On the chaotic properties of the von Foerster-Lasota equation

In this article we present a different approach to some of the results published in our recent paper (Brzeźniak and Dawidowicz in Semigroup Forum, 78(1):118–137, 2009). This new approach is based on a deep result from a paper (Ergod. Theory Dyn. Syst. 17(4):793–819, 1997) by Desch Schappacher and Webb.

the present paper we use a different approach to the same problem. This approach is based on applying some deep results from the paper [7]. Our main result in the most general form, see Theorem 5.5 can be summarized as follows.
Theorem 5. 5 Assume, that p ∈ [1, ∞) and that h : [0, 1] → C is a continuous function such that Then the C 0 -semigroup on L p (0, 1) generated by the following first order PDE is chaotic in the sense of Definition 2.2.
In Theorem 5.7 we formulate an analogous result for the Hölder spaces. The proof of both Theorems 5.5 and 5.7 is based on applying similar results when function h is constant, see Theorem in Sect. 4 An anonymous referee has asked whether our method can be used to get alternative prove of the main result from the paper [16] by Takeo. Our results from Sect. 5 not only give an affirmative answer to this question but in fact are stronger than the corresponding ones from [16].
In a forthcoming publication we plan to generalize results presented he to problems in multidimensional domains.
Notation By C, respectively R, we will denote the set of complex, resp. real, numbers. By C * , respectively R * , we will denote the set of nonzero complex, resp. real, numbers. A subset Z of C * will be called bounded if there exists r > 0 such that r ≤ |z| ≤ 1 r for all z ∈ Z. A function f with values in C * will be called bounded, if the range of f is a bounded subset of C * .

Formulation of the problem
Let consider the following differential equation together with the initial condition The paper [2] contains results about stability and chaos of the dynamical systems generated by this equation in various Banach spaces of functions on the interval [0, 1].
In particular we showed that for each such a space there is a threshold λ c such that the semigroup generated by (2.1) is chaotic or asymptotically stable depending whether λ > λ c or λ ≤ λ c . Our approach to the chaos was based on a classical Avez method used earlier by Lasota and Pianigiani [12] and the second named authour [4] to study the problem of the existence of chaos in the space of Lipschitz functions. The main aim of the present paper is to present an alternative proof of our results from [2] based on the aforementioned paper [7] by Desch, Schappacher and Webb. In that paper the authours proved the following fundamental result.
Let us now assume that X is a certain Banach space of functions defined on the interval [0, 1] and an operator A defined informally by the following formula For a given complex number γ let us define a function u γ by the following equality The function u γ is a candidate for the eigenvector u γ from Theorem 2.1. Clearly if u γ ∈ X and Φ ∈ X * , then the function F Φ introduced in Theorem 2.1 has the following representation where, for k ∈ N, Hence to prove the chaos property of the semigroup generated by the operator A (whose domain is denoted by D(A)), it is sufficient to show that the following three conditions are satisfied.
(i) The set {γ ∈ C : u γ ∈ D(A)} contains an open set with nonempty intersection with the imaginary axis.
In the following sections we will show how this program will be realized in the cases when

The case of the space L p
In the first we consider our problem in the space L p (0, 1), p ∈ [1, ∞). As in [17] it can be shown that in this case x (λ−Re (γ )) p dx < ∞.
The last condition is equivalent to the following one ) and in view of the equality (3.1) we infer that u γ ∈ D(A). Hence we proved that u γ ∈ D(A) iff condition (3.2) is satisfied.
We are ready to state the main result of this section.
Proof of Theorem 3.1 Let us fix λ ∈ R such that λ > − 1 p . Then the condition (i) is an obvious consequence of condition (3.2).
Next we shall show that the set of functions of the form is linearly dense in L p ((0, 1]). We begin the proof by observing, that each of these functions belong to L p (0, 1]. Indeed, since λp + 1 > 0 we have For a function f ∈ L p (0, 1) define a function If by the formula One can easily show that If belongs to L p (0, ∞) and that the map I : Using the change of variables z = −(λp + 1) ln x we obtain the following equality, for k ∈ N, Hence, we have which concludes the proof of condition (iii). In view of Theorem 2.1, Theorem 3.1 follows.

