Well-Posedness for Some Third-Order Evolution Differential Equations: a Semigroup Approach3

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arXiv:2106.03564v1 [math.AP] 7 Jun 2021 (1.1) time in equation evolution third-order following ; Date .Rmk ntecase the on order Remaks the reducing by 5. Analysis 4.2. Sectoriality equations differential 4.1. Parabolic problems 4. Ill-posed approach 3. New setting 2.2. Natural Framework2.1. Functional 2. Introduction 1. nti ae,w ics h elpsdeso h acyproblem Cauchy the of well-posedness the discuss we paper, this In equations References Moore-Gibson-Thompson-type 6. eerhprilyspotdb Nq#332/054adb FAP by and 303929/2015-4 # CNPq by supported partially Research ue8 2021. 8, June : e od n phrases: and words Key ahmtc ujc lsicto 2010 Classification Subject Mathematics regularity. Abstract. where and ihtetidodreouineuto ntime in equation evolution third-order the with etra prtrwt opc eovn,adfrsome for and resolvent, compact with operator sectorial ELPSDESFRSM HR-RE EVOLUTION THIRD-ORDER SOME FOR WELL-POSEDNESS LN .M EER,AEADEN CARVALHO N. ALEXANDRE BEZERRA, M. D. FLANK IFRNILEUTOS EIRU APPROACH SEMIGROUP A EQUATIONS: DIFFERENTIAL f : η D ą p A 0, nti ae,w ics h elpsdeso h acyproblem Cauchy the of well-posedness the discuss we paper, this In 1 3 Ă q X sasprbeHletspace, Hilbert separable a is X Ñ η prxmtos rcinlpwr;scoiloeao;semigr operator; sectorial powers; fractional Approximations; u X “ ttt u sannierfnto ihsial odtoso rwhand growth of conditions suitable with function nonlinear a is 1 ttt ` ` Au Au 1. ` ` Introduction ηA Contents ηA 40,4D6 70,35K10. 47D03, 47D06, 34A08, : 3 1 3 1 u u 1 tt tt ` ` A ηA ηA : 2 3 3 2 u D u t p t “ A “ Ă q f λ 0 p ; f u N UA .SANTOS A. LUCAS AND , p ą q u X q S 031020 Brazil. 2003/10042-0, # ESP ehv Re have we 0 Ñ X sa unbounded an is soitdwt the with associated σ associated p A ą q oups. λ 0 14 10 25 23 21 9 7 5 3 3 1 2 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS where η 0, X is a separable Hilbert space and A : D A X X is an unbounded ą p q Ă Ñ sectorial operator with compact resolvent, and for some λ0 0 we have Reσ A λ0, that ą p qą is, Reλ λ0 for all λ σ A , where σ A is the spectrum of A. This allows us to define the fractionalą power A´α Pofp orderq α 0,p1 qaccording to [2, Formula 4.6.9] and [15, Theorem 1.4.2], as a closed linear operatorP on p itsq domain with inverse Aα. Denote by Xα D Aα for α 0, 1 , taking A0 : I on X0 : X when α 0. Recall that Xα is dense“ in Xp forq all α P r0, 1 q, for details see“ [2, Theorem“ 4.6.5]. The“ fractional power space Xα endowed with theP pnorms α Xα : A X }¨} “} ¨} is a Banach space. It is not difficult to show that Aα is the generator of a strongly continuous analytic semigroup on X, that we will denote by e´tAα : t 0 , see [15] for any α 0, 1 . With this notation, we have X´α Xα 1 for alltα 0, seeě [2]u for the characterizationPr ofs the negative scale. “ p q ą α Let X 1 denote the extrapolation space of X generated by A, and let X 1 : α 0 the ´ t ´ ě u fractional power scale generated by operator A in X´1, see [2] and [3] for more details. 1 Here f : D A 3 X X is a nonlinear function with suitable growth conditions and regularity in (1.1)p q for Ă differentÑ cases of η 0; namely, we consider: ą If 0 η 1, then we prove that the Cauchy problem defined by the linear equation ‚ associatedă ă with (1.1) is ill-posed, consequently, the Cauchy problem defined by (1.1) is ill-posed for any nonlinear function f, under the point of view of the theory of strongly continuous semigroups of bounded linear operators; If η 1, then we assume that f is twice continuously Fréchet differentiable and ‚ Lipschitz“ continuous on bounded sets; 1 If η 1, then we assume that f is an ǫ-regular map relative to the pair X 3 ,X for ‚ ą p1 q ǫ 0; that is, there exist constants c 0, ρ 1, γ ǫ with ρǫ γ ǫ 3 such that ě 1 ą ą p q ď p qă f : X 3 `ǫ Xγpǫq and Ñ ρ´1 ρ´1 1 (1.2) f φ1 f φ2 Xγpǫq c φ1 φ2 `ǫ 1 φ1 1 φ2 1 , } p q´ p q} ď } ´ }X 3 p `} }X 3 `ǫ `} }X 3 `ǫ q 1 `ǫ for any φ1,φ2 X 3 , see [3, Definition 2], [7] and [8] for more details. P 1 For a better understanding of the ǫ-regular map relative to the pair X 3 ,X for ǫ 0, we construct the following diagram. p q ě

f . . 1 X 3 X ¨ ¨ ¨ f ¨ ¨ ¨ . . 1 γ ǫ X 3 `ǫ X p q ¨¨¨ ¨¨¨

1 1 Figure 1. X 3 `ǫ X 3 and Xγpǫq X Ă Ă WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH3

The evolution equations of third order in time have been studied extensively in the Hilbert setting and much progress has been achieved. In [1, 6, 12, 16, 17, 18, 19, 21] and [22], and references therein, the Moore-Gibson-Thompson (MGT) equation is studied in diﬀerent contexts and results of existence, stability and regularity of solutions are obtained by the spectral theory of the MGT operator. In [14] the abstract linear equations of third order in time is analyzed and results on (non)existence of solution are obtained. In [4] the abstract linear equations of third order in time is analyzed and results on (non) existence, stability and regularity of solution are obtained via theory of fractional powers of closed and densely deﬁned operators. The article is organized in the following way. In Section 2 we present general facts on spectral behavior of the our problem. In Section 3 we consider the case 0 η 1 and we obtain the result that shows that the problem (1.1) is ill-posed under theď pointă of view of the theory of strongly continuous semigroups of bounded linear operators. In Section 4 we consider the case η 1 and we obtain a result of existence, stability and regularity of solutions for (1.1). In Sectioną 5 we consider the case η 1 via theory of strongly continuous groups of bounded linear operators. Finally, in Section 6“ we explore our results on the Moore- Gibson-Thompson-type equations, according to the references [1, 6, 12, 16, 17, 18, 19, 21] and [22].

