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Regularity of solutions for a third order differential equation in Hilbert spaces
Applied Mathematics and Computation 217 (2011) 8522–8533
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Regularity of solutions for a third order differential equation in Hilbert spaces Claudio Fernández a, Carlos Lizama b,⇑, Verónica Poblete c a
Pontificia Universidad Católica de Chile, Facultad de Matemáticas, Avenida Vicuña Mackenna 4860, Santiago, Chile Universidad de Santiago de Chile, Departamento de Matemática, Facultad de Ciencias, Casilla 307-Correo 2, Santiago, Chile c Universidad de Chile, Departamento de Matemática, Facultad de Ciencias, Las Palmeras 3425 Ñuñoa, Santiago, Chile b
a r t i c l e
i n f o
Keywords: Third order Cauchy problems Regularity of solutions Mild and strong solutions Energy functional
a b s t r a c t We study regularity of mild and strong solutions of an abstract mathematical model of a flexible space structure under appropriate initial conditions. We apply our results showing qualitative properties of the trajectories in the case of the negative Laplacian operator. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Let H be a Hilbert space and A an unbounded, self-adjoint, elliptic operator in H. Consider the problem
8 000 au ðtÞ þ u00 ðtÞ þ bAuðtÞ þ cAu0 ðtÞ ¼ f ðtÞ; > > > < uð0Þ ¼ u ; 0 0 > u ð0Þ ¼ u > 1; > : 00 u ð0Þ ¼ u2 :
t P 0; ð1:1Þ
Here f : Rþ ! H; a; b; c 2 Rþ ; and u0, u1, u2 2 H are given data. In the case A is the negative Laplacian in H = L2(X) equipped with suitable boundary conditions, problem (1.1) was first derived by Bose and Gorain [4] to model flexible structural systems possessing internal damping. For further information and a complete description of the model, see [2,3]. Maximal regularity properties for problem (1.1) in Lebesgue spaces has been recently studied in [9]. In this paper, we will study existence, uniqueness and regularity properties for strong and mild solutions of problem (1.1). A popular approach is to reduce the third order problem to a first order system in a suitable phase space and use operator semigroup theory. The disadvantage of this approach is that finding an ideal space is generally difficult and the structure of the phase space (if any) may be complicated, so that it is inconvenient for computational applications; also, it is well known that some inherent properties of problems of order greater than or equal to two cannot always be reflected precisely from the corresponding first order systems (see for example [8,14]). Our idea is to establish a more inclusive theory about third order problems and therefore we give a direct treatment of problem (1.1). It is also important to note that abstract differential equations of order greater than two are, in general, ill-posed (see [14]). Consequently, in our case we necessarily have to assume ab < c. pffiffiffi In Section 3, it is shown that under these assumption, for any u0 2 DðAÞ; u1 2 Dð AÞ; u2 2 H and f 2 L1 ð0; þ1; HÞ, the solution u satisfies
⇑ Corresponding author. E-mail addresses: [email protected] (C. Fernández), [email protected] (C. Lizama), [email protected] (V. Poblete). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.056
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pffiffiffi u 2 C 1 ð½0; 1Þ; Dð AÞÞ \ Cð½0; 1Þ; HÞ: pffiffiffi pffiffiffi For smoother data, u0 2 DðA3=2 Þ; u1 2 DðAÞ; u2 2 Dð AÞ; and f 2 L1 ð½0; þ1Þ; Dð AÞÞ; we show that the solution u is more regular, that is,
pffiffiffi u 2 C 3 ð½0; 1Þ; HÞ \ C 2 ð½0; 1Þ; Dð AÞÞ \ C 1 ð½0; 1Þ; DðAÞÞ \ Cð½0; 1Þ; DðAÞÞ: The paper is organized as follows. Section 2 is mainly devoted to the construction of a resolvent operator family for problem (1.1) with f 0. In passing, we define an energy functional naturally associated to third order problems, by means of the expression
EðtÞ ¼
i pffiffiffi pffiffiffi pffiffiffi 1h kaS00 ðtÞx þ S0 ðtÞxk2 þ bka AS0 ðtÞx þ ASðtÞxk2 þ aðc abÞk AS0 ðtÞxk2 : 2
In Section 3, we state our main results on regularity of mild and strong solutions of problem (1.1), as well as important a priori-estimates, necessary for applications in nonlinear problems. In Section 4, we give some practical examples of application of our results, including the Voigt model of viscoelasticity and the Klein–Gordon equation. 2. The homogeneous problem Let H be a real Hilbert space with scalar product h, i. In this section, we consider the homogeneous equation
au000 ðtÞ þ u00 ðtÞ þ bAuðtÞ þ cAu0 ðtÞ ¼ 0; for t P 0;
ð2:1Þ
where A is an unbounded, self-adjoint linear operator with domain D(A) dense in H. The domain D(A) will be regarded as a Hilbert space endowed with the graph-norm. We further assume that A is elliptic, or in other words there exists a constant M > 0 such that
hAx; xi P Mkxk2 ;
for all x 2 DðAÞ:
