Devaney Chaos and Distributional Chaos in the Solution of Certain Partial Differential Equations

and Applied Analysis 3 2. Criteria to Determine Devaney Chaos and Distributional Chaos The following statement of the Hypercyclicity Criterion for C0-semigroups is inspired by its version for operators in 9 . Theorem 2.1 Hypercyclicity Criterion for C0-semigroups; see 10, Th. 2.1 , 11, Crit. 3.1 , and 1, Th. 7.26 . Let T be a C0-semigroup in L X . If there are a sequence {tn}n ⊂ R with limn→∞tn ∞, dense subsets Y,Z ⊂ X and maps Stn : Z → X, n ∈ N such that i limn→∞Ttny 0 for all y ∈ Y , ii limn→∞Stnz 0 for all z ∈ Z, and iii limn→∞TtnStnz z for all z ∈ Z, then T is hypercyclic. Sometimes the Hypercyclicity Criterion is hard to be applied, and in fact, it only provides one of the ingredients of Devaney chaos. Moreover, in many situations, we can have the infinitesimal generator of a C0-semigroup but we do not have the explicit representation of its operators. This is quite common when we deal with the solution C0-semigroups associated to certain partial differential equations. Desch et al. gave a criterion which permits us to state the Devaney chaos of a C0-semigroup in terms of the abundance of eigenvectors of the infinitesimal generator. Theorem 2.2 Desch-Schappacher-Webb Criterion; see 12, 13 . Let X be a complex separable Banach space, and let T be a C0-semigroup in L X with infinitesimal generator A,D A . Assume that there exist an open connected subset U ⊂ C and a weak holomorphic function f : U → X, such that i U ∩ iR/ ∅, ii f λ ∈ ker λI −A for every λ ∈ U, and iii for any x∗ ∈ X∗, if 〈f λ , x∗〉 0 for all λ ∈ U, then x∗ 0. Then the semigroup T is chaotic. For the case of distributional chaos, Albanese et al. obtained the following sufficient condition, inspired by the result for the discrete case given by Bermúdez et al. in 4 . Theorem 2.3 Dense Distributionally Irregular Manifold Criterion; see 8, Cor. 2 . Let T be a C0-semigroup in L X . Assume that there exist i a dense subset X0 ⊂ X such that limt→∞Ttx 0 for each x ∈ X0, and ii a Lebesgue measurable subset B ⊂ R 0 withDens B 1 satisfying either ∫ B ‖Tt‖ −1dt <∞, or X is a complex Hilbert space and ∫ B ‖Tt‖ −2dt <∞. Then T has a dense manifold whose nonzero vectors are distributionally irregular vectors. (When this happens, one says that T has a dense distributionally irregular manifold). Furthermore, they also proved that a C0-semigroup T is distributionally chaotic if, and only if, T has a distributionally irregular vector 8, Th. 3.4 . Therefore, Theorem 2.3 can be also understood as a criterion for distributional chaos. In the sequel, we will apply this criterion several times in order to determine that certain C0-semigroups are distributionally chaotic. 4 Abstract and Applied Analysis 3. Distributionally Chaotic C0-Semigroups In this section, we consider several examples of C0-semigroups that are already known to be Devaney chaotic, and we will study when they exhibit distributional chaos. These examples will be considered on the following spaces:


Introduction
During the last years, several notions have been introduced for describing the dynamical behavior of linear operators on infinite-dimensional spaces, such as hypercyclicity, chaos in the sense of Devaney, chaos in the sense of Li-Yorke, subchaos, mixing and weakly mixing properties, and frequent hypercyclicity, among others. These notions have been extended, as far as possible, to the setting of C 0 -semigroups of linear and continuous operators. The recent monograph by Grosse-Erdmann and Peris Manguillot 1 is a good reference for researchers interested in the study of linear dynamics. In particular, it contains a chapter dedicated to analyze the dynamics of C 0 -semigroups. See also 2 , which contains additional information on further topics in the area.
In the sequel, let X be an infinite-dimensional separable Banach space. A C 0 -semigroup T is a family of linear and continuous operators {T t } t≥0 ⊂ L X such that T 0 Id, T t T s T t s for all t, s ≥ 0, and for all s ≥ 0, we have lim t → s T t T s pointwise on X.
We say that a C 0 -semigroup T is Devaney chaotic if it is transitive and it has a dense set of periodic points. On the one hand, a C 0 -semigroup T is transitive if for any pair of nonempty open sets U, V ⊂ X there is some t > 0 such that T t U ∩ V / ∅. In this setting, transitivity is equivalent to the existence of some x ∈ X with dense orbit in X, that is, {T t x, t ≥ 0} X, see for instance 1, Th. 1.57 . This phenomenon is usually known in operator theory as hypercyclicity, and such a vector x is said to be a hypercyclic vector for T. On the other hand, a vector x ∈ X is said to be a periodic point for T if there is some t > 0 such that T t x x.
Other definitions of chaos, such as the one introduced by Li-Yorke and the one of distributional chaos introduced by Schweizer and Smítal, have been also considered. The relationships between these two notions in the Banach space setting have been recently studied in 3 . We recall that a C 0 -semigroup T is said to be Li-Yorke chaotic if there exists an uncountable subset Γ ⊂ X, called the scrambled set, such that for every pair x, y ∈ Γ of distinct points, we have that Clearly, every hypercyclic C 0 -semigroup is Li-Yorke chaotic: we just have to fix a hypercyclic vector x ∈ X and consider Γ : {λx; |λ| ≤ 1} as a scrambled set, as it is indicated in 4, page 84 . Distributional chaos is inspired by the notion of Li-Yorke chaos. In order to define it, given a subset B ⊂ R 0 , we define its upper density as Dens B : lim sup t → ∞ 1/t μ B ∩ 0, t , where μ stands for the Lebesgue measure on R 0 .
are an uncountable set S ⊂ X and δ > 0, so that for each ε > 0 and each pair x, y ∈ S of distinct points, we have The set S is called the scrambled set. If S is dense in X, then T is said to be densely distributionally chaotic.

1.3
Such vectors were considered in 5 so as to get a further insight into the phenomenon of distributional chaos, showing the equivalence between a distributionally chaotic operator and an operator having a distributionally irregular vector.
The first systematic approach to distributional chaos for linear operators was taken in 6 , where this phenomenon was studied in detail for backward shift operators. Later, Peris and Barrachina proved that for translation C 0 -semigroups on weighted L p spaces, 1 ≤ p < ∞, Devaney chaos implies distributional chaos. However, the converse does not hold. They also provide an example of a translation C 0 -semigroup that is distributionally chaotic but it is neither Devaney chaotic nor hypercyclic 7 .
Hypercyclicity and Devaney chaos are hard to observe directly from the definition. The Hypercyclicity Criterion, in any of its forms, and the Desch-Schappacher-Webb Criterion have turn out to be powerful tools in order to verify these properties. Very recently, Albanese et al. have stated a criterion in order to show that a C 0 -semigroup is distributionally chaotic and has a dense distributionally irregular manifold 8 . Our goal is to study distributional chaos for some C 0 -semigroups that are already known to be Devaney chaotic. The dynamics exhibited by these C 0 -semigroups will motivate us to pose some open questions.

Criteria to Determine Devaney Chaos and Distributional Chaos
The following statement of the Hypercyclicity Criterion for C 0 -semigroups is inspired by its version for operators in 9 .
Theorem 2.1 Hypercyclicity Criterion for C 0 -semigroups; see 10, Th. 2.1 , 11, Crit. 3.1 , and 1, Th. 7.26 . Let T be a C 0 -semigroup in L X . If there are a sequence {t n } n ⊂ R with lim n → ∞ t n ∞, dense subsets Y, Z ⊂ X and maps S t n : Z → X, n ∈ N such that i lim n → ∞ T t n y 0 for all y ∈ Y , ii lim n → ∞ S t n z 0 for all z ∈ Z, and iii lim n → ∞ T t n S t n z z for all z ∈ Z, then T is hypercyclic.
Sometimes the Hypercyclicity Criterion is hard to be applied, and in fact, it only provides one of the ingredients of Devaney chaos. Moreover, in many situations, we can have the infinitesimal generator of a C 0 -semigroup but we do not have the explicit representation of its operators. This is quite common when we deal with the solution C 0 -semigroups associated to certain partial differential equations. Desch et al. gave a criterion which permits us to state the Devaney chaos of a C 0 -semigroup in terms of the abundance of eigenvectors of the infinitesimal generator.

Theorem 2.2 Desch-Schappacher-Webb Criterion; see 12, 13 . Let X be a complex separable Banach space, and let T be a C 0 -semigroup in L X with infinitesimal generator A, D A . Assume that there exist an open connected subset U ⊂ C and a weak holomorphic function
Then the semigroup T is chaotic.
For the case of distributional chaos, Albanese et al. obtained the following sufficient condition, inspired by the result for the discrete case given by Bermúdez et al. in 4 .