2. Functional Framework 2 1 We ﬁrst introduce some notations, we consider Z X 3 X 3 X endowed with the norm given by “ ˆ ˆ 2 u 2 2 2 u v v u 2 v 1 w X, Z. w Z “} }X 3 `} }X 3 `} } @ w P ›” ı› ” ı 2.1. Natural setting.› We› can rewrite the initial value problem associated with equation › › (1.1) as the Cauchy problem in Z

d u u u (2.1) v A η v F v , t 0, dt w ` p q w “ p w q ą and ” ı ” ı ” ı u u0 (2.2) v 0 v0 , w p q“ w0 ” ı ” ı where v ut and w vt and the unbounded linear operator Apηq : D Apηq Z Z is defined by“ “ p q Ă Ñ 1 2 1 3 3 (2.3) D A η X X X p p qq“ ˆ ˆ and 0 Í 0 ´v u u u 1 2 1 A v 0 0 Í v ´w v 3 3 (2.4) pηq : 2 1 2 1 , X X X . w “ A ηA 3 ηA 3 w “ Au`ηA 3 v`ηA 3 w @ w P ˆ ˆ ” ı ” ı” ı ” ı ” ı The nonlinearity F given by u 0 (2.5) F v 0 , p w q“ fpuq ” ı ” ı 4 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

1 where f : D A 3 X X is a Lipschitz continuous function on bounded sets. From nowp on,qĂ we denoteÑ 1 1 2 1 3 3 Z D A η X X X . “ p p qq“ ˆ ˆ u0 We also consider the following notion of mild solution for (2.1)-(2.2). Given v0 Z w0 P u u ” ı we say that v is a local mild solution of (2.1)-(2.2) provided v C 0, τu0,v0,w0 ,Z , w w P pr q q ” ı u ” ı f u C 0, τu0,v0,w0 ,X and, for t 0, τu0,v0,w0 , v satisfies the integral equation p q P pr q q Pp q w uptq t ” ı 0 Á t u0 Á pt´sq (2.6) vptq e pηq v0 e pηq 0 ds. w0 wptq “ ` 0 fpupsqq „  ” ı ż ” ı for some τu0,v0,w0 0. ą Lemma 2.1. Let Apηq be the unbounded linear operator defined in (2.3)-(2.4). The following conditions hold.

i The linear operator A η is closed and densely defined; q p q ii Zero belongs to the resolvent set ρ Apηq ; namely, the resolvent operator of Apηq is the q 1 p q bounded linear operator A´ : Z Z given by pηq Ñ 1 2 ´ ´ ´1 A´1 ηA 3 ηA 3 A pηq Í 0 0 . “ 0 Í 0 ” ı Moreover, Apηq has compact resolvent; iii The spectrum set of A η , σ A η , is given by q ´ p q p´ p qq 1 1 1 3 3 3 σ A η λ C : λ σ A zηλ C : λ σ A zηλ C : λ σ A , p´ p qq “ t P P p´ quYt P P p´ quYt P P p´ qu 1 1 where σ A 3 denote the spectrum set of A 3 and p´ q ´ 1 2 zη η 1 i 3 2η η “ 2 p ´ q´ ` ´ and ” a ı 1 2 zη η 1 i 3 2η η . “ 2 p ´ q` ` ´

” aun un ı Proof: To prove part i we take a sequence vn , Apηq vn is the graph of Apηq, which q wn wn φ ν converges in Z to ϕ , µ . From this we´” easilyı conclude” ı¯ that ν ϕ, µ ψ, and ψ ζ “´ “´ ´” ı ” ı¯ 2 2 2 3 3 3 un φ in X A un A φ in X, Ñ 1 ùñ 1 Ñ 1 3 3 3 vn ϕ in X A vn A ϕ in X, Ñ ùñ Ñ and 2 1 3 3 Aun ηA vn ηA wn ζ in X. ` ` Ñ Since 2 1 2 1 3 3 3 3 A un ηA vn ηwn A φ ηA ϕ ηψ in X, ` ` Ñ ` ` WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH5

1 2 1 1 2 1 the closedness of A 3 imples that A 3 φ ηA 3 ϕ ηψ X 3 and Aφ ηA 3 ϕ ηA 3 ψ ζ, and φ ` ` P φ ν ` ` “ ϕ ϕ µ we obtain the result; that is, D Apηq and Apηq . ψ P p q ψ “ ζ For the proof of ii the result” ı easily follows from A”´1 ı: Z ” Zı to be given by q pηq Ñ 1 2 ´ ´ ´1 A´1 ηA 3 ηA 3 A pηq Í 0 0 “ 0 Í 0 2 ”1 ı 1 2 1 which takes bounded subsets of X 3 X 3 X into bounded subsets of X X 3 X 3 . The latter space is compactly embeddedˆ in Zˆbecause the inclusions Xβ Xγ âre compact,ˆ for β γ 0, provided A has compact resolvent. Finally, after consideringĂ the eigenvalue ą ě problem for the operator A η , ´ p q A η u λu, ´ p q “ and after straightforward calculations, part iii follows immediately from the fact that Apηq has compact resolvent. q

2.2. New approach. Let (1.1) be the original equation of third-order. Note that using the change variable 1 3 v ut A u, “ ` and the function

w vt, “ we can rewrite the (1.1) as follows, a ﬁrst-order evolution equation in time for w 1 2 3 3 (2.7) wt η 1 A w A v f u . `p ´ q ` “ p q The initial value problem associated with equation (2.7) as the Cauchy problem in Z

d u u u (2.8) v B η v F v , t 0, dt w ` p q w “ p w q ą and ” ı ” ı ” ı u u0 (2.9) v 0 v0 , w p q“ w0 ” ı ” ı where v ut and w vt and the unbounded linear operator Bpηq : D Bpηq Z Z is defined by“ “ p q Ă Ñ 1 2 1 3 3 (2.10) D B η X X X , p p qq“ ˆ ˆ and 1 1 3 0 3 u A Í u A u´v u 1 2 1 (2.11) B v : 0 0 Í v ´w , v X X 3 X 3 . pηq w 2 1 w 2 1 w “ 0 A 3 pη´1qA 3 “ A 3 v`pη´1qA 3 w @ P ˆ ˆ ” ı ” ı” ı ” ı ” ı The nonlinearity F given by (2.5). Lemma 2.2. The following conditions hold.

i The linear operator B η is closed and densely deﬁned; q p q 6 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

ii Zero belongs to the resolvent set ρ Bpηq ; namely, the resolvent operator of Bpηq is the q 1 p q bounded linear operator B´ : Z Z given by pηq Ñ 1 2 ´ 3 1 ´ 3 ´1 1 A pη´ qA A B´ 1 2 pηq 0 pη´1qA´ 3 A´ 3 . “ 0 0 ” ´I ı Moreover, Bpηq has compact resolvent; iii The spectrum set of Bpηq, σ Bpηq , is given by q ´ 1 p´ q 1 1 3 3 3 σ Bpηq λ C : λ σ A cηλ C : λ σ A dηλ C : λ σ A , p´ q “ t P 1 P p´ quYt P P 1 p´ quYt P P p´ qu where σ A 3 denote the spectrum set of A 3 and p´ q ´ 1 2 cη η 1 η 2η 3 “ 2 p ´ q` ´ ´ and ” a ı 1 2 dη η 1 η 2η 3 , “ 2 p ´ q´ ´ ´

” aun unı v v Proof: To prove part i we take a sequence n , Bpηq n is the graph of Bpηq, which q wn wn φ ν converges in Z to ϕ , µ . Then we have´” ı ” ı¯ ψ ζ

´” ı ” ı¯ 2 3 un φ in X Ñ 1 3 vn ϕ in X Ñ wn ψ in X Ñ and

1 2 3 3 A un vn ν in X ´ Ñ 1 3 wn µ in X 2 ´ Ñ 1 3 3 A vn η 1 A wn ζ in X `p ´ q Ñ From this, we easily conclude that

1 ψ µ X 3 . 1 “´ P From the closedness of A 3 , we obtain

1 1 1 1 A 3 ϕ η 1 ψ X 3 and A 3 A 3 ϕ η 1 ψ ζ, `p ´ q P p `p ´ q q“ that is, 2 2 1 ϕ X 3 and A 3 ϕ η 1 A 3 ψ ζ 1 P `p ´ q “ From the closedness of A 3 , we also obtain 2 1 1 1 2 1 2 A 3 φ A 3 ϕ X 3 and A 3 A 3 φ A 3 ϕ A 3 ν ´ P p ´ q“ that is, 1 1 φ X and A 3 φ ϕ ν P ´ “ WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH7

φ φ ν ϕ ϕ µ From this, we conclude that D Bpηq and Bpηq . ψ P p q ψ “ ζ For the proof of ii the result” easilyı follows from B´”1 : ıZ ”Zıto be given by q pηq Ñ 1 2 ´ 3 1 ´ 3 ´1 1 ηA p ´ηqA A B´ 1 2 pηq 0 p1´ηqA´ 3 A´ 3 . “ 0 0 ” ´I ı Finally, after considering the eigenvalue problem for the operator B η , ´ p q B η u λu, ´ p q “ and after straightforward calculations, part iii follows immediately from the fact that Bpηq has compact resolvent. q

3. Ill-posed problems In this section we consider the case 0 η 1. We prove that the Cauchy problem deﬁned by the linear equation associated withď (1.1)ă is ill-posed in Z, consequently, the Cauchy problem deﬁned by (1.1) is ill-posed for any nonlinear function f in Z, under the point of view of the theory of strongly continuous semigroups of bounded linear operators.