ð2:2Þ
pffiffiffi Since D(A) is dense in Dð AÞ, we easily obtain the abstract version of Poincare’s inequality with Dirichlet boundary conditions
pffiffiffi k Axk P Mkxk;
pffiffiffi for all x 2 Dð AÞ:
ð2:3Þ
In this section, we will denote the solution of Eq. (2.1) by means of a strongly continuous family of bounded, linear operators, that we denote by {S(t)}tP0. We define the abstract energy functional associated to Eq. (2.1) and get a priori estimates. Recall that a family fCðtÞgt2R of continuous linear operators on H is called a strongly continuous operator cosine family if C(0) = I, 2C(t)C(s) = C(t + s) + C(t s) and limt?0 C(t)x = x for all x 2 H. For further information on cosine families, see [1,8], and the references given there. The following observation, stated as a Lemma, will be the key to associate to Eq. (2.1) a family of operators which will then provide the solution of the non-homogeneous problem (1.1) by means of a kind of variation of parameters formula. Lemma 2.1. If A is a self-adjoint, strictly positive operator, then – A generates a strongly continuous operator cosine family. pffiffiffi Proof. By Stone’s theorem, there exists a strongly continuous group of unitary operators fUðtÞgt2R with generator i A: Define for t 2 R and x 2 H
CðtÞx ¼
UðtÞx þ UðtÞx : 2
Then we obtain a strongly continuous operator cosine family with generator C00 (0)x = Ax, x 2 D(A). h Now we define an appropriate concept of resolvent family for the homogeneous problem. Definition 2.2. A family {S(t)}tP0 of bounded linear operators in H is called a regular resolvent for Eq. (2.1) if the following conditions are satisfied: (S1) (S2) (S3) (S4)
S(t) is strongly continuous on [0, 1) and S(0) = I. For any x 2 D(A) and t P 0, we have that S(t)x 2 D(A) and AS(t)x = S(t)Ax. For any x 2 D(A), we have that S()x is twice continuously differentiable in H on [0, 1). For any x 2 D(A2) and t P 0, we have that S0 (t)x 2 D(A), AS0 ()x is continuous and S()x is three times continuously differentiable in H on [0, 1). Moreover, the equation
aS000 ðtÞx þ S00 ðtÞx þ cAS0 ðtÞx þ bASðtÞx ¼ 0 holds.
ð2:4Þ
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In order to establish existence of a regular resolvent for Eq. (2.1), we will use the theory of integral evolutionary equations [13]. For this, note that by integrating three times Eq. (2.1), we find that it is equivalent to the perturbed integral Volterra equation
uðtÞ ¼ gðtÞ þ ða ðAÞuÞðtÞ þ ðb uÞðtÞ;
ð2:5Þ
where
aðtÞ ¼
b 2 c t þ t a 2a
gðtÞ ¼
t c 2 t2 t2 u0 ð0Þ þ u00 ð0Þ: 1þ þ t A uð0Þ þ t þ a 2a 2a 2
and bðtÞ ¼
1
ð2:6Þ
a
and
ð2:7Þ
Our main result of this section is the following. Theorem 2.3. Eq. (2.1) admits a regular resolvent {S(t)}tP0 which satisfies the following:
aS0 ðtÞx þ SðtÞx þ cAð1 SÞðtÞx þ bAðt SÞðtÞx ¼ 0
ð2:8Þ
aS00 ðtÞx þ S0 ðtÞx þ cASðtÞx þ bAð1 SÞðtÞx ¼ 0 for all x 2 DðAÞ:
ð2:9Þ
and
Proof. It follows from Lemma 2.1 that A is the generator of a cosine family, say C(t). Hence the equation
v ðtÞ ¼ gðtÞ þ
c a
Z
t
ðt sÞðAÞv ðsÞds
ð2:10Þ
0
is well-posed and (t ⁄ C)(t) 2 D(A) for all t P 0 (see [13, Definition 1.3 and Proposition 1.1]). Consider Eq. (2.5) where a(t) is given in (2.6). Define cðtÞ ¼ ac t, kðtÞ ¼ bc : Note that Eq. (2.5) can be rewritten as
uðtÞ ¼ gðtÞ þ
Z
t
aðt sÞðAÞuðsÞds þ 0
¼ gðtÞ þ
b 2a
Z
t
bðt sÞuðsÞds
0
Z
t
ðt sÞ2 ðAÞuðsÞds þ
0
c a
Z
t
ðt sÞðAÞuðsÞds þ ðb uÞðtÞ
0
¼ gðtÞ þ ðc k ðAÞuÞðtÞ þ ðc ðAÞuÞðtÞ þ ðb uÞðtÞ: Hence
uðtÞ ¼ gðtÞ þ ½ðc þ c kÞ ðAÞuðtÞ þ ðb uÞðtÞ:
ð2:11Þ
Since it is a perturbed form of (2.10), it follows from [13, Theorem 1.2] that Eq. (2.5) admits a resolvent family. Furthermore, c u 2 Cð½0; þ1Þ; DðAÞÞ if and only if ðc þ c kÞ u 2 Cð½0; þ1Þ; DðAÞÞ (see the remark after formula (2.16) of [13, p.39]). Then there exists a family {S(t)}tP0 of bounded linear operators in H such that (i) S(0) = I and S(t) is strongly continuous on [0, 1); (ii) S(t) commutes with A, which means that S(t)D(A) D(A) and AS(t)x = S(t)Ax, for all x 2 D(A) and t P 0; (iii) for t P 0 and x 2 D(A), the resolvent equation
SðtÞx ¼ x þ ða ðAÞSÞðtÞx þ ðb SÞðtÞx
ð2:12Þ
holds. By [13, Proposition 1.1], for any x 2 H, the function (a ⁄ S)()x belongs to Cð½0; 1Þ; DðAÞÞ; Aða SÞ is strongly continuous in H and
SðtÞx ¼ x Aða SÞðtÞx þ ðb SÞðtÞx;
for x 2 H:
ð2:13Þ
Note that, for all x 2 D(A), from (2.12) and the fact that S(t) commutes with A, we conclude that (b ⁄ S)(t) x = S(t)x x + (a ⁄ S)(t)Ax and hence (1 ⁄ S)(t) is leaving D(A) invariant and A(1 ⁄ S)() is strongly continuous. Moreover, from the proof of [13, Theorem 1.2] we deduce that (t ⁄ S)(t) maps H into D(A) and A(t ⁄ S)() is strongly continuous. By (i) and (ii), it is clear that (S1) and (S2) are satisfied. For x 2 D(A), differentiating (2.12) we obtain (2.8). Again, differentiating (2.12) we obtain (2.9), for all x 2 D(A). We conclude that (S3) holds.
