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

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Well-Posedness for Some Third-Order Evolution Differential Equations: a Semigroup Approach3 WELL-POSEDNESS FOR SOME THIRD-ORDER EVOLUTION DIFFERENTIAL EQUATIONS: A SEMIGROUP APPROACH FLANK D. M. BEZERRA, ALEXANDRE N. CARVALHO;, AND LUCAS A. SANTOS Abstract. In this paper, we discuss the well-posedness of the Cauchy problem associated with the third-order evolution equation in time 1 2 uttt ` Au ` ηA 3 utt ` ηA 3 ut “ fpuq where η ą 0, X is a separable Hilbert space, A : DpAq Ă X Ñ X is an unbounded sectorial operator with compact resolvent, and for some λ0 ą 0 we have ReσpAq ą λ0 1 and f : DpA 3 q Ă X Ñ X is a nonlinear function with suitable conditions of growth and regularity. Mathematics Subject Classification 2010: 34A08, 47D06, 47D03, 35K10. Key words and phrases: Approximations; fractional powers; sectorial operator; semigroups. Contents 1. Introduction 1 2. Functional Framework 3 2.1. Natural setting 3 2.2. New approach 5 3. Ill-posed problems 7 4. Parabolic differential equations 9 4.1. Sectoriality 10 4.2. Analysis by reducing the order 14 5. Remaks on the case η 1 21 6. Moore-Gibson-Thompson-type“ equations 23 References 25 arXiv:2106.03564v1 [math.AP] 7 Jun 2021 1. Introduction In this paper, we discuss the well-posedness of the Cauchy problem associated with the following third-order evolution equation in time 1 2 3 3 (1.1) uttt Au ηA utt ηA ut f u ` ` ` “ p q Date: June 8, 2021. ;Research partially supported by CNPq # 303929/2015-4 and by FAPESP # 2003/10042-0, Brazil. 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 Fr´echet 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 ´I 0 ´v u u u 1 2 1 A v 0 0 ´I 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 ´A t u0 ´A 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 ´I 0 0 . “ 0 ´I 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 .
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