Lemma 3.1. Let A η be the unbounded linear operator deﬁned in (2.3)-(2.4). If 0 η p q ď ă 1, then the unbounded linear operator Apηq with Apηq : D Apηq Z Z is not the inﬁnitesimal generator of a strongly continuous´ semigroup in Zp . q Ă Ñ

A ´ pηqt Proof: If Apηq generates a strongly continuous semigroup e : t 0 in Z, it follows from [2] that´ there exist constants ω η 0 and M η 1 sucht that ě u p qě p qě

A ´ pηqt ωpηqt (3.1) e L Y M η e for t 0. } } p q ď p q ě Moreover, from [2] we have

(3.2) λ C : Reλ ω η ρ A η t P ą p qu Ă p´ p qq where ρ Apηq denotes the resolvent set of the operator Apηq. From Lemma 2.1(iii) we have p´ q ´ 1 1 1 3 3 3 σ A η λ C : λ σ A zηλ C : λ σ A zηλ C : λ σ A , p´ p qq “ t P P p´ quYt P P p´ quYt P P p´ qu where 1 2 zη η 1 i 3 2η η “ 2 p ´ q` ` ´ and ´ a ¯ 1 2 zη η 1 i 3 2η η , “ 2 p ´ q´ ` ´ where 3 2η η2 0 for any 0 η´ 1, see Figurea 2. ¯ ` ´ ą ď ă 8 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

Im 1 zησ A 3 p´ q

1 σ A 3 θη p´ q Re

1 zησ A 3 p´ q π Figure 2. Semi-lines contained the eigenvalues of A η and 0 θη , ´ p q ă ă 2 0 η 1. ď ă

1 1 Since σ A 3 µn : n N with µn σ A 3 for each n N and µn as n and 0 ηp´ 1,q we “ conclude t´ thatP u P p q P Ñ 8 Ñ 8 ď ă σ A η λ C : Reλ ω η p´ p qq X t P ą p qu‰H This contradicts (3.4) and therefore Apηq can not be the inﬁnitesimal generator of a strongly continuous semigroup in Z. ´ Theorem 3.2. If 0 η 1, then the Cauchy problem ď ă d u u 0 v A η v 0 , t 0, dt w ` p q w “ 0 ą and ” ı ” ı ” ı u u0 v 0 v0 , w p q“ w0 where v ut and w vt is ill-posed on” Zı. ” ı “ “ Proof: The result easily follows from Lemma 3.1.

u0 Theorem 3.3. The Cauchy problem (2.1)-(2.2) is ill-posed on Z. More precisely, let v0 w0 P u ” ı Z does not exist v C 0, τu0,v0,w0 ,Z with f u C 0, τu0,v0,w0 ,X such that (2.6) w P pr q q p q P pr q q holds, for any τu0,v”0,wı0 0. ą Proof: The result easily follows from Theorem 3.2. Remark 3.4. If η 0 then thanks to the results in [4] the Cauchy linear problem associated with (2.1) is ill-posed“ in Z, and therefore, the Cauchy problem (2.1) is ill-posed in Z. On the new approach present in Subsection 2.2 we have WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH9

Theorem 3.5. Let 0 η 1 and let Bpηq be the unbounded linear operator deﬁned in (2.10)-(2.11). Then theď problemă (2.8)-(2.9) is ill-posed in the sense that it does not generate a strongly continuous semigroup of bounded linear operators on the state space Z.

B ´ pηqt Proof: If Bpηq generates a strongly continuous semigroup e : t 0 on Z, it follows from Pazy´ [20, Theorem 1.2.2] that there exist constants ω t 0 and Mě 1u such that ě ě

B ´ pηqt ωt (3.3) e L Y Me for 0 t . } } p q ď ď ă8

Moreover, from Pazy [20, Remark 1.5.4] we have

(3.4) λ C : Reλ ω ρ B η . t P ą uĂ p´ p qq where ρ B η denotes the resolvent set of the operator B η . p´ p qq ´ p q From Lemma 2.2 we can consider a sequence λkzη k σ Bpηq , for k 1, 2, 3,... , with 1 p q P p´ q “ λk σ A 3 and λk as k . Note that P p´ q | |Ñ8 Ñ8

2 η2 2η 3 arg λkzη arctan ´ ` ` p q“ 1 η ˜ a ´ ¸ and since 0 η 1, we have ď ă

0 arg λkzη π 2 ă p qă { for every k 1 and λkzη as k . Then we conclude that ě | |Ñ8 Ñ8

σ B η λ C : Reλ ω . p´ p qq X t P ą u ‰ H

This contradicts (3.4) and therefore Bpηq can not be the inﬁnitesimal generator of a strongly continuous semigroup on Z. ´

4. Parabolic differential equations In this section we consider the case η 1. Namely, thanks to the Lemma 2.2 we have the ą following illustration of the eigenvalues of B η . ´ p q 10 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

1 zησ A 3 p´ q

1 θ σ A 3 η p´ q Re

1 zησ A 3 p´ q π Figure 3. Semi-lines contained the eigenvalues of B η and 0 θη , η 1. ´ p q ă ă 2 ą

4.1. Sectoriality. Initially, we prove the following theorem on the sectoriality of the operator B η for η 1. p q ą Theorem 4.1. Let η 1. The unbounded linear operator Bpηq defined in (2.10)-(2.11) is a sectorial operator. ą Proof: In this proof, M will denote a positive constant, not necessarily the same one. Let λ C, then P 1 λIÁ 3 I 0 B 0 λI I λI pηq 2 1 . ´ “ 0 Á 3 λI´pη´1qA 3 ” ı From Lemma2.2 it follows that 1 1 1 3 3 3 (4.1) ρ B η ρ A ρ cηA ρ dηA . p p qq“ p qX p qX p q Note that for λ ρ B η we have P p p qq 1 1 1 3 3 3 pλIćηA qpλI´dη A q ´pλI´pη´1qA q I ´1 ´1 1 1 1 (4.2) λI Bpηq Dη λ 0 pλIÁ 3 qpλI´pη´1qA 3 q ´pλIÁ 3 q p ´ q “ p q 1 2 1 « 0 pλIÁ 3 qA 3 λpλIÁ 3 q ff where 1 1 1 3 3 3 Dη λ λI A λI cηA λI dηA p q“p ´ qp ´ qp ´ q with

1 2 (4.3) cη η 1 η 2η 3 “ 2 ´ ` ´ ´ 1 ´ a 2 ¯ (4.4) dη η 1 η 2η 3 . “ 2 ´ ´ ´ ´ ´ a ¯ WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH11