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Now, for x 2 D(A2), from (2.8) it follows that S0 (t)x 2 D(A) and AS0 () is strongly continuous. Finally, differentiating (2.9) we obtain that the resolvent equation in (S4) holds. h The next Theorem gives a number of bounds and inequalities that are the key pieces for the main results obtained later in this paper. Theorem 2.4. Suppose ab < c. Then the regular resolvent {S(t)}tP0 for Eq. (2.1) satisfies the following properties: (i) For any x 2 H, the functions
t ! ð1 SÞðtÞx ¼
Z
t
SðsÞxds and t ! ðt SÞðtÞx ¼
pffiffiffi 0 belong to Cð½0; 1Þ; Dð AÞÞ: In addition, for t P 0, we have
Z
t
ðt sÞSðsÞxds;
0
pffiffiffi k Að1 SÞðtÞxk 6 Ckxk
ð2:14Þ
pffiffiffi k Aðt SÞðtÞxk 6 Ckxk:
ð2:15Þ
and pffiffiffi pffiffiffi (ii) For any x 2 Dð AÞ; SðÞx belongs to Cð½0; 1Þ; Dð AÞÞ: Moreover, for t P 0, we have
pffiffiffi pffiffiffi k ASðtÞxk 6 Ck Axk:
ð2:16Þ
pffiffiffi Rt Rt (iii) For any x 2 Dð AÞ; the functions t ! ð1 SÞðtÞx ¼ 0 SðsÞxds and t ! ðt SÞðtÞx ¼ 0 ðt sÞSðsÞxds belong to Cð½0; 1Þ; DðAÞÞ. In addition, for t P 0, we have
pffiffiffi kAð1 SÞðtÞxk 6 Ck Axk
ð2:17Þ
pffiffiffi kAðt SÞðtÞxk 6 Ck Axk:
ð2:18Þ
and pffiffiffi (iv) For any x 2 Dð AÞ; SðÞx is continuously differentiable in H on [0, 1). Moreover, for t P 0, we have
pffiffiffi kS0 ðtÞxk 6 Cðkxk þ k AxkÞ: (v) For any x 2 D(A), S()x belongs to Cð½0; 1Þ; DðAÞÞ. Moreover, for t P 0, we have
kASðtÞxk 6 CkAxk: pffiffiffi (vi) For any x 2 D(A), S0 ()x belongs to Cð½0; 1Þ; Dð AÞÞ: Moreover, for t P 0, we have
pffiffiffi k AS0 ðtÞxk 6 CkAxk: Here C denotes a suitable positive constant which depends on a, b and c. Proof. (i) Let x 2 D(A) and ð1 SÞðtÞx ¼
Rt 0
SðsÞxds: Multiplying (2.8) by aS(t)x + (1 ⁄ S)(t)x we obtain
2
a hS ðtÞx; SðtÞxi þ akSðtÞxk þ abhAðt SÞðtÞx; SðtÞxi þ achAð1 SÞðtÞx; SðtÞxi þ ahS0 ðtÞx; ð1 SÞðtÞxi þ hSðtÞx; ð1 SÞðtÞxi þ bhAðt SÞðtÞx; ð1 SÞðtÞxi þ chAð1 SÞðtÞx; ð1 SÞðtÞxi ¼ 0 2
0
or equivalently
2 ac d pffiffiffi 1 d 0 kSðtÞxk2 þ akSðtÞxk2 þ abhAðt SÞðtÞx; SðtÞxi þ kð1 SÞðtÞxk2 Að1 SÞðtÞx þ ahS ðtÞx; ð1 SÞðtÞxi þ 2 dt 2 dt 2 dt 2 b d pffiffiffi þ Aðt SÞðtÞx þ chAð1 SÞðtÞx; ð1 SÞðtÞxi ¼ 0: 2 dt
a2 d
Integrating the last identity on [0, t], we have
Z t Z t 2 a2 ac pffiffiffi kSðsÞxk2 ds þ ab hAðt SÞðsÞx; SðsÞxids þ Að1 SÞðtÞx kSðtÞxk2 kSð0Þxk2 þ a 2 2 2 0 0 Z t Z t 2 1 b pffiffiffi þ a hS0 ðsÞx; ð1 SÞðsÞxids þ kð1 SÞðtÞxk2 þ Aðt SÞðtÞx þ c hAð1 SÞðsÞx; ð1 SÞðsÞxids ¼ 0: 2 2 0 0
a2
Integrating by parts the fourth and sixth terms on the left-hand side of the preceding equality, we obtain
Z t Z t pffiffiffi 2 2 ac pffiffiffi kSð0Þxk2 þ a kSðsÞxk2 ds þ abhAðt SÞðtÞx;ð1 SÞðtÞxi ab Að1 SÞðsÞx ds þ Að1 SÞðtÞx 2 2 2 0 0 Z t Z t pffiffiffi 2 2 1 b pffiffiffi þ ahSðtÞx;ð1 SÞðtÞxi a kSðsÞxk2 ds þ kð1 SÞðtÞxk2 þ Aðt SÞðtÞx þ c Að1 SÞðsÞx ds ¼ 0: 2 2 0 0
a2
kSðtÞxk2
a2
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Taking into account the Hilbert space structure, we can reorganize the above equality as follows:
1 kaSðtÞx þ ð1 SÞðtÞxk2 þ abhAðt SÞðtÞx; ð1 SÞðtÞxi þ ðc abÞ 2 2 a2 b pffiffiffi þ Aðt SÞðtÞx ¼ kSð0Þxk2 ; 2 2
Z t pffiffiffi 2 2 ac pffiffiffi Að1 SÞðsÞx ds þ Að1 SÞðtÞx 2 0
or equivalently
Z t pffiffiffi 2 2 pffiffiffi 1 b pffiffiffi kaSðtÞx þ ð1 SÞðtÞxk2 þ Aðt SÞðtÞx þ a Að1 SÞðtÞx þ ðc abÞ Að1 SÞðsÞx ds 2 2 0 2 a2 ac pffiffiffi a2 b pffiffiffi 2 þ k Að1 SÞðtÞxk2 Að1 SÞðtÞx ¼ kSð0Þxk : 2 2 2 Hence, we obtain the identity
Z t pffiffiffi pffiffiffi pffiffiffi kaSðtÞx þ ð1 SÞðtÞxk2 þ bka Að1 SÞðtÞx þ Aðt SÞðtÞxk2 þ 2ðc abÞ k Að1 SÞðsÞxk2 ds 0 pffiffiffi þ aðc abÞk Að1 SÞðtÞxk2 ¼ a2 kxk2 :
ð2:19Þ
Since ab < c, we have that the sum on the left-hand side of the equality preceding is positive for all x non-zero. Then
pffiffiffi k Að1 SÞðtÞxk 6 Ckxk;
for any t P 0;
x 2 DðAÞ:
Again, for t P 0, from (2.19) we obtain
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi Aðt SÞðtÞx ¼ Að1 SÞðtÞx þ Aðt SÞðtÞx Að1 SÞðtÞx 6 Að1 SÞðtÞx þ Aðt SÞðtÞx þ Að1 SÞðtÞx 6 Ckxk; where C is a generic constant which depends on a, b and c. The proof of (i) follows using the density of D(A) in H. (ii) Fix x 2 D(A). As in the proof of (i), multiplying (2.8) by aS0 (t)x + S(t)x and integrating on [0, t],t P 0, we obtain the identity
pffiffiffi 2 pffiffiffi 2 pffiffiffi 2 pffiffiffi kaS0 ðtÞx þ SðtÞxk2 þ ba ASðtÞx þ Að1 SÞðtÞx þ aðc abÞ ASðtÞx ¼ ac Ax :
ð2:20Þ
pffiffiffi pffiffiffi pffiffiffi Since ab < c, we have ASðtÞx 6 C Ax; for any t P 0 and x 2 D(A). Using the density of D(A) in Dð AÞ; the result follows. (iii) Fix x 2 D(A). According to (2.8), we have
cAð1 SÞðtÞx ¼ aS0 ðtÞx þ SðtÞx þ bAðt SÞðtÞx: Then
kAð1 SÞðtÞxk 6
1
c
b kaS0 ðtÞx þ SðtÞxk þ kAðt SÞðtÞxk:
c
pffiffiffi From (2.20) we have kaS ðtÞx þ SðtÞxk 6 Ck Axk and by (2.15) we obtain that 0
pffiffiffi pffiffiffi pffiffiffi kAðt SÞðtÞxk ¼ k Aðt SÞðtÞ Axk 6 Ck Axk: We conclude that
pffiffiffi kAð1 SÞðtÞxk 6 Ck Axk: pffiffiffi pffiffiffi Finally, note that the previous inequality holds for any x 2 Dð AÞ thanks to the density of D(A) in Dð AÞ: This proves (2.17) and (2.18). (iv) Fix x 2 D(A). It follows from (2.8) that, for t P 0, we have
kS0 ðtÞxk 6
1
a
pffiffiffi b c kSðtÞxk þ kAðt SÞðtÞxk þ kAð1 SÞðtÞxk 6 Cðkxk þ k AxkÞ:
a
a
pffiffiffi Using the density of D(A) in Dð AÞ again, the conclusion follows. (v) Let x 2 D(A). It follows from (ii) that
pffiffiffi pffiffiffi kASðtÞxk ¼ k ASðtÞ Axk 6 CkAxk: (vi) Fix x 2 D(A2). Multiplying (2.4) by a S00 (t)x + S0 (t)x, we obtain
C. Fernández et al. / Applied Mathematics and Computation 217 (2011) 8522–8533
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ac d pffiffiffi 0 2 1 d 0 kS00 ðtÞxk2 þ abhASðtÞx; S00 ðtÞxi þ kS ðtÞxk2 þ ahS000 ðtÞx; S0 ðtÞxi AS ðtÞx þ 2 dt 2 dt 2 dt 2 b d pffiffiffi 0 0 þ ASðtÞx þ chAS ðtÞx; S ðtÞxi ¼ 0: 2 dt
akS00 ðtÞxk2 þ
a2 d
Integrating on [0, t], we obtain
a
Z