The case of the Hölder spaces
Let us assume that α ∈ (0, 1]. By C α ([0, 1]) we denote the Banach space of all complex valued Hölder continues functions with exponent α equipped with the standard norm: As in [2] we put The restriction of the function H 1,α to the space V α is a norm on that space which is equivalent to the norm induced by the original norm induced from the space c α ([0, 1]).
As in [17] it can be shown that in this case Before we formulate the main result in this section let us introduce some auxiliary notation. We begin with recalling that the Sobolev space W where v n = u − u n . Since the functions u n belong to W 1,p for every p ≥ 1 and by (4.2) H 1,α (u n − u) → 0, the proof of our claim is complete.
Proof Analogously to proof of the Let us define a function g λ n by g λ n (x) = x λ (− ln x) n , x ∈ (0, 1].
We can notice, that It is easy, to notice, that Hence, for all x, y ∈ [0, 1] α is convex and vanishing in 0, Therefore, for all x, y ∈ [0, 1], Hence This completes the proof.

The case of non-constant function h
In the paper [16] a theorem similar to our Theorem 3.1 has been proved but with the constant λ being replaced by a continuous function h : [0, 1] → C satisfying the following two conditions.

There exists a number δ > 0 such that Re
Takeo's result is different from ours, because for λ < 0, a constant function h = λ does not satisfy condition (1). We will prove a result stronger than Theorem 3.5 in [16]. For this purpose let us consider the following first order partial differential equation. ∂u At first we will show that the asymptotic behavior of the C 0 -semigroup generated by equation (5.1) depends only on the behavior of the function h in the neighborhood of 0. To be precise we will prove the following.
Consider also the following differential equation Assume, that both equations (5.1) and (5.2) generate C 0 -semigroups on X denoted by, respectively, {T t } t≥0 and { T t } t≥0 . Then, there exist such t 0 > 0 and a continuous function g : [0, 1] → C * , such that and Proof Let us begin with the observation that for every u ∈ X we have Let us next choose t 0 > 0 such that e −t 0 < δ. Let us then take t > t 0 . Then we have the following train of equalities for every x ∈ [0, 1].
h(x e s )ds

ds T t u(x).
This implies that with a function g defined by the formula we have, for every t > t 0 , The proof is complete.

Remark 5.2
If the operation of multiplication by the function g is an isomorphism of X, then the chaos property of the system generated by (5.1) is equivalent to the chaos property of the system generated by (5.2). Then the C 0 -semigroup generated by equation ( (1) and (2) from the beginning of this section, then h also satisfies our assumption (H). Hence our theorem 5.5 is stronger, than that in [16] Proof Let us choose λ and δ > 0 so that assumption (H) holds. Hence, the function ρ : [0, 1] → C defined by

Justification of the claim made in
is well-defined, continuous and ρ(0) = 0. In particular, ρ is a bounded function. Therefore, the multiplication by ρ defines a bounded, injective linear operator R on the space L p . It is easy to verify that if u is a solution to problem (2.1), then u defined by the following formula is the solution to problem (5.1). Moreover, the diagram , is the dynamical system generated by (5.1), resp. (2.1), is commuting.
To complete the proof it is sufficient to show that the range of R, i.e. the set R(L p ), is dense in L p . To prove this let us notice that for every ε > 0 there exits δ > 0 such that ρ(x) ∈ [δ, 1 δ ] for all x ∈ [ε, 1]. Take now u ∈ L p and let define an L p (0, 1)-valued sequence u n ∞ n=1 by Clearly lim n→∞ u n − u L p = 0 and by the property of ρ the functions u n ρ belong to L p . Therefore, u n = R( u n ρ ) ∈ R(L p ) what completes the proof. The above Theorem 5.5 can be generalized to the framework analogous to the one studied in [16].
Theorem 5.7 Assume that α ∈ (0, 1]. Assume that h : (0, 1] → C is a Lebesgue measurable function satisfying the following condition. (H2) There exist a real number λ such that λ > α that and hence, in view of (5.11), both functions ρ and 1 ρ are bounded and Lipschitz on [0, 1]. Hence by Proposition A.1 the operator R of multiplication by ρ is an isomorphism of V α . We can complete the proof by following the same argument as in proof of Theorem 5.5.
The last inequality concludes the proof.
Remark A.2 As far as we are aware Proposition A.1 is known in the case of spaces C α , see for instance [1]. Another proof of Proposition A.1 would be to use this result together with the fact that the spaces c α and V α are closures in the C α space of appropriate spaces of smooth functions.