1 1 Since Re cη 0 and Re dη 0, for η 1, cηA 3 and dηA 3 are sectorial operators. Let p q ą p q ą ą 1 1 1 1 1 1 1 3 1 3 1 3 S 3 ,S 3 ,S 3 be sectors such that S 3 ρ A , S 3 ρ cηA , S 3 ρ dηA A cηA dηA A Ă p q cηA Ă p q dη A Ă p q and

1 M 3 λI A L X , for each λ S 1 } ´ } p q ă λ P A 3 | | 1 M 3 λI cηA L X , for each λ S 1 } ´ } p q ă λ P A 3 | | 1 M 3 λI dηA L X , for each λ S 1 } ´ } p q ă λ P A 3 | | for some M 0. We will prove that Bη is a sectorial operator using the sector ą

1 1 1 SB η : S S S . p q “ A 3 X cηA 3 X dηA 3 u It is immediate that SB ρ B η . If λ SB and u v Z with u Z 1, then pηq Ă p p qq P pηq “ w P } } ď writing ” ı ϕ1 ´1 λI B η u ϕ2 p ´ p qq “ ϕ3 ” ı where (4.5) 2 1 2 1 1 1 1 3 3 ´ 3 ´ ´ ϕ1 λ I η 1 λA A Dη λ u λI η 1 A Dη λ v Dη λ w, “p ´p ´ q ´ q p q ` p´ `p ´ q q p q ` p q 2 1 2 1 1 1 3 3 ´ 3 ´ ϕ2 λ I ηλA η 1 A Dη λ v λI A Dη λ w, “p ´ `p ´ q q p q ` p´ ` q p q 2 1 2 1 1 3 ´ 3 ´ ϕ3 λA A Dη λ v λ I λA Dη λ w. “p ´ q p q `p ´ q p q In order to conclude that M ϕ1 2 ϕ2 1 ϕ3 X } }X 3 `} }X 3 `} } ă λ | | 1 2 ´1 ´1 2 ´1 We only need to show that λA 3 Dη λ , A 3 Dη λ , λ Dη λ L X and are bounded p q ´1 p q p q P p q by M λ , which is clear because Dη λ L X and {| | p q P p q

´1 M Dη λ L X } p q } p q ă λ 3 | |

Lemma 4.2. Let Z´1 denote the extrapolation space of Z generated by Bpηq. Then

1 1 ´ Z 1 X 3 X X 3 . ´ “ ˆ ˆ 12 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

B´1 Proof: The extrapolation space of Z is the completion of the normed space Z, pηq Z . u p } } q Since for v Z we have w P ” ı 1 2 ´ 3 1 ´ 3 ´1 ´1 u ηA u`p ´ηqA v`A w B v 1 2 , pηq w p1´ηqA´ 3 v`A´ 3 w “ v ” ı ” ı and consequently

1 u B´ v pηq 2 1 w X 3 ˆX 3 ˆX 1 2 1 2 › ” ´ı › ´ ´1 ´ ´ › ηA 3 u› 1 η A 3 v A w 2 1 η A 3 v A 3 w 1 v › › X 3 X 3 X “} 1 `p ´ q ` } ` }p1 ´ q ` } `} } 3 ´ 3 η A u X max 1, η 1 v X 2 A w X ď } } ` t ´ u} } ` } } u C1η v 1 2 , ď w XˆX´ 3 ˆX´ 3 › ” ı › › › where C1η max› 2, η› 0. By other“ hand t uą

u v 1 1 w X 3 ˆXˆX´ 3 › ” uı › 1 v w 1 › › 3 X ´ 3 ›“} }›X `} } `} }X 1 1 ´ 3 ηA u 2 v X w 1 “ η } }X 3 `} } `} }X´ 3 1 1 2 1 1 1 ´ 3 ´ 3 ´ ηA u 1 η A v A w 2 2 v X 1 w 1 ď η } `p ´ q ` }X 3 ` ´ η } } ` η ` } }X´ 3 ´ ¯ ´ ¯ 1 1 2 1 2 1 1 2 ´ 3 ´ 3 ´ ´ 3 ´ 3 ηA u 1 η A v A w 2 2 η v X 1 1 η A v A w 1 , ď η } `p ´ q ` }X 3 ` ` ´ η } } ` η ` }p ´ q ` }X 3 1 u ´ ¯ ´ ¯ B´ v C2η pηq 2 1 , ď w X 3 ˆX 3 ˆX › ” ı › › › 2 for some› C2η 2› η 0. “ ` ´ η ą Hence, there exist C1η 0 and C2η 0 such that ą ą 1 u u 1 u B´ v v B´ v pηq 2 1 C1η 1 2 C1ηC2η pηq 2 1 , w X 3 ˆX 3 ˆX ď w XˆX´ 3 ˆX´ 3 ď w X 3 ˆX 3 ˆX › ” ı › › ” ı › › ” ı › 2 1 1 2 1 1 › › 3 3 B›´ ›2 1 3 › 3 ›B´ 1 1 completions› of X› X X, › › and X X› X, › ´ p ˆ ˆ } pηq ¨}X 3 ˆX 3 ˆX q p ˆ ˆ } pηq ¨}X 3 ˆXˆX 3 q coincide. Consider the closed extension of Bpηq to Z´1 (see [2, Page 262]) and still denote it by Bpηq. 1 2 1 Then B η is a sectorial and positive operator in Z 1 with the domain Z 1 X 3 X 3 X; p q ´ ´ “ ˆ ˆ the imaginary powers of Bpηq are bounded. Our next concern will be to obtain embeddings α B of the spaces from the fractional power scale Z´1, α 0, generated by pηq,Z´1 . Before we can proceed we need the following generalě interpolation result.p q WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH13

Proposition 4.3. Let Xi, Yi, Mi, i 1, 2, 3 be the Banach spaces such that X1 X0, “ Ă Y1 Y0, M1 M0 topologically and algebraically. Then Ă Ă X0 Y0 M0, X1 Y1 M1 θ X0, X1 θ Y0, Y1 θ M0, M1 θ, r ˆ ˆ ˆ ˆ s “r s ˆr s ˆr s for any θ 0, 1 . Pp q Proof: The proof is an immediate consequence of the deﬁnition of complex interpolation spaces in [23, Section 1.9.2]. The following result also may be established by a straightforward extension of [7, Theorem 2] so we omit its proof.

α Lemma 4.4. Denote by Z 1 : α 0, 1 the partial fractional power scale generated by t ´ P r su operator Bpηq in Z´1. Then 1 1 k`α k` `α k`α k´ `α Z 1 X 3 X 3 X 3 , α 0, 1 , k N. ´ “ ˆ ˆ Pr s P For better understanding the relation of the fractional power scale-spaces of the operator Bpηq, we construct the following diagram.

Bpηq

1 1 1 ¨ ¨ ¨ Z Z Z 1 Z´ ¨ ¨ ¨ Bpηq q “

α 1 ¨ ¨ ¨ Z´1 Z´1 Z´1 ¨ ¨ ¨

Figure 4. Fractional power scale generated by operator B η in Z 1 and α 1. p q ´ ą In this section, we consider the case η 1. Now we prove some of the main results of this paper. ą

Proposition 4.5. Let Z´1 denote the extrapolation space of Z generated by Aη. Then 1 1 ´ Z 1 X 3 X X 3 . ´ “ ˆ ˆ Proof: Again, following [2] and [8] we recall that the extrapolation space of Z is the com- ´1 u pletion of the normed space Z, A Z . Since for v Z p } η } q w P 1 2 ´ ´ 1 ” ı ´1 u ηA 3 u`ηA 3 v`A´ w u A v 2 1 u 2 1 3 1 η v 1 1 , η w ´ w ´ X 3 ˆX 3 ˆX “ ´v X 3 ˆX 3 ˆX ď p ` q X 3 ˆXˆX 3 › ” ı› ›” ı› ›” ı› and› › › › › › › › u › › 1 u › › v A´ v 1 1 3 1 η η 2 1 , w X 3 ˆXˆX´ 3 ď p ` q w X 3 ˆX 3 ˆX 2 ›” 1 ı› 1 › ”2 ı› 1 3 3 A´ 2 1 3 3 1 1 completions of X X› ›X, and› X ›X X, ´ coincide, p ˆ› ˆ› } η ¨}X 3 ˆX 3 ˆX q › p ˆ› ˆ }¨}X 3 ˆXˆX 3 q see the Figure 4.1. 14 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

Consider the closed extension of Aη to Z´1 (see [2, Page 262]) and still denote it by Aη 2 1 1 3 3 with the domain Z´1 X X X. For better understanding“ ˆ the relationˆ of the fractional power scale-spaces of the operator Aη, we construct the following diagram.