Z t Z t 2 1 ac pffiffiffi kS00 ðtÞxk2 þ ab hASðsÞx; S00 ðsÞxids þ AS0 ðtÞx þ kS0 ðtÞxk2 þ a hS000 ðsÞx; S0 ðsÞxids 2 2 2 0 0 0 Z t 2 b pffiffiffi b pffiffiffi a2 00 1 0 ac pffiffiffi 0 2 0 2 2 0 þ k ASðtÞxk þ c hAS ðsÞx; S ðsÞxids ¼ kS ð0Þxk þ kS ð0Þxk þ AS ð0Þx þ k ASð0Þxk2 : 2 2 2 2 2 0 t
kS00 ðsÞxk2 ds þ
a2
Integrating by parts the third and sixth terms on the left-hand side of the above equality, we obtain
Z t pffiffiffi pffiffiffi 2 pffiffiffi 2 2 pffiffiffi 1 0 kaS00 ðtÞx þ S0 ðtÞxk2 þ ba AS0 ðtÞx þ ASðtÞx þ aðc abÞ AS0 ðtÞx þ ðc abÞ AS ðsÞx ds 2 0 2 pffiffiffi pffiffiffi a2 1 ac pffiffiffi b pffiffiffi ¼ kS00 ð0Þxk2 þ kS0 ð0Þxk2 þ k AS0 ð0Þxk2 þ ASð0Þx þ ahS0 ð0Þx; S00 ð0Þxi þ abh AS0 ð0Þx; ASð0Þxi: 2 2 2 2 From (2.8) and (2.9), we have that S0 ð0Þx ¼ a1 x; S00 ð0Þx ¼ a12 x ac Ax: Replacing in preceding identity, we obtain
Z t pffiffiffi pffiffiffi 2 pffiffiffi 2 2 pffiffiffi 0 kaS00 ðtÞx þ S0 ðtÞxk2 þ ba AS0 ðtÞx þ ASðtÞx þ aðc abÞ AS0 ðtÞx þ 2ðc abÞ AS ðsÞx ds
c ab pffiffiffi 2 2 2 ¼ Ax þ c kAxk : a
0
ð2:21Þ
pffiffiffi Since ab < c, and since A is elliptic, we obtain AS0 ðtÞx 6 CkAxk; for any t P 0 and x 2 D(A2). Using the density of D(A2) in D(A) completes the proof. h Corollary 2.5. Under the assumptions of Theorem 2.4, we have: (a) For x 2 H, there exists C > 0 such that, for t P 0 we have
kðt SÞðtÞxk 6 Ckxk and kð1 SÞðtÞxk 6 Ckxk: (b) For x 2 H, there exists C > 0 such that, for t P 0, we have
kSðtÞxk 6 Ckxk: pffiffiffi (c) For x 2 Dð AÞ; there exists K > 0 such that, for t P 0, we have
pffiffiffi kSðtÞxk 6 Kk Axk: pffiffiffi (d) Equality (2.8) holds for all x 2 Dð AÞ: Proof pffiffiffi (a) For x 2 H, according to Theorem 2.4 (i), we deduce that (t ⁄ S)(t)x and (1 ⁄ S)(t)x belong to Dð AÞ: Assertion (a) now follows from (2.3), (2.14) and (2.15). (b) For x 2 H and t P 0, note that
akSðtÞxk 6 kaSðtÞx þ ð1 SÞðtÞxk þ kð1 SÞðtÞxk: Assertion (b) now pffiffiffifollows from (2.19) and assertion (a). pffiffiffi (c) For x 2 Dð AÞ; according to Theorem 2.4 (ii), we have that S(t)x belongs to Dð AÞ: Assertion (c) now follows from (2.3) and (2.16). pffiffiffi Assertion (d) follows from Theorem 2.4 (iii), (2.8), and the density of D(A) in Dð AÞ:
h
Based on the proof of the previous theorem, we define the following energy functional associated to our problem. We consider a P 0. Definition 2.6. For x 2 D(A) and ab 6 c, we define the abstract energy functional associated to Eq. (2.1) by
EðtÞ ¼
2 2 pffiffiffi pffiffiffi pffiffiffi 1 kaS00 ðtÞx þ S0 ðtÞxk2 þ ba AS0 ðtÞx þ ASðtÞx þ aðc abÞ AS0 ðtÞx ; 2
for t P 0:
ð2:22Þ
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Note that by the previous Theorem E : ½0; 1Þ ! R is a well defined, positive and continuous function. Moreover, note that in the case a = 0, c = 0 and b = 1 we obtain
EðtÞ ¼
2 1 0 1 pffiffiffi kS ðtÞxk2 þ ASðtÞx ; 2 2
which is the usual expression of the energy functional for the second order equation
