H ¨OLDER REGULARITY FOR THE MOORE-GIBSON-THOMPSON EQUATION WITH INFINITE DELAY

. We characterize the well-posedness of a third order in time equation with inﬁnite delay in H¨older spaces, solely in terms of spectral properties concerning the data of the problem. Our analysis includes the case of the linearized Kuznetzov and Westerwelt equations. We show in case of the Laplacian operator the new and surprising fact that for the standard memory kernel g ( t ) = t ν − 1 Γ( ν ) e − at the third order problem is ill-posed whenever 0 < ν ≤ 1 and a is inversely proportional to the damping term of the given model.


Introduction
Our concern in this article is the study of well-posedness for the following abstract integrodifferential equation of third order in time where κ, b, c, τ are nonnegative real numbers, f is a vector-valued function, A is a closed linear operator with domain D(A) defined on a Banach space X, where w(s) = u(s), (Type I) (1.2) w(s) = u t (s), (Type II) (1.3) w(s) = µu(s) + u t (s), µ = 0 (Type III) (1.4) and the memory kernel g fulfills suitable assumptions.The well-posedness of higher order abstract Cauchy problems is a topic that has been studied for a long time, see e.g.[8,6,23,32,37,40] and references therein.On the other hand, the equation (1.1) of type I, II and III were recently introduced by I. Lasiecka and X. Wang in an interesting series of articles [33,34].From then on, it began to attract the attention of an increasing number of researchers [17,18].
In case g ≡ 0, the linear part of the third order equation (1.1) is referred to as the Moore-Gibson-Thomson equation (MGT-equation) [29].This model represent powerful applications in different fields of practical interest such as high intensity ultrasound and vibrations of flexible materials, see e.g.[26,28,35].In such cases, the operator A is usually the Laplacian and f (t, u, u t , u tt ) = K((u t ) 2 ) t for a suitable constant K > 0. In high intensity ultrasound, u denotes the potential velocity of the acoustic phenomenon described on some bounded R 3 -domains [29].
An operator-theoretical semigroup method to study the linear case g ≡ 0 was employed by Marchand-Devitt and Triggiani [35] and Kaltenbacher-Lasiecka and Marchand [28].These authors rewrite the MGT-equation as a first order abstract Cauchy problem u = Au + F in several state spaces.Then, well-posedness and exponential decay were obtained.It was proved (see [28,Theorem 1.1]) that if A is unbounded then, for b = 0 the operator matrix A cannot be the generator of a C 0 -semigroup on the state space H := D(A 1/2 ) × D(A 1/2 ) × H and hence the problem is ill-posed, in the sense that the matrix operator A does not generate a strongly continuous semigroup on H. On the other hand, a direct approach to well-posedness for the abstract MGT-equation without reduction of order was undertaken by Fernández, Lizama and Poblete [24].However, this approach has been controversial [35].More recently, in [25, Section 2] and [18] the MGT-equation has been studied from a different perspective, by means of the theory of integral equations.As a consequence, well-posedness and qualitative properties can be studied from a representation of the solution of the linear part of the MGT-equation by means of certain variation of constant formula.In the context of a Banach space X, this study seems to be possible using the theory of resolvent families deeply studied by Prüss [38].It is worthwhile to mention that qualitative properties such as the existence of global attractors for the MGTequation have also been studied, see for instance [11].Results concerning the regularity of strong and mild solutions as well as compactness of trajectories have been provided in [25].The study of the existence of asymptotically almost periodic and almost automorphic solutions can be found in [19] and [2], respectively.Existence of mild solutions with non local initial conditions by methods of Hausdorff measure of non compactness, has been established in [2] and concerning ill-posedness, the presence of chaos for the linear MGT-equation has been discovered in [14].A variation of the MGT-equation in order to include non-degenerate equations was recently introduced by Cai and Bu [9].
In our case of study g = 0 and A is typically the Laplacian operator.Depending on the properties of the environment surrounding sound propagation, the memory kernel can exhibit several structures by selecting different values of g which yield to different possible configurations [17].In the abstract case, −A is a non-negative and self-adjoint operator defined on a Hilbert space H.There is little literature concerning the precise abstract model (1.1), and much work is left to be done in the qualitative analysis of (1.1) with memory term, specially concerning its treatment on general Banach spaces.In particular, the well-posedness of (1.1) in several classes of spaces of functions remains largely open.As it is well known, these results are necessary for the treatment of nonlinear problems [20].