Aη

1 1 1 ¨ ¨ ¨ Z Z Z 1 Z´ ¨ ¨ ¨ Apη q “

α 1 ¨¨¨Z´1 Z´1 Z´1 ¨¨¨

Figure 5. Fractional power scale generated by operator Aη in Z 1 and α 1. ´ ą

4.2. Analysis by reducing the order. Namely, we consider the reduction of order 1 3 (4.6) v ut A u “ ` for positive time, where u is an unknown function to be determined, and we obtain the following equation of second order 2 1 3 3 (4.7) vtt A v η 1 A vt g v ` `p ´ q “ p q for positive time, where η 0 and ą (4.8) g v f u . p q“ p q Note that we can reviews (4.7) as follows

v 0 I v 0 v 0 v0 1 (4.9) 2 ´ 1 , p q X 3 X. vt ` A 3 η 1 A 3 vt “ g v vt 0 “ w0 P ˆ „ t „ p ´ q  „  „ p q „ p q „  Λ v0 v t, v0,w0 1 1 Solving (4.9) we ﬁnd e´ t p q X 3 X where Λ : D Λ X 3 X w0 “ vt t, v0,w0 P ˆ p qĂ ˆ Ñ 1 „ 2 „ p1 q X 3 X is deﬁned by, D Λ X 3 X 3 ˆ p q“ ˆ v0 0 I v0 w0 Λ 2 ´ 1 ´2 w0 “ A 3 η 1 A 3 w0 “ A 3 v0 „  „ p ´ q  „  „  1 ` It follows that R t v t, v0,w0 X 3 is a continuous function. We can then try to solve Q ÞÑ p q P 1 3 (4.10) ut A u v t, v0,w0 , u 0 u0 ` “ p q p q“ through the variation of constants formula

1 t 1 (4.11) u t eÁ 3 tu 0 eÁ 3 pt´sqv s ds. p q“ p q` 0 p q ż WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH15 for positive time. We next solve (4.7) following the same ideas in [8, 9, 10] and [11]. Next, we use (4.11) to obtain a local solution to our original differential equation. A similar argument has been used in [5] and [16] for the classical MGT equation. To better present our results we introduce some notations. We consider the space Y 1 “ X 3 X equipped the norm given by ˆ 2 v 2 2 v v 1 w X, Y. w Y “} }X 3 `} } @ w P › „  › „  › › We can rewrite the initial value› problem› associated with equation (4.7) as the Cauchy problem in Y d v v v (4.12) Λ G , t 0, dt w ` w “ p w q ą „  „  „  and

v v0 (4.13) 0 w p q“ w0 „  „  where w vt and the unbounded linear operator Λ : D Λ Y Y is defined by “ 2 1 p qĂ Ñ D Λ X 3 X 3 p q“ ˆ and v 0 I v w v 2 1 Λ : 2 ´ 1 2 ´ 1 , X 3 X 3 . w “ A 3 η 1 A 3 w “ A 3 v η 1 A 3 w @ w P ˆ „  „ p ´ q  „  „ `p ´ q  „  From now on, we denote 1 2 1 Y D Λ X 3 X 3 . “ p q“ ˆ The nonlinearity G is given by v 0 (4.14) G p w q“ g v „  „ p q where g is given by (4.8). The following result may be established by a straightforward extension of [8, Lemma 1] so we omit its proof. Lemma 4.6. The following conditions hold. i The linear operator Λ is closed and densely defined; q 1 2 ii 0 ρ Λ with Λ´1 pη´1qA´ 3 A´ 3 . Moreover, Λ has compact resolvent; q P p q “ Í 0 iii The spectrum of Λ consists” of isolatedı eigenvalues λ˘ given by q n ˘ 2 3 λ η η 1 ?µn, µn σ A ; n “p ˘ ´ q P p q iv The linear operator Λ generatesa in Y a C0 analytic semigroup e´Λt : t 0 ; vq The semigroup e´Λt´: t 0 in Y are compact and asymptoticallyt decaying;ě u q t ě u 16 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

v vi For each Y 1, we have q w P „  1 v v v Λ η . η w Y 1 ď w Y ď w Y 1 › „  › › „  › › „  › The Lemma 4.6 allows us› to deﬁne› the› fractional› power› Λ›´α of order α 0, 1 . Denote › › › › › › by Y α D Λα for α 0, 1 , taking Λ0 : I on Y 0 : Y when α 0. RecallP p thatq Y α is dense in“Y pforq all α P0, r1 ,q for details see“ [2, Theorem“ 4.6.5]. The“ fractional power space Xα endowed with theP norm p s α Y α : Λ Y }¨} “} ¨} is a Banach space. 1 Remark 4.7. Let X´1 denote the extrapolation space of X generated by A 3 . Consider the 1 1 1 3 3 3 closed extension of A to X´1 (see [2, Page 262]) and still denote it by A . Then A is a 1 1 3 3 sectorial and positive operator in X´1 with the domain X´1; the imaginary powers of A are bounded, see e.g. [7, Proposition 5]. Our next concern will be to obtain embeddings of the α 1 3 spaces from the fractional power scale X 1, α 0, generated by A 3 ,X 1 . ´ ě p ´ q For better understanding the relation of the fractional power scale-spaces of the operator 1 A 3 , we construct the following diagram.

1 A 3

1 1 1 3 3 1 ´ 3 ¨ ¨ ¨ X X X1 X ¨ ¨ ¨ Ap 3 q “

α 1 3 3 X 1 ¨¨¨X´1 X´1 ´ ¨¨¨

1 Figure 6. Fractional power scale generated by operator A 3 in X 1 and α 1. ´ ą The following result also may be established by a straightforward extension of [7, Propo- sition 3] or [9, Theorem 2.3] so we omit its proof. Theorem 4.8. For each α 0, 1 , the fractional power spaces Y α associated to the operator 1`α αPr s Λ coincide with X 3 X 3 with equivalent norms. ˆ Lemma 4.9. Let Y´1 denote the extrapolation space of Y generated by Λ. Then 1 ´ Y 1 X X 3 . ´ “ ˆ Proof: Following [2] and [8] we recall that the extrapolation space of Y is the completion of

´1 v the normed space Y, Λ Y . Since for Y p } } q w P „  ´1 v v 2 ´1 v Λ 1 η 1 η Λ 1 , w X 3 ˆX ď w XˆX´ 3 ď w X 3 ˆX › „  › › „  › › „  › › › › › › › › › › › › › WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH17

1 1 ´1 ´1 completions of X 3 X, Λ 1 and X 3 X, Λ 1 coincide, see the Figure p ˆ } ¨}X 3 ˆX q p ˆ } ¨}XˆX´ 3 q 4.2. Consider the closed extension of Λ to Y´1 (see [2, Page 262]) and still denote it by Λ. Then 1 1 3 Λ is a sectorial and positive operator in Y´1 with the domain Y´1 X X; the imaginary powers of Λ are bounded, see e.g. [7, Proposition 5]. Our next“ concernˆ will be to obtain α embeddings of the spaces from the fractional power scale Y´1, α 0, generated by Λ,Y´1 . The following result also may be established by a straightforwardě extension of [7, Theoremp q 2] so we omit its proof.