u00 ðtÞ þ AuðtÞ ¼ f ðtÞ: Compare also with Sforza [5, p. 160]. Moreover, note that (2.22) is the abstract version of the energy functional defined by Bose-Gorain [3] in the case A = D, the Laplacian. We conclude this section with the following result on an important property of the energy functional E(t). Theorem 2.7. Suppose ab 6 c. For any x 2 D(A), E(t) is non-increasing. In particular, E(t) 6 E(0). Proof. Let x 2 D(A). If a > 0, note that
Eð0Þ ¼
pffiffiffi 2 pffiffiffi pffiffiffi 1 c ab pffiffiffi 2 c2 2 kaS00 ð0Þx þ S0 ð0Þxk2 þ ba AS0 ð0Þx þ ASð0Þx þ aðc abÞk AS0 ð0Þxk2 ¼ Ax þ kAxk : 2 2a 2
From (2.21) we have that
EðtÞ þ ðc abÞ
Z t pffiffiffi 2 0 AS ðsÞx ds ¼ Eð0Þ:
ð2:23Þ
0
For ab < c we obtain that hðtÞ ¼ ðc abÞ
2 Rt pffiffiffi 0 AS ðsÞx ds is a positive and increasing function, we conclude that E(t) is non0
increasing and E(t) 6 E(0) for any t P 0. for any x 2 D(A) and t P 0, hence E(t) is non-increasing, in particular E(t) 6 E(0). h From (2.23) we directly obtain the following Corollary. Corollary 2.8. Suppose ab 6 c. For any x 2 D(A), we have
pffiffiffi 2 E0 ðtÞ ¼ ðab cÞ AS0 ðtÞx :
ð2:24Þ
3. Regularity of strong and mild solutions In this section we will assume that A is a self-adjoint linear operator which satisfies the ellipticity condition (2.2). The existence of a regular resolvent {S(t)}tP0 proved in the preceding section allows us to solve the non-homogeneous problem
8 000 au ðtÞ þ u00 ðtÞ þ bAuðtÞ þ cAu0 ðtÞ ¼ f ðtÞ; > > > < uð0Þ ¼ u ; 0 0 > ð0Þ ¼ u u > 1; > : 00 u ð0Þ ¼ u2
t P 0; ð3:25Þ
for which we will now state the notions of mild and strong solutions. Definition 3.9. Let f 2 Cð½0; 1Þ; HÞ. We say that u is a strong solution of problem (3.25) in ½0; 1Þ u 2 C 3 ð½0; 1Þ; HÞ \ C 1 ð½0; 1Þ; DðAÞÞ, and u satisfies problem (3.25). Let f 2 L1 ð0; 1; HÞ; u0 2 DðAÞ and u1, u2 2 H. The mild solution of problem (3.25) in [0, 1) with initial conditions
uð0Þ ¼ u0 ;
u0 ð0Þ ¼ u1 ;
if
u00 ð0Þ ¼ u2
is the function u 2 Cð½0; 1Þ; HÞ defined by
1 c 1 1 uðtÞ ¼ SðtÞu0 þ ð1 SÞðtÞu0 þ ðt SÞðtÞAu0 þ ð1 SÞðtÞu1 þ ðt SÞðtÞu1 þ ðt SÞðtÞu2 þ ðt S f ÞðtÞ:
a
a
a
a
We begin with the following result. Theorem 3.10. If u is a strong solution of problem (3.25), then u is a mild solution of problem (3.25). Proof. A strong solution u of problem (3.25) is also a strong solution of the integral equation
Z t t c 2 t2 t2 1 u1 þ u2 þ ð1 t f ÞðtÞ þ uðtÞ ¼ 1 þ þ t A u0 þ t þ Aðt sÞuðsÞds; a 2a 2a 2 a 0
ð3:26Þ
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where AðtÞ ¼ ð2ba t2 þ ac tÞA a1 I: According to [13, Proposition 6.3], we have
uðtÞ ¼ ¼
d dt Z t
# 1 c ðt sÞ2 ðt sÞ2 1 1 þ ðt sÞ þ ðt sÞ2 A u0 þ ðt sÞu1 þ u1 þ u2 þ ð1 t f Þðt sÞ ds a 2a 2a 2 a 0 1 c 1 1 SðsÞ þ ðt sÞA u0 þ u1 þ ðt sÞu1 þ ðt sÞu2 þ ðt f Þðt sÞ ds þ SðtÞu0
Z
t
"
SðsÞ
a
a
a a c 1 1 ¼ SðtÞu0 þ ð1 SÞðtÞu0 þ ðt SÞðtÞAu0 þ ð1 SÞðtÞu1 þ ðt SÞðtÞu1 þ ðt SÞðtÞu2 þ ðt S f ÞðtÞ: a a a a 0
1