In particular, the study of the well-posedness of the Moore-Gibson-Thomson equation (1.1) in Hölder spaces remains open.
The main objective of this article is to provide a complete answer to this open problem.We have succeeded in solving it by giving a complete characterization of C α well-posedness, in the sense of temporal maximal regularity for the full model (1.1) in the vector-valued space of Hölder continuous functions C α (R, X), 0 < α < 1 and for all types I, II and III before mentioned.We note that C α well-posedness of (1.1) without memory term has been characterized in [15] whereas L p well-posedness has been analyzed in [24].Hölder regularity has been also treated for other clases of abstract evolution equations [16].
More precisely, we prove in Theorem 3.12 that if g ∈ L 1 loc (R + ) satisfies certain conditions of regularity then, the following assertions are equivalent Here ρ(A) denotes the resolvent set of A and g denotes the Fourier transform of g whenever it exists.An analogous result holds in the particular case τ = 0. See Theorem 3.15 below.The main difference is that the term (iη) 3 must be replaced by (iη) 2 in front of the resolvent operator (• − A) −1 , taking into account the higher order derivative term.This last model contains, as particular examples, the linear Westervelt or Kuznetsov equation, the viscoelastic membrane equation and the viscoelastic plate equation with memory in case A = ∆, the Laplacian operator, see [30] and [36].It is worthwhile to observe that, as an immediate consequence of our characterization, the following estimate holds.Here C is a constant independent of f.In order to achieve our main results, we use a method based on operator-valued Fourier multipliers.The main advantage of this method is that it does not require the representation of the solution by means of families of bounded linear operators as done, for instance, using the theory of C 0 -semigroups of linear operators in [29] or other classes of resolvent families [25].The key tool is a very general Fourier multiplier theorem due to Arendt, Batty and Bu [4,Theorem 5.3].This method has been successfully used for the treatment of well-posedness of abstract evolution equations by S. Bu and collaborators [7,9,10] among other authors.See also [12,21] and references therein.
Several examples are presented in the last section of this work.For instance, we consider the problem (1.5) with Dirichlet boundary conditions.Suppose first that τ = 0 in equation (1.5).Under the assumptions 0 < ν ≤ 1, c 2 > 1 a ν and −τ c 2 + κb = 0 it is proved using our abstract characterization that the equation (1.5) is C α -ill posed on the space C α (R, C 0 (Ω)) for any 0 < α < 1.This is a new and surprising fact that has not been observed before.
On the other hand, if we suppose τ = 0, then we prove that the equation (1.5) is C α -well posed on the space C α (R, C 0 (Ω)).This way the presence of the term of third order in time becomes crucial for C α -well-posedness in case that A = ∆, the Laplacian operator.In contrast, an example of closed operator A is provided in the context of a Hilbert space showing that for τ = 0, g(t) = e −at with a > 1 and κ = b = c 2 = 1 the corresponding MGTM equation is always C α -well-posed.
It is worthwhile to mention that our characterization is very flexible with respect to the class of closed linear operators A that are admissible.Indeed, we provide an example that shows that A does not need to be the generator of any C 0 -semigroup in order to obtain C α well-posedness of (1.1).
We finish this paper with a criterion (see Theorem 5.4 below) that ensures the existence of a unique solution with C α -temporal regularity for a nonlinear version of (1.1) given by where φ 1 , φ 3 :

Preliminaries
Let X, Y be Banach spaces and let 0 < α < 1 be fixed.We denote by Ċα (R, X) the space Ċα (R, X) = {f : R → X : where Let k ∈ N, and C α+k (R, X) be the Banach space of all u ∈ C k (R, X) such that u (k) ∈ C α (R, X), equipped with the norm Let Ω ⊆ R be an open set.We denote by C ∞ c (Ω) the space of all C ∞ -functions in Ω having compact support in Ω.By Ff or f we denote the Fourier transform of a function f ∈ L 1 (R, X), given by The following notion of multiplier is originally due to Arendt, Batty and Bu.
The following operator-valued Fourier multiplier theorem, stated in [3,Theorem 5.3], which provides sufficient conditions to ensure when M ∈ C 2 (R\{0}, B(X, Y )) is a Ċα -multiplier, plays a key role in the proof of the main results of the paper.
Then M is a Ċα -multiplier.