α Lemma 4.10. Denote by Y 1 : α 0, 1 the partial fractional power scale generated by t ´ P r su operator Λ in Y´1. Then 1 k`α k`α k´ `α Y 1 X 3 X 3 , α 0, 1 , k N. ´ “ ˆ Pr s P For better understanding the relation of the fractional power scale-spaces of the operator Λ, we construct the following diagram.

1 1 1 ¨ ¨ ¨ Y Y Y 1 Y ´ ¨ ¨ ¨ Λp q “

α 1 ¨ ¨ ¨ Y´1 Y´1 Y´1 ¨ ¨ ¨

Figure 7. Fractional power scale generated by operator Λ in Y 1 and α 1. ´ ą ´Λt Since the semigroup e : t 0 generated by Λ in Y 1 is analytic and the linear t ě u ´ ´ v0 Cauchy problem 4.12-4.13 with Y´1 has a unique solution w0 P „  v v0 (4.15) t e´Λt , t 0. w p q“ w0 ě „  „  In the following theorem we explain the smoothing action of the solution to the linear Cauchy problem 4.12-4.13, it follows from [9, Theorem 2.4] and [11, Theorem 2.1.1].

v0 Theorem 4.11. If t 0 and Y´1, then ą w0 P „  v ´Λt v0 α t e Y´1, for each α 0. w p q“ w0 P ě „  „  v0 1 In particular, if t 0 and Y´1, then ą w0 P „  v ´Λt v0 1 t e Y´1. w p q“ w0 P „  „  18 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

Moreover,

v 1 α C 0, ,Y 1 , for each α 0, 1 . w p¨q P pp 8q ´ q Pr q „ 

v0 1 In particular, if Y´1, then w0 P „ 

v 1 C 0, ,Y 1 . w p¨q P pr 8q ´ q „  1 Theorem 4.12. Assume that f is ǫ regular map relative to the pair X 3 ,X for ǫ 0, in the 1 α ´ 3 p q ě3 sense of (1.2), then g is ǫ regular map relative to the pair X´1,X´1 , where X´1 : α 0 ´ 1p q t ě u 3 denotes the fractional power scale generated by operator A in X´1.

1 Proof: We will prove that there exist constants c 0, ρ 1, γ ǫ 0 with ρǫ γ ǫ 3 1 ą ą p q ą ď p q ă 3 `ǫ γpǫq such that g : X 1 X 1 and ´ Ñ ´ ρ´1 ρ´1 1 g φ1 g φ2 γpǫq c φ1 φ2 `ǫ 1 φ1 1 φ2 1 , X 1 3 3 `ǫ 3 `ǫ ´ X 1 } p q´ p q} ď } ´ } ´ p `} }X´1 `} }X´1 q

1 3 `ǫ for any φ1,φ2 X´1 , see the ﬁgures below. P 1 3 For a better understanding of the ǫ-regular map relative to the pair X´1,X´1 for ǫ 0, we construct the following diagram. p q ě

.1 . 3 X X´1 ¨ ¨ ¨ ´1 g ¨ ¨ ¨

. 1 . 3 `ǫ γpǫq X X 1 ¨¨¨´1 ´ ¨¨¨

1 1 3 `ǫ 3 γpǫq Figure 8. X 1 X 1 and X 1 X 1 ´ Ă ´ ´ Ă ´

1 Since f is an ǫ-regular map relative to the pair X 3 ,X for ǫ 0, we have the following: p q ě if g φi f ψi , where p q“ p q 1 t 1 Á 3 t Á 3 pt´sq ψi e u 0 e φi s ds, i 1, 2, “ p q` 0 p q “ ż 1 and eÁ 3 t; t 0 denotes the C0 semigroup corresponding to (4.6) with v 0 subject to t ě u 1 ´ 1 ” `ǫ Á 3 t initial condition u 0 u0 X 3 then for each t 0, g f e is an ǫ-regular map p q “ P ě “ ˝ WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH19

1 3 relative to the pair X 1,X 1 for ǫ 0 because by Remark 4.7 we have p ´ ´ q ě

1 2 γpǫq g φ g φ X } p q´ p q} ´1 c0 f ψ1 f ψ2 γpǫq ď } p q´ p q}X ρ´1 ρ´1 1 c0 ψ1 ψ2 `ǫ 1 ψ1 1 ψ2 1 X 3 3 `ǫ X 3 `ǫ ď } ´ } p `} }X´1 `} } q t 1 t 1 Á 3 pt´sq Á 3 pt´sq c1 e φ1 s ds e φ2 s ds 1 ǫ “ 0 p q ´ 0 p q X 3 ` ˆ ż ż › 1 t 1 ρ´1 › 1 t 1 ρ´1 › Á 3 t Á 3 pt´sq › Á 3 t Á 3 pt´sq 1 1› e u 0 e φ1 s ds `ǫ › e u 0 e φ2 s ds 1 3 3 `ǫ ˆ ` p q` 0 p q X 1 ` p q` 0 p q X ż ´ ż ´ t› 1 › › › ¯ › Á 3 pt´sq › › › c1 ›e φ1 φ2 s ds 1 › › › ǫ “ 0 p ´ qp q X 3 ` ˆ ż › 1 t 1 › ρ´1 1 t 1 ρ´1 › Á 3 t Á 3 pt´s›q Á 3 t Á 3 pt´sq 1 1› e u 0 e ›φ1 s ds `ǫ e u 0 e φ2 s ds 1 3 3 `ǫ ˆ ` p q` 0 p q X 1 ` p q` 0 p q X ż ´ ż ´ › ρ´1 ρ´›1 › › ¯ 1 c2 φ1› φ2 `ǫ 1 φ1 1 φ2 ›1 › › 3 3 `ǫ 3 `ǫ › X 1 › › › ď } ´ } ´ p `} }X´1 `} }X´1 q for some positive constants c0,c1 and c2. Thanks to [7, Corollary 2] we have the following on the ǫ regular map G defined in (4.14). ´ 1 Proposition 4.13. For ǫ 0, G is ǫ regular map relative to the pair Y´1,Y´1 ; that is, ě ´ p q 1`ǫ there exist constants C 0, ρ 1, γ ǫ with ρǫ γ ǫ 1 such that such that G : Y´1 γpǫq ą ą p q ď p qă Ñ Y´1 and

φ φ1 φ φ1 φ ρ´1 φ1 ρ´1 (4.16) G G C 1 , 1 γpǫq 1 1`ǫ 1`ǫ 1 1`ǫ p ϕ q´ p ϕ q Y´1 ď ϕ ´ ϕ Y´1 ` ϕ Y´1 ` ϕ Y´1 › „  „  › › „  „  › ´ › „  › › „  › ¯ › › › › › › › › › 1 › › › › › › › φ φ 1`ǫ for any , Y 1 . ϕ ϕ1 P ´ „  „  1 φ φ 1`ǫ Proof: Let , Y 1 . Thanks to Lemma 4.10 we have ϕ ϕ1 P ´ „  „ 

1 φ φ 0 1 G G 1 1 g φ g φ ´1`γpǫq γpǫq γpǫq X 3 p ϕ q´ p ϕ q Y´1 “ g φ g φ Y´1 “} p q´ p q} › „  „  › › „ p q´ p q › › › › › › › › › 1 and from fractional power scale generated by operator A 3 in X´1 we obtain