In what follows, we will assume the condition ab < c. Our next regularity result establishes conditions on the initial data under which mild solutions are obtained. We also state a priori bounds that will be useful later. pffiffiffi Theorem 3.11. If f 2 L1 ð½0; 1Þ; HÞ, u0 2 DðAÞ; u1 2 Dð AÞ; and u2 2 H, then the mild solution of problem (3.25) belongs to p ffiffiffi C 1 ð½0; 1Þ; Dð AÞÞ \ C 1 ð½0; 1Þ; HÞ: Moreover, for any t 2 [0, 1), the following estimate holds:
pffiffiffi pffiffiffi pffiffiffi ku00 ðtÞk þ ku0 ðtÞk þ k Au0 ðtÞk þ kuðtÞk þ k AuðtÞk 6 Cðku0 k þ ku1 k þ ku2 k þ kAu0 k þ k Au1 k þ kf k1 Þ:
ð3:27Þ
pffiffiffi Proof. It follows from (i) and (ii) of Theorem 2.4 that u 2 Cð½0; 1Þ; Dð AÞ \ Cð½0; 1Þ; HÞ: According to 2.4 and Remark 2.5 we have
1 c 1 kuðtÞk 6kSðtÞu0 k þ kð1 SÞðtÞu0 k þ kðt SÞðtÞAu0 k þ kð1 SÞðtÞu1 k þ kðt SÞðtÞu1 k þ kðt SÞðtÞu2 k
a
a
a
1
þ kðt S f ÞðtÞk a pffiffiffi 6 Cðk Au0 k þ ku0 k þ kAu0 k þ ku1 k þ ku2 k þ kf k1 Þ 6 Cðku0 k þ kAu0 k þ ku1 k þ ku2 k þ kf k1 Þ with in the last inequality by Definition (2.2). On the other hand, according to Theorem 2.4, we also have
pffiffiffi pffiffiffi 1 pffiffiffi c pffiffiffi pffiffiffi 1 pffiffiffi AuðtÞ 6 ASðtÞu0 þ Að1 SÞðtÞu0 þ Aðt SÞðtÞAu0 þ Að1 SÞðtÞu1 þ Aðt SÞðtÞu1 a a a pffiffiffi 1 pffiffiffi þ Aðt SÞðtÞu2 þ Aðt S f ÞðtÞ
a
6 Cðku0 k þ kAu0 k þ ku1 k þ ku2 k þ kf k1 Þ: Furthermore, note that
1 c 1 1 u0 ðtÞ ¼ S0 ðtÞu0 þ SðtÞu0 þ ð1 SÞðtÞAu0 þ SðtÞu1 þ ð1 SÞðtÞu1 þ ð1 SÞðtÞu2 þ ð1 S f ÞðtÞ:
a
a
a
a
pffiffiffi pffiffiffi Since u0 2 D(A) and u1 2 Dð AÞ; Theorem 2.4 implies that u 2 C 1 ð½0; 1Þ; Dð AÞ \ Cð½0; 1Þ; HÞ: The second derivative of u is
1 c 1 1 u00 ðtÞ ¼ S00 ðtÞu0 þ S0 ðtÞu0 þ SðtÞAu0 þ S0 ðtÞu1 þ SðtÞu1 þ SðtÞu2 þ ðS f ÞðtÞ:
a
a
a
a
Combining (2.21), (2.20), Corollary 2.5 (c), and Theorem 2.4, we have u 2 C 2 ð½0; 1Þ; HÞ. Applying Theorem 2.4, we obtain estimate (3.27). h The main result of this section, Theorem 3.12, deals with the regularity of strong solutions of the non-homogeneous problem (3.25). pffiffiffi pffiffiffi 3=2 Theorem 3.12. Let f 2 L1 ð½0; 1Þ; pffiffiffiDð AÞÞ; u0 2 DðA Þ; u1 2 DðAÞ, and u2 2 Dð AÞ: Then the mild solution of problem (3.25) is 2 strong, belongs to C ð½0; 1Þ; Dð AÞÞ, and for any t P 0,the estimate
pffiffiffi ku000 ðtÞk þ ku00 ðtÞk þ ku0 ðtÞk þ kuðtÞk þ Au00 ðtÞ þ kAu0 ðtÞk þ kAuðtÞk pffiffiffi 6 Cðku0 k þ ku1 k þ ku2 k þ kA3=2 u0 k þ kAu1 k þ k Au2 k þ kf k1 Þ
holds, where C is a positive constant depending on a, band c. Proof. We consider the functions
v ðtÞ ¼
1
a
ðt S f ÞðtÞ;
wðtÞ ¼ SðtÞu0 ;
1 pðtÞ ¼ ð1 SÞðtÞu1 þ ðt SÞðtÞu1
a
rðtÞ ¼
1
a
c a
ð1 SÞðtÞu0 ; þ ðt SÞðtÞAu0 ;
and qðtÞ ¼ ðt SÞðtÞu2 :
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We will show that each one of the functions v, w, r, p and q is a strong solution of the problem (3.25) and that the corresponding estimates holds. The claim pffiffiffiof Theorem will follows by the superposition principle. Let u0 = u1 = u2 = 0. Since f ðtÞ 2 Dð AÞ for all t P 0, using Theorem 2.4 (iv) we obtain that v is three times differentiable, moreover, it follows from the closedness of A and Theorem 2.4 (iii) that v and v0 belong to Cð½0; þ1Þ; DðAÞÞ. We differentiate v(t), and obtain