2) is replaced by the following weaker condition Observe that a function a satisfying the above conditions belongs to It follows from (2.4) that the above integral is well defined.Moreover, The Laplace transform of a function f ∈ L 1 loc (R + , X) will be denoted by whenever the integral is absolutely convergent for Re λ > ω.The relation between the Laplace transform of f ∈ L 1 (R, X) satisfying f (t) = 0 for t < 0, and its Fourier transform is We denote by û the Carleman transform of u By the above definition and the dominated convergence theorem we have see [38, p.19].For more details about the Carleman transform see [4,Chapter 4] or [38].For σ > 0, we also consider the operator L σ defined by The following result on properties of L σ complement those given in [31,Appendix].
Proof.It is a consequence of (2.8) and the following well-known property:

A characterization of C α -well-posedness: Type I
In this section, we first state necessary conditions for the well-posedness of the Moore-Gibson-Thompson equation with memory (1.1) (MGTM-equation) of type I in vector-valued Hölder continuous spaces.Let f ∈ C α (R, X) and A be a closed linear operator in X.In the following, we consider the non-homogeneous MGTM-equation ) and τ, κ, c 2 , b are positive real numbers.Recall that the domain of A, D(A), is a Banach space when endowed with the graph norm.Definition 3.1.We say that the equation In [33,Theorem 1.4] the authors proved that is a necessary condition to get exponentially stability of the standard energy for the weak solution of the homogeneous MGTM-equation of type I in a Hilbert space.This assumption implies that c 2 + bηi − g(η) = 0 for all η ∈ R and hence, by the Riemann-Lebesgue theorem, we get that the function belongs to C 0 (R).The functions γ and β given by will play a key role in the characterization of the well-posedness of equation (3.1).
Our first result in this section is the following theorem.Its proof follows some ideas provided in [31] and [3].
Proof.In order to prove (i), we first assume that there exist η ∈ R and x ∈ D(A) such that Ax = β(η)x.It is easy to see that u is a solution of (3.1) for f = 0, and then x = 0 by uniqueness.Secondly, let η ∈ R and y ∈ X.Note that f (t) = e iηt y ∈ C α (R, X), so u = Lf is the unique solution of (3.1) associated to f , where ) is the bounded linear operator which associates to each f ∈ C α (R, X) the solution of (3.1).One can easily check that for a fixed s ∈ R, are both solutions of (3.1) with forcing term e iηs f (t).By uniqueness it follows that u(t) = e iηt u(0).Taking x = u(0) ∈ D(A) and replacing u as above in (3.1) we get In order to prove (ii), we observe that e iηt x α = K α |η| α x , where K α = 2 sup t>0 t −α sin(t/2), see [3,Section3].Then we get Also, using that γ ∈ C 0 (R) and the continuity of η → (iη) 3 γ(η)(β(η) − A) −1 at η = 0, we conclude (ii).
Remark 3.3.Note that condition (i) of the above theorem establishes that we need invertibility of the operator A to have C α -well-posedness of (3.1).Definition 3.4.[31] We say that h loc (R + ) be of subexponential growth and n ∈ N. In [38, Definition 3.3], the following concept of n-regularity is considered: where C is a positive constant.Observe that if we further assume that h ∈ L 1 (R + , t n dt), this concept implies that Lemma 8.1].Therefore the function h is n-regular on R in the sense of Definition 3.4 too.
Remark 3.6.Observe that if h is 2-regular on R, then h is a Ċα -multiplier.
The following Lemma provides several useful properties on the scalar function γ that will be crucial for the forthcoming section.Lemma 3.8.Let g be a 2-regular function on R such that c 2 + bsi − g(s) = 0 for all s ∈ R. Then the following statements hold.
(ii) The function γ satisfies Proof.By the Riemann-Lebesgue theorem we get that γ(s) ∈ C 0 (R) and sγ(s) ∈ BC(R) since c 2 + bsi − g(s) = 0 for all s ∈ R. In addition, using the 2-regularity of the function g and the identities for all s ∈ R, we get the boundedness for the remaining functions.In order to prove (ii) it is enough to use (i) and sup s∈R |s(g(s In what follows, we will denote by id(s) := is for s ∈ R and The next Lemma shows that the particular structure of M gives immediate properties on additional operators related with M .Lemma 3.9.Let g be 2-regular on R, and assume that c 2 + bsi − g(s) = 0 for all s ∈ R, and then (id) 3 M and (id) 2 M are Ċα -multipliers in B(X), and idM, M and gM are Ċα -multipliers in B(X, D(A)).