1 φ φ 1 G G c g φ g φ γpǫq 1 γpǫq 3 p ϕ q´ p ϕ q Y´1 ď } p q´ p q}X 1 › „  „  › ´ › › › › 20 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS for some c 0, and by Theorem 4.12 we get ą 1 φ φ ρ´1 ρ´1 1 1 1 G G C φ φ `ǫ˜ 1 φ 1 φ 1 1 γpǫq 3 3 `ǫ˜ 3 `ǫ˜ p ϕ q´ p ϕ q Y 1 ď } ´ }X´1 p `} }X `} }X q „  „  ´ ´1 ´1 › › 1 ρ´1 1 ρ´1 › › C φ φ ǫ˜ 1 φ ˜ φ ˜ › › ď } ´ }X p `} }Xǫ `} }Xǫ q 1 φ φ ρ´1 1 ρ´1 C 1 φ ǫ˜ φ ǫ˜ 1 1`ǫ X X ď ϕ ´ ϕ Y´1 p `} } `} } q › „  „  › › γpǫq 1› ǫ for some C 0, ρ 1 and γ ǫ 0 with›ρǫ˜ 3 3› ǫ˜ 3 . Hence, byą Lemmaą 4.10 wep obtainqą (4.16) andď theă proofp of“ theq result is complete.

v0 1 v 1 Definition 4.14. Let ǫ 0, τ 0, Y´1. We say that : t0, τ Y´1 is ě ą w0 P w r s Ñ „  „  v 1 an ǫ regular mild solution (ǫ solution for shor) to (4.12)-(4.13) if C t0, τ ,Y 1 ´ ´ w P pr s ´ qX „  1`ǫ v C t0, τ ,Y 1 , and t satisfies pp s ´ q w p q „  t v t v0 v s p q e´Λpt´t0q e´Λpt´sqG p q ds. w t “ w0 ` p w s q „ p q „  żt0 „ p q Thanks to [3, Theorem 1], see also [7, Theorem 3], we have the following on existence of ǫ-regular solution to (4.12)-(4.13) on certain interval 0, τ . r s v0 1 Theorem 4.15. Let ǫ 0, τ 0, Y´1. Then exists a unique ǫ-regular solution to ě ą w0 P „  (4.12)-(4.13) on certain interval 0, τ . This solution satisfies r s

v 1 1`θ C 0, τ ,Y 1 C 0, τ ,Y 1 , 0 θ γ ǫ , w P pp s ´ qX pp s ´ q ď ă p q „  and

θ v t 1 t p q Y `θ 0, as t 0, 0 θ γ ǫ . } w t } ´1 Ñ Œ ă ă p q „ p q From this, we establish local well posedness for the Cauchy problem (2.1)-(2.2). We would like to study this problem in the phase space Z. To pose the problem in the mentioned space we will need to consider the nonlinear term F as a map with values in the extrapolated space Z´1 associated to Aη in Z.

u0 1 u 1 Deﬁnition 4.16. Let ǫ 0, τ 0, v0 Z 1 and v : 0, τ Z 1. We say that “ ą w0 P ´ w p¨q r s Ñ ´ u v is an ǫ regular solution to (2.1)” -(2.2)ı on 0, τ ”if andı only if w p¨q ´ r s ” ı u 1 1`ǫ i) v C 0, τ ,Z 1 C 0, τ ,Z 1 ; w p¨q P pr s ´ qX pp s ´ q ” ı WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH21

u ii) v satisfies the Cauchy integral formula: w p¨q ” ı t u Á t u0 Á pt´sq u (4.17) v t e η v0 e η F v s ds, t 0, τ . w0 w p q“ ` 0 w p q Pr s ” ı ” ı ż ´” ı ¯ u0 1 Theorem 4.17. Let ǫ 0, τ 0, v0 Z 1. Then there exists a unique ǫ regular “ ą w0 P ´ ´ solution to (2.1)-(2.2) on certain interval” ı0, τ . In addition, we have r s u 1 1`θ v C 0, τ ,Z 1 C 0, τ ,Z 1 , 0 θ γ ǫ , w P pp s ´ qX pp s ´ q ď ă p q and ” ı u t θ p q t v t 1`θ 0, as t 0, 0 θ γ ǫ . Z´1 } »wptqfi } Ñ Œ ă ă p q p q – fl Λ v0 tÑ0 1 Proof: We also know that, for ǫ 0, tǫ e´ t 0. Knowing that Y ǫ X 3 `ǫ ą w0 ǫ ÝÑ “ ˆ › „ ›Y ǫ ǫ tÑ0 X , we have that t v t, v0,w0 1 › 0. Then› the integral defining u t, u0, v0,w0 is } p q}X 3 `ǫ ÝÑ› › p q convergent and the resulting function is continuous› › at t 0. In fact “ 1 t 1 Á 3 t Á 3 pt´sq 0 2 0 0 2 1´ǫ 1 u u 3 e u u 3 e LpX,X q v s 3 `ǫ ds } ´ }X ď} ´ }X ` 0 } } } p q}X ż 1 t A 3 t 1 ǫ ǫ ǫ ´ 2 ´ ´ 1 e u0 u0 3 M t s s s v s 3 `ǫ ds ď} ´ }X ` 0 p ´ q } p q}X ż 1 1 A 3 t 1 ǫ ǫ ǫ ´ 2 ´ ` ´ 1 e u0 u0 3 M 1 s s ds sup s v s 3 `ǫ ď} ´ }X ` 0 p ´ q 0 } p q}X ż sPr ,ts This ensures the continuity. Finally, the result follows from (4.6) and Theorem 4.15.

5. Remaks on the case η 1 “ In this section we consider the case η 1. We note that the initial value problem associated with equation (2.7) as the Cauchy problem“ in Z

d u u u (5.1) v B 1 v F1 v , t 0, dt w ` p q w “ p w q ą and ” ı ” ı ” ı u u0 (5.2) v 0 v0 , w p q“ w0 ” ı ” ı where v ut and w vt and the unbounded linear operator Bp1q : D Bp1q Z Z is defined by“ “ p q Ă Ñ 1 2 1 (5.3) D B 1 X X 3 X 3 , p p qq“ ˆ ˆ 22 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS and 1 1 u A 3 Í 0 u A 3 u´v u 1 2 1 (5.4) B 1 v : 0 0 Í v ´w , v X X 3 X 3 . p q w 2 w 2 w “ 0 A 3 0 “ A 3 v @ P ˆ ˆ ” ı ” ı” ı ” ı 2 ” ı The nonlinearity F1 given by (2.5), where f : D A 3 X X is a Lipschitz continuous function on bounded sets. p qĂ Ñ Thanks to Lemma 2.2, in particular, we have

Lemma 5.1. Let Bp1q be the unbounded linear operator defined in (5.3)-(5.4). The following conditions hold. i The unbounded linear operator B 1 is closed and densely defined; q p q ii Zero belongs to the resolvent set ρ Bp1q ; namely, the resolvent operator of Bp1q is the q 1 p q bounded linear operator B´ : Z Z given by p1q Ñ 1 ´ 1 1 A 3 0 A´ B´ 2 (5.5) p1q 0 0 A´ 3 . “ 0 Í 0 ” ı Moreover, Bp1q has compact resolvent; iii The spectrum set of B1, σ Bp1q , is given by q ´ 1 p´ q 1 1 3 3 3 σ Bp1q λ C : λ σ A λi C : λ σ A λi C : λ σ A , p´ q “ t P 1 P p´ quYt P P 1p´ quYt´ P P p´ qu where σ A 3 denote the spectrum set of A 3 . p´ q ´

