorem, satisfies (1) as well, this contradicts to (P). Thus and we can suppose that has a subsequence converging to a bounded positive solution of (1) satisfying for all [respectively, for all ]. Since [respectively, ] is impossible due to Proposition 1 and the second step of the proof, we conclude that [respectively, ]. Let [respectively, ] be such that , then has a subsequence converging to a positive solution of (1) such that . Now, let us consider . Each satisfies

(3)

where . We claim that has a subsequence converging to a continuous solution of equation

(4)

Indeed, the sequence is equicontinuous because of

where was defined on step 1. In addition,

so that, by the Lebesgue’s dominated convergence theorem, we can pass to the limit (as ) in (3). Hence, our claim is proved.

4. The proof of Theorem 1.2 will be finalized, if we show that (4) cannot have any nontrivial continuous solution . Since

there exists such that

Integrating equation (4) between and , we obtain

from which

Therefore . Now we easily get a contradiction by integrating (4) over the real line:

Hence, the dichotomy principle of Theorem 1.2 is established at . The other case can be reduced to the previous one by doing the change of variables and considering equation (2) with instead of (1) with .

3 The uniform permanence property

3.1 The uniform boundedness of solutions.

It should be noted that, in general, equation (1) might have unbounded continuous solutions. Corresponding examples can be constructed by taking appropriate linear . Nevertheless, as we show in the continuation, with conditions (N) and being assumed, it is easy to avoid eventual troubles with unbounded solutions in the following two ways:
Modification of the convolution equation. Consider

and

Then is a strictly increasing function. Indeed, , and we know that strictly increases in . Furthermore, for , we have where is a constant. Hence is strictly increasing on . If we set , we find that for .

Let us consider now a modified convolution equation

Each its solution is bounded;

The latter estimate assures that simultaneously satisfies (1).

Subexponential solutions. Assume additionally that

(5)

If, for some , satisfies (1) and , then

(6)

where . Suppose, in addition, that

and . The first inequality holds automatically if because of . Similarly, since , the second inequality holds whenever is finite.

Lemma 3.1.

If (5) holds, and is a finite number then each solution of (1) is bounded. In fact,

Proof.

Using in (6) and arguing by induction, we find that

Then, by passing to the limit as , we obtain the required estimate. We recall here that , and . The inequality follows from Lemma 3.2 proved in continuation. ∎

3.2 The proof of the uniform persistence (Theorem 1.4)

Let a bounded positive solution of the equation (1). Set

Lemma 3.2.

.

Proof.

Let be such that . We have

Thus . Similarly, . ∎

Now, assumption , and yield , cf. Fig. 1. Hence, due to the positivity of , there exists such that . Then, applying Theorem 1.2 and Corollary 1, we find that and . Making use of our standard limiting solution argument, we see that, for some , the sequence is converging in the compact-open topology of to some function solving equation (1). By Lemma 3.2, we have which implies .

Remark 2.

The last argument in the proof of Lemma 3.2 shows also that , where and .

4 The proof of the existence

Throughout all this section, we are assuming that holds, and

(7)

1. For a moment, let us suppose additionally that

is bounded and uniformly linear in some right neighborhood of the origin: , , .

Let be the leftmost positive solution of equation , and set

where and are such that . We want to prove the existence of fixed points , , , to the operator

A formal linearization of along the trivial steady state is given by

On the other hand, , . Indeed, we have, for a fixed ,

Lemma 4.1.

is a closed, bounded, convex subset of and is a completely continuous map.

Proof.

It is clear that is a closed, bounded, convex subset of . To prove that , we observe first that, for ,

Next, if for some we have that , then so that , which implies that . If then . In either case,

Now, we claim that is a precompact subset of . Indeed, the convergence in is the uniform convergence on compact subsets of . On the other hand, the set of functions from restricted on every fixed compact interval is obviously uniformly bounded and is also equicontinuous in virtue of the estimation (uniform with respect to ):

(8)

Finally, the continuity of in can be easily established by using the dominated convergence theorem and the compactness property of . ∎

Then Lemmas 3.1, 3.2, 4.1 and the Schauder’s fixed point theorem yields

Theorem 4.2.

Assume and let be the leftmost positive zero of . Then has at least one fixed point in . If is a finite number then is also finite and . Moreover, if the point is globally attracting with respect to the map then .

It should be noted that the last statement of this theorem is a straightforward consequence of Remark 2 (see also [11] where various conditions assuring the global stability property of are given).

2. Next we show how to reduce the general situation to the case studied in the first part of this section. Consider the sequence of measurable functions

all of them continuous in for each fixed and satisfying hypothesis with . Note that converges uniformly to on for every fixed . Next, set and consider continuous increasing functions

Since , the sequence is monotone. Now, for each fixed , we have that