Proof.First of all, we observe that sup s∈R (is) j M (s) < ∞ for j = 0, 1, 2, since sup s∈R s 3 M (s) < ∞ and the continuity of s → M (s) in s = 0 holds.Also, we have the identities By Lemma 3.8, we get for |s| > ε > 0. We denote M j := (id) j M for j = 0, 1, 2, 3.By the continuity of s → M (s) in s = 0, it is enough to prove that sup |s|>ε sM j (s) < ∞ and sup |s|>ε s 2 M j (s) < ∞ for each j = 0, 1, 2, 3. Let j = 3.First we have ), where we have applied Lemma 3.8 and (3.3).It shows that sup s∈R sM 3 (s) < ∞.Secondly, we have the inequality It is clear that the first two summands in the previous inequality are uniformly bounded.Using again Lemma 3.8 and (3.3) we get It implies that sup s∈R s 5 M (s) < ∞.As a consequence, by Theorem 2.3, we obtain that M 3 is a Ċα -multiplier in B(X).The cases j = 0, 1, 2 are analogous.Observe that the 2-regularity of g implies that gM is a Ċα -multiplier.Finally, the identity shows that M is a Ċα -multiplier in B(X, D(A)).Also, (id)M is a Ċα -multiplier in B(X, D(A)), since (id)γ is a Ċα -multiplier (Lemma 3.8 (i)).It follows by the 2-regularity of g that gM is a Ċα -multiplier in B(X, D(A)), and we conclude the result.
Remark 3.10.By Lemma 3.8 (i), the condition sup s∈R s 3 M (s) < ∞ in the above theorem could be replaced by sup The following theorem is one of the main results of this paper.It characterizes the C α -wellposedness of equation (3.1) in Hölder continuous spaces, under the 2-regularity assumption on the function g and the following hypothesis: There exists ε > 0 such that Remark 3.11.It is interesting to observe that the assumption (3.2) i.e. c 2 > G(+∞), was used in [33] to get exponentially stability of the energy for weak solutions of the MGTM-equation of type I on Hilbert spaces.Note that it also implies condition (3.4).Therefore it seems natural to impose this condition in order to obtain C α -well-posedness of (3.1).
We now present our main result.
Theorem 3.12.Let A be a closed linear operator defined on a Banach space X.Suppose g ∈ L 1 loc (R + ) is 2-regular on R and (3.4) holds.Then the following assertions are equivalent.
Proof.The implication (i) ⇒ (ii) is a straightforward consequence of Theorem 3.2.
Note that the operator M (s) satisfies the following identity, for all φ ∈ C 1 c (R\{0}).By (3.5)-(3.11)and the previous identity we have Since g(0) = c 2 and 0 ∈ ρ(A) we can take It is easy to see that u solves (3.1).
Remark 3.14.The proof of the above theorem states that u , u , Au , Au, g * Au ∈ C α (R, X).Moreover, using the closed graph theorem, we can conclude that there exists a positive constant In what follows we analyze the special case τ = 0, namely, Abstract integro-differential equations of second order with a memory term naturally appear in the theory of viscoelasticity.For instance, the model (3.14) with b = 0 and A = ∆ corresponds to the viscoelastic membrane equation.In [1], decay estimates for the solutions of this abstract model defined on Hilbert spaces is studied.Moreover, for A = −∆ 2 equation (3.14) models the viscoelastic plate equation with memory [36], [39].We say that equation (3.14).For this equation, the function γ defined previously , η ∈ R, does not change.However, the function β takes the form when τ = 0.By the continuity of s → M (s) = γ(s)(β(s) − A) −1 at s = 0, we note that (ii) of Theorem 3.2, i.e., sup Therefore, using the same ideas in the proof of Theorem 3.12, we obtain the following characterization of C α -well-posedness of the MGTM equation of Type I with τ = 0.
Theorem 3.15.Let A be a closed linear operator defined on a Banach space X.Suppose g ∈ L 1 loc (R + ) is 2-regular on R and (3.4) holds.Then the following assertions are equivalent.
We introduce the following definition.We will use the same tools as the ones in the previous section in order to get C α -well-posedness for the MGTM-equation of types II and III.We will omit unnecessary steps and we will write the main theorems directly.We first treat the MGTM equation of Type II.