1 iσ A 3 p´ q

1 σ A 3 p´ q Re

1 iσ A 3 ´ p´ q

Figure 9. Semi-lines contained the eigenvalues of B1. ´

Proof: The proof of i easily follows from (5.4). For the proof of ii the result easily follows from (5.5). Finally, forq the proof of iii the result easily follows fromq (5.4). q Theorem 5.2. Let Bp1q be the unbounded linear operator deﬁned in (5.3)-(5.4). The unbounded linear operator B 1 is not a dissipative operator on the state space Z. ´ p q WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH23

u 1 1 1 Proof: Indeed, let u X be such that u 0 and let 2A 3 u Z . Note that P ‰ 0 P u u 1 1 1 ” ı 3 Bp1q 2A 3 u , 2A 3 u A u,u 2 0 0 “ ´x yX 3 and consequently A ” ı ” ıE u u 1 1 Re Bp1q 2A 3 u , 2A 3 u 0 ´ 0 0 ą and the prove is complete. A ” ı ” ıE Theorem5.2 ensures that the linear operator Bp1q is not an inﬁnitesimal generator of a speciﬁc type of strongly continuous semigroup in´ Z; that is, the strongly continuous semigroup of contractions in Z.

6. Moore-Gibson-Thompson-type equations In this subsection we present boundary-initial value problems associated with a Moore- Gibson-Thompson equation with fractional damped and strongly damped linear wave equation, where our results from previous section can be applied. Namely, let Ω RN , N 3, be a bounded smooth domain, and the initial-boundary value problems Ă ě 1 2 uttt ∆u η ∆ 3 utt η ∆ 3 ut f u , t 0, x Ω, ´ ` p´ q ` p´ q “ p q ą P (6.1) u 0, x ϕ x , ut 0, x ξ x , utt 0, x ψ x , x Ω, $ p q“ p q p q“ p q p q“ p q P &’u t, x 0, t 0, x Ω, p q“ ě PB where η 1.’ ą % The nonlinearity f : R R in (6.1) is a continuously diﬀerentiable function satisfying for some 1 ρ 3N`4 the growthÑ condition ă ă 3N´8 (6.2) f 1 s C 1 s ρ´1 . | p q| ď p ` | | q Here we consider X L2 Ω and the negative Laplacian operator “ p q Au ∆u, “´ with domain 2 1 D A H Ω H0 Ω p q“ p qX p q which is a sectorial operator and it bounded imaginary powers, and consequently the spaces Xα, α 0, 1 , are characterized with the aid of complex interpolation as Pr s α 2 1 2 2α X H Ω H0 Ω , L Ω α H Ω “r p qX p q p qs “ tIup q and X´α H2α Ω 1 “p tIup qq where , α denotes the complex interpolation function (see [2] and [13]). In particular 1 1 r¨0 ¨s 2 1 ´ 1 1 1 2 1 X X L Ω , X 2 H0 Ω , X 2 H0 Ω and X H Ω H0 Ω . “ “ p q “ p q “p p qq “ p qX p q With this set-up we will consider problem (6.1) in the form (2.8)-(2.9) with u0 ϕ, v0 1 1 “ “ A 3 ϕ ξ, and w0 A 3 ξ ψ. ` “ ` 24 F.D.M.BEZERRA,A.N.CARVALHO,ANDL.A.SANTOS

1 α Let F : Z´1 Z´1, α 0, be a locally Lipschitz continuous map, as well as in (2.5). Ñ ě u u0 1 Recall that a mild solution of (2.8)-(2.9) on 0, τ is a function v , v0 C 0, τ ,Z 1 r s w ¨ w0 P pr s ´ q which satisfies ” ı´ ” ı¯ t u u0 ´B t u0 ´B pt´sq u u0 v t, v0 e η v0 e η F v s, v0 ds, w0 w0 w0 w “ ` 0 w ” ı´ ” ı¯ ” ı ż ´” ı´ ” ı¯¯ 1 u0 1 for t 0, τ . We say that (2.8)-(2.9) is locally well posed in Z 1 is for any v0 Z 1 there Pr s ´ w0 P ´ is a unique mild solution ” ı u u0 t v t, v0 ÞÑ w w0 of (2.8)-(2.9) defined on a maximal interval” ı´ of existence” ı¯ 0, tu,v0,w0 and depending continu- u0 r q ously on the initial data v0 . w0 As a consequence of the” Sobolevı embeddings we obtain the following result cf. [11, Propo- sition 1.3.8]. Proposition 6.1. Let Ω RN be a bounded domain of class Cm and A, D A be a sectorial operator in Lp Ω , 1 pĂ , with D A W 2m,p Ω for some m p 1. Thenp qq for α 0, 1 the following inclusionp q ă holds.ă8 p qĂ p q ě Pr s Xα W s,q Ω Ă p q if 2mα N s N , 1 p q , s 0. ´ p ě ´ q ă ď ă8 ě From we have 1 Theorem 6.2. The problem (2.8)-(2.9) with η 1 is locally posed in Z´1 whenever f 3 4 ą satisfies (6.2) for some 1 ρ N` ă ă 3N´8 1 Proof: The map F defined as in (2.5) is Lipschitz continuous on bounded sets from Z´1 1`σ σ ´1`σ 3 σ 3 3 3 into Z´1 X X X whenever 0 σ σ˜, andσ ˜ min 1, ρ 1 2 4 N 1 2 “6Npρ´1q ˆ ˆ ă ď “ tu1 p ú2 qp ´ q` u 8 4 2 1 (X 3 ã L ` p σq Ω ). Indeed, if B is a bounded subset of Z 1 and v1 , v2 B, we have Ñ p q ´ w1 w2 P u1 u2 ” ı ” ı F v1 F v2 c1 f u1 f u2 ´1`σ . w1 w2 σ X 3 ´ Z´1 ď } p q´ p q} › ´” 2p1ı¯´σq ´” ı¯› 6N › ´1`›σ Since L 3N`4p1´σq Ω ›ã H 3 Ω X 3 › we obtain p q Ñ tIu p q“ u1 u2 F v1 F v2 c2 f u1 f u2 6N w1 w2 σ 3N`4p1´σq Ω ´ Z´1 ď } p q´ p q}L p q › ´” ı¯ ´” ı¯› and thanks to (6.2)› there exists C 0› such that › ą › ρ´1 ρ´1 s1,s2 R, f s1 f s2 C s1 s2 1 s1 s2 @ P | p q´ p q| ď | ´ |p ` | | ` | | q and consequently u1 u2 ρ´1 ρ´1 1 2 F v F v c3 u1 u2 2 1 u1 6 1 u2 6 1 1 2 3 Npρ´ q Npρ´ q w w σ X 8 4 1 8 4 1 ´ Z´1 ď } ´ } `} }L ` p ´σq pΩq `} }L ` p ´σq pΩq › ´” ı¯ ´” ı¯› ´ ¯ › › u1 u2 c4 v1 v2 . › › w1 w2 1 ď ´ Z´1 ›” ı ” ı› › › › › WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH25

The proof now follows from [15].

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(F. D. M. Bezerra) Departamento de Matematica,´ Universidade Federal da Para´ıba, 58051- 900 Joao˜ Pessoa PB, Brazil. Email address: [email protected]

(A. N. Carvalho) Departamento de Matematica,´ Instituto de Cienciasˆ Matematicas´ e de Computac¸ao,˜ Universidade de Sao˜ Paulo-Campus de Sao˜ Carlos, Caixa Postal 668, 13560- 970 Sao˜ Carlos SP, Brazil. Email address: [email protected]

(L. A. Santos) Instituto Federal da Para´ıba, 58780-000 Itaporanga PB, Brazil. Email address: [email protected]