Theorem 2.3 Dense Distributionally Irregular Manifold Criterion
Then T has a dense manifold whose nonzero vectors are distributionally irregular vectors. (When this happens, one says that T has a dense distributionally irregular manifold).
Furthermore, they also proved that a C 0 -semigroup T is distributionally chaotic if, and only if, T has a distributionally irregular vector 8, Th. 3.4 . Therefore, Theorem 2.3 can be also understood as a criterion for distributional chaos. In the sequel, we will apply this criterion several times in order to determine that certain C 0 -semigroups are distributionally chaotic.

Distributionally Chaotic C 0 -Semigroups
In this section, we consider several examples of C 0 -semigroups that are already known to be Devaney chaotic, and we will study when they exhibit distributional chaos. These examples will be considered on the following spaces: where I is an interval on R and ρ a weight function. If ρ x 1, then we will simply denote it as L p I, C , 1 ≤ p < ∞. The hypothesis on ρ may be different on each example.
In 14 , Takeo considered the following first order abstract Cauchy problem on where ζ and h are bounded continuous functions defined on I. This ordinary differential equation has been used to model the dynamics of a population of cells under simultaneous proliferation and maturation 15 . When ζ x is constant and equal to 1 and

Theorem 3.1. If h x is a real function and there is a measurable set
Proof. If we define ρ x exp −p x 0 h s ds , then the operators of {T t } t≥0 can be rewritten as This function ρ x is an admissible weight function in the sense of 12, Def. 4.1 , which ensures that the left translation semigroup {τ t } t≥0 defined as Abstract and Applied Analysis 5 Let us define φ f x ρ x 1/p f x and consider the following commutative diagram: The hypothesis on B let us conclude that {τ t } t≥0 is distributionally chaotic on L p ρ R 0 , C , see 7, Th. 2.3 . Therefore, the conclusion is obtained since distributional chaos is preserved under conjugacy 6, Th. 2 . On the one hand, if h x is constant and equal to 1, then we have that {T t } t≥0 is Devaney chaotic and distributionally chaotic on L p R 0 , C . On the other hand, taking B 0, 2 ∪ n∈N n 2 It is also hypercyclic since ρ n 2 e − n 2 −2n 4 p for n ≥ 2, which yields that lim inf x → ∞ ρ x 0 14, Th. 2.2 . However, it cannot be Devaney chaotic since R 0 ρ x dx ∞. To sum up, we have an example of a C 0 -semigroup that is hypercyclic, distributionally chaotic, but it is not Devaney chaotic. This example can be compared with the example provided in 7, Ex. 2. of a distributionally chaotic translation C 0 -semigroup that is neither hypercyclic nor chaotic. Now, let us consider another example of a C 0 -semigroup whose dynamical behavior was already discussed in 14 : Let ρ : 0, 1 → R be a continuous function such that there exist constants M ≥ 1, ω ∈ R, and γ < 0 such that With such a function ρ, we can consider the spaces L p ρ 0, 1 , C , for 1 ≤ p < ∞. The family of operators {S t } t≥0 with S t f x f e γt x , t ≥ 0 defines a C 0 -semigroup on them 14 .
Proof. Let us apply Theorem 2.3. Take X 0 {f ∈ C 0, 1 , C : f 0 0}. This set is dense in L p ρ 0, 1 , C and, clearly, lim t → ∞ S t f 0 for every f ∈ X 0 , which fulfills condition i in Theorem 2.3.
Let us prove that p,ρ dt is finite: Fix t > 0 and a continuous function g on 0, 1 with g p,ρ 1, for instance g x 1/ρ x 1/p .

Abstract and Applied Analysis
There is some t 0 > 0 such that for t > t 0 , we have e γt Since g t p,ρ ≤ t −2 and S t g t g, then S t p,ρ ≥ t 2 for t ≥ t 0 . So that p,ρ dt is convergent, which yields the conclusion.

Remark 3.4.
The assumption γ < 0 forces w > 0: If not, take any x ∈ 0, 1 . Taking limits when t → ∞ in the inequality ρ x /ρ e γt x ≤ Me ωt we have ρ x /ρ 0 ≤ 0, which is a contradiction because ρ is a positive continuous function.
We return to the initial value problem stated in 3.2 . Consider the case when I 0, 1 , ζ x : γx, γ < 0, and h ∈ C 0, 1 , C . Under these hypotheses, the C 0 -semigroup { T t } t≥0 defined as gives the solution C 0 -semigroup to 3.2 on L p 0, 1 , C , 1 ≤ p < ∞ 14, Th. 3.4 . The particular case when γ −1 and h x −1/2 was studied using the Wiener measure in 15 .
Proof. We apply again Theorem 2.3: Condition i holds in the same way as in the proof of Theorem 3.3 taking X 0 {f ∈ C 0, 1 , C : f 0 0}.