In a similar way as in the above section, one can easily observe that the assumption implies that c 2 + bηi − iηg(η) = 0 for all η ∈ R.Moreover, the function belongs to C 0 (R).Observe that, instead of (4.3), we could suppose that the function g satisfies the condition g ∈ L 1 (R + ).This condition is verified for 2-regular derivable functions which are well defined at t = 0, for instance g ν,a for ν ≥ 1.In this case, the following assumption , belongs to C 0 (R).If any of the above assumptions for γ 2 hold, the function is well-defined.We introduce the following notion of regularity.
Remark 4.3.Let h be a strongly 2-regular function.Then h is also 2-regular.Moreover, we have sup In particular, h and s h(s) are Ċα -multipliers.If we assume that h is differentiable, then h is 2-regular as well.
Example 4.4.The functions g ν,a (t) = t ν−1 Γ(ν) e −at are strongly 2-regular on R for a > 0 and ν ≥ 1 since g ν,a (s) = 1 (is+a) ν for all s ∈ R. The next result characterizes the C α -well-posedness of equation (4.1) under the assumption of strong 2-regularity on the function g.Both, conditions (4.3) and (4.4), imply that there exists ε > 0 such that (4.5) sup 0≤σ<ε, ρ∈R Theorem 4.5.Let A be a closed linear operator defined on a Banach space X.Suppose g ∈ L 1 loc (R + ) is strongly 2-regular on R and (4.5) holds.Then the following assertions are equivalent.
For the particular case τ = 0, (4.6).For this equation, the function γ 2 does not change, and the function β 2 is It is not difficult to observe that we can obtain the following characterization of C α -wellposedness of the MGTM equation of Type II with τ = 0.
Theorem 4.6.Let A be a closed linear operator defined on a Banach space X.Suppose g ∈ L 1 loc (R + ) is strongly 2-regular on R and (4.5) holds.Then the following assertions are equivalent.
Finally, we study the MGTM-equation of Type III.Observe that if the following inequality holds = 0 for all η ∈ R.Moreover, the function , belongs to C 0 (R).We note that condition (4.7) implies exponential stability of the standard energy for the weak solution of the homogeneous MGTM-equation of type III in a Hilbert space, see [33,Theorem 1.8].On the other hand, if the function g satisfies that g ∈ L 1 (R + ), then the condition It is a straightforward consequence that is well-defined if (4.7) or (4.8) holds.Also, the previous conditions imply that there exists ε > 0 such that (4.9) sup 0≤σ<ε, ρ∈R The characterization of C α -well-posedness of MGTM equations of type III reads as follows.
Theorem 4.7.Let A be a closed linear operator defined on a Banach space X.Suppose g ∈ L 1 loc (R + ) is strongly 2-regular on R and (4.9) holds.Then the following assertions are equivalent.
(ii) {β 3 (s)} s∈R ⊆ ρ(A) and Similarly to the previous cases, for τ = 0, (4.10) the equation (4.10) is said C α -well-posed if for each f ∈ C α (R, X) there exists a unique function u ∈ C α+1 (R, D(A))∩C α+2 (R, X) satisfying (4.10).In this case, the function γ 3 does not change, and the function β 3 is Then we get the following characterization of C α -well-posedness of the MGTM equation of Type III with τ = 0.
Theorem 4.8.Let A be a closed linear operator defined on a Banach space X.Suppose g ∈ L 1 loc (R + ) is strongly 2-regular on R and (4.9) holds.Then the following assertions are equivalent.

Examples
In this section we illustrate some of the main results in this paper with some examples of classes of operators A. Recall that we assume τ, κ, b, µ > 0.
First, we will prove that if 0 < ν ≤ 1 and c 2 > 1 a ν = G(+∞), the corresponding MGTM equation of type I is C α -ill-posed for τ = 0. See Remark 3.13 for the precise meaning of illposedness.However, the Kuznetsov and Westervelt equations of Type I (MGTM equation for τ = 0) are C α -well-posed.

4 .
Characterizations of C α -well-posedness: Type II and Type III In this section, our aim is to characterize the C α -well-posedness of the MGTM-equation of Types II and III.Let f ∈ C α (R, X) and A be a closed linear operator in X.The non-homogeneous MGTM equations of Type II and III are given by (4.1)