Abstract and Applied Analysis 7
In order to verify condition ii , let α ∈ R be such that min{ h x : x ∈ 0, 1 } > α > γ/p. For every t > 0, we define f t as a function with ||f t || p 1 and supp f t ⊂ 0, e γt . Using it, we have the following estimations for || T t || p : Under the hypothesis of the last theorem, Takeo proved that { T t } t≥0 is Devaney chaotic by applying the Desch-Schappacher-Webb Criterion 14 . Independently, Brzeźniak and Dawidowicz also proved that { T t } t≥0 is Devaney chaotic when γ −1 and h x λ ∈ R with λ > −1/p, that is known as the von Foerster-Lasota equation 18, Theorems 8.3 and 8.4 . Furthermore, they also showed that for λ ≤ −1/p the orbits of all elements tend to 0, which makes chaos disappear. Therefore, we can affirm that Devaney chaos coincides exactly with distributional chaos for the same values of λ. As we will see later, this is due to the fact that Devaney chaos can be obtained here from the Desch-Schappacher-Webb Criterion. This can be easily seen if we reformulate Theorem 2.3 in terms of the infinitesimal generator of the C 0 -semigroup. The following result is a continuous version of 4, Cor. 31 . i there is a dense subset X 0 ⊂ X with lim t → ∞ T t x 0, for each x ∈ X 0 , and ii there is some λ ∈ σ p A with λ > 0, then T has a dense distributionally irregular manifold. In particular, T is distributionally chaotic.
Proof. Fix t > 0. On the one hand, if condition i holds, then we have lim n → ∞ T n t x 0 for every x ∈ X 0 . On the other hand, by the point spectral mapping theorem for C 0 -semigroups, since λ ∈ σ p A , then e λt ∈ σ p T t . Therefore r T t ≥ |e λt | > 1 and, by 4, Cor. 31 , T t admits a dense distributionally irregular manifold. By 8, Rem. 2 , this is equivalent to say that T admits a dense distributionally irregular manifold. Furthermore, T is distributionally chaotic 8, Prop. 2 .
Remark 3.8. Clearly, the conditions in Theorem 3.7 hold whenever the Desch-Schappacher-Webb Criterion can be applied. Therefore, among others, the following C 0 -semigroups that are known to be Devaney chaotic are also distributionally chaotic and have a dense distributionally irregular manifold : 19  Finally, Brzeźniak and Dawidowicz also studied in 18 Devaney chaos for the case γ −1 and h x λ ∈ R in certain subspaces of Hölder continuous functions on 0, 1 . For α ∈ 0, 1 , 0 < r ≤ 1, we define the space C α r 0, 1 of functions f : 0, 1 → R such that f α,r : sup For α ∈ 0, 1 , let us consider V α 0, 1 the space of functions f ∈ C α 1 0, 1 : lim In 18 , it is shown that V α 0, 1 is a separable Banach space endowed with the norm ||f|| α,1 . Furthermore, following a constructive approach, it is proved that if γ −1 and h x λ > α, then { T t } t≥0 is Devaney chaotic there. We will prove that in this case { T t } t≥0 is also distributionally chaotic.
Proof. We will apply Theorem 2.3 again. Since { T t } t≥0 is Devaney chaotic, then there is a dense set of points with bounded orbit. Therefore { T t } t≥0 is weakly mixing 1, Th. 7.23 , and any non-trivial operator T t is weakly mixing, too 11, Th. 2.4 . Fix t > 0. By 9, Th. 2.3 , T t satisfies the Hypercyclicity Criterion. So that, there is a dense set X 0 ⊂ V α 0, 1 such that lim n → ∞ T n t x 0 for all x ∈ X 0 . Using the local equicontinuity of { T t } t≥0 , we have lim t → ∞ T t x 0 for every x ∈ X 0 and condition i holds.

Discussion and Conclusions
Consider the initial value problem of 3.2 on L 1 R 0 , C with ζ x 1 and h x kx k−1 / 1 x k . Here, the solution C 0 -semigroup {T t } t≥0 is defined as The C 0 -semigroup {T t } t≥0 defined in 4.1 is distributionally chaotic on L 1 R 0 , C by Theorem 3.1. The hypercyclicity of this C 0 -semigroup for k 2 was obtained by El Mourchid in 24 and the Devaney chaos by Grosse-Erdmann and Peris in 1, Prop. 7.34 . In this case, the point spectrum of the infinitesimal generator is the closed left half plane. This inhibits the Desch-Schappacher-Webb Criterion to be applied in the way it has been formulated. Nevertheless, El Mourchid observed that the hypercyclic behavior of this C 0 -semigroup is essentially due to the imaginary eigenvalues of its infinitesimal generator 24 , see also 1, Ex. 7.5.1 . In fact, the Desch-Schappacher-Webb Criterion can be strengthened and reformulated as follows.
Theorem 4.1 see 24, Th. 2.1 and 1, Th. 7.31 . Let X be a complex separable Banach space, and let T be a C 0 -semigroup on X with infinitesimal generator A, D A . If there are a < b and continuous functions f j : a, b → X, j ∈ J, with 1 f j s ∈ ker is I − A for every s ∈ a, b , j ∈ J, and 2 span{f j s ; s ∈ a, b , j ∈ J} is dense in X, then the semigroup T is Devaney chaotic.
To sum up, we have seen that even when we apply this stronger version of the Desch-Schappacher-Webb Criterion for the C 0 -semigroup in 4.1 , asking only for an abundance of eigenvalues of real part equal to zero, then we can also prove that there is a dense distributionally irregular manifold. Therefore, we can pose the following problem. Problem 1. Do the hypothesis in Theorem 4.1 imply the existence of a dense distributionally irregular manifold for T? If not, is there at least a distributionally irregular vector for T?
By the equivalence between a C 0 -semigroup with a distributionally irregular vector and a distributionally chaotic C 0 -semigroup, 8, Th. 3.4 , the former problem can also be presented as follows.
Problem 2. Do the hypothesis in Theorem 4.1 imply that T is distributionally chaotic?
These questions could have a positive answer, but it is still unknown whether Devaney chaos implies distributional chaos on C 0 -semigroups.

Problem 3.
Are there examples of Devaney chaotic C 0 -semigroups which are not distributionally chaotic?
A C 0 -semigroup is said to be frequently hypercyclic if there exists some x ∈ X such that for every nonempty open set U ⊂ X, the set U x : {s ≥ 0 : T s x ∈ U} has positive lower density, that is lim inf t → ∞ 1/t μ U x ∩ 0, t is positive. In 25 , Mangino and Peris observed that with the same arguments used in 12, 24 , one can show that the Desch-Schappacher-Webb Criterion implies frequent hypercyclicity. They also provide the Frequent Hypercyclicity Criterion for C 0 -semigroups 25, Th. 2.2 . So that, one can raise the following question. The hypothesis in Theorem 4.1 also yields the mixing property for the C 0 -semigroup T, see 1 . We recall that a C 0 -semigroup is topologically mixing if for any pair of nonempty open sets U, V ⊂ X there is some t 0 > 0 such that T t U ∩V / ∅ for all t ≥ t 0 . Clearly, topological mixing implies transitivity i.e., hypercyclicity , but it is strictly stronger than it. Topologically mixing translation C 0 -semigroups on the weighted L p ρ -spaces considered in this paper are characterized by the condition lim t → ∞ ρ t 0 26, Th. 4.3 .
On the one hand, the aforementioned example of Peris and Barrachina 7, Ex. 2.7 provides an example of a distributionally chaotic C 0 -semigroup that it is not topologically mixing. On the other hand, in 27 , there is an example of a backward shift operator on a weighted sequence space p v , 1 ≤ p < ∞, that is topologically mixing but it is not distributionally chaotic. This operator will provide us an analogous counterexample in the frame of C 0 -semigroups. We thank A. Peris for this counterexample.
Example 4.2. Consider the sequence n k k defined as n k k! 3 , k ∈ N, and define the function ρ : R 0 → R 0 as ρ t 1 if 0 ≤ t. This function is an admissible weight in the sense of 12, Def. 4.1 and makes the translation semigroup {τ t } t≥0 to be a C 0 -semigroup. On the one hand, since lim t → ∞ ρ t 0, then the translation C 0 -semigroup is topologically mixing. On the other hand, if the translation C 0 -semigroup was distributionally chaotic, by 7, Th. 2.10 , the backward shift operator, defined as B x 1 , x 2 , . . .
x 2 , x 3 , . . . , would be distributionally chaotic on the space 1 v : { x n n : j∈N |x j |v j < ∞} with v n n ρ n n , which is a contradiction as it is indicated in 27 .