1 b c av 000 ðtÞ þ v 00 ðtÞ þ bAv ðtÞ þ cAv 0 ðtÞ ¼ f ðtÞ þ ðS0 f ÞðtÞ þ ðS f ÞðtÞ þ Aðt S f ÞðtÞ þ Að1 S f ÞðtÞ a a a Z Z Z
t 1 t 1 t S0 ðt sÞf ðsÞds þ Sðt sÞf ðsÞds þ bAt Sðt sÞf ðsÞds ¼ f ðtÞ þ a 0 a 0 0 Z t 1 þ cAð1 SÞðt sÞf ðsÞds a 0 Z 1 t 0 ¼ f ðtÞ þ ½aS ðt sÞ þ Sðt sÞ þ bAðt SÞðt sÞ þ cAð1 SÞðt sÞf ðsÞds:
a
0
pffiffiffi Since f ðtÞ 2 Dð AÞ; Corollary 2.5 (c) implies that [aS0 (t s) + S(t s) + bA(t ⁄ S)(t s) + cA(1 ⁄ S)(t s)]f(s) = 0, and hence problem (3.25) holds. Now we show that the estimative of Theorem holds for v. According to Corollary 2.5(a) and (b), we have kv ðtÞk 6 Ckf kL1 ; kv 0 ðtÞk 6 Ckf kL1 ; and kv 00 ðtÞk 6 Ckf kL1 : It follows from Theorem 2.4 (iv) that kv 000 ðtÞk 6 Ckf kL1 : Applying pffiffiffi Theorem 2.4 (iii) we obtain kAv ðtÞk þ kAv 0 ðtÞk 6 Ckf kL1 : Again, according to Theorem 2.4 (ii), we have v 00 ðtÞ 2 Dð AÞ and pffiffiffi 00 k Av ðtÞk 6 Ckf kL1 : Hence
pffiffiffi kv 000 ðtÞk þ kv 00 ðtÞk þ kv 0 ðtÞk þ kv ðtÞk þ kAv ðtÞj þ kAv 0 ðtÞk þ k Av 00 ðtÞk 6 Ckf kL1
and thus the claim is proved for v. Let u1 = u2 = 0 and f = 0. By hypothesis, u0 2 D(A3/2), and hence w(t) 2 D(A). It follows fromp(2.8) that w0 ðtÞ ¼ a1 SðtÞu0 ab ðt SÞðtÞAu0 ac ð1 SÞðtÞAu0 . ffiffiffi Since Au0 2 Dð AÞ; Theorem 2.4 (iii), (v) implies that w0ðÞ 2 Cð½0; þ1Þ; DðAÞÞ. Using (2.9), it now follows that
1 b c w00 ðtÞ ¼ w0 ðtÞ ð1 SÞðtÞAu0 SðtÞAu0 :
a
a
a
pffiffiffi Due to Theorem 2.4 (i), (ii) and (vi), we have w00 ðÞ 2 Cð½0; þ1Þ; Dð AÞÞ: Note that
1 b c w000 ðtÞ ¼ w00 ðtÞ SðtÞAu0 S0 ðtÞAu0 :
a
a
a
It now follows from Theorem 2.4 (iv) and Corollary 2.5 (b) that w000 ðÞ 2 Cð½0; þ1Þ; HÞ. Finally, is easy to verify that w(t) satisfies problem (3.25). Now we show that the estimate holds for w. From Corollary 2.5 (b) and Theorem 2.4 (v), it follows that
pffiffiffi kwðtÞk 6 Ck Au0 k and kAwðtÞk 6 CkAu0 k: pffiffiffi By Corollary 2.5(a) and (b) we obtain kw0 ðtÞk 6 CðkAu0 k þ k Au0 kÞ: Using Theorem 2.4 (iii), (v) we have kAw0 (t)k 6 C(kA3/ 2 u0k + kAu0k). Considering again Theorem 2.4 and Corollary 2.5 we obtain
pffiffiffi pffiffiffi kw00 ðtÞk þ k Aw00 ðtÞk 6 Cðku0 k þ k Au0 k þ kA3=2 u0 k þ kAu0 kÞ and
pffiffiffi kw000 ðtÞk 6 Cðku0 k þ k Au0 k þ kA3=2 u0 k þ kAu0 kÞ: Let u1 = u2 = 0 and f = 0. Thanks to Theorem 2.4 (iii), (v), we obtain that r() pffiffiffiand their first derivative belong to Cð½0; þ1Þ; DðAÞÞ. Moreover, by (vi), their second derivative belongs to Cð½0; þ1Þ; Dð AÞÞ: The third derivative of r is given by
r000 ðtÞ ¼
1
a
c a
w00 ðtÞ þ S0 ðtÞAu0 ;
pffiffiffi and belongs to Cð½0; þ1Þ; HÞ because w00 2 Cð½0; þ1Þ; Dð AÞÞ and S0 ðÞAu0 2 Cð½0; þ1Þ; HÞ, by Theorem 2.4 (iv). It follows from Corollary 2.5(a) and (b) that
pffiffiffi krðtÞk þ kr 0 ðtÞk 6 Cðku0 k þ k Au0 k þ kAu0 kÞ: By Theorem 2.4 and Corollary 2.5,
pffiffiffi kr 00 ðtÞk þ kr 000 ðtÞk 6 CðkA3=2 u0 k þ k Au0 k þ kAu0 k þ ku0 kÞ and
pffiffiffi kAr 0 ðtÞk þ kArðtÞk 6 CðkA3=2 u0 k þ k Au0 k þ kAu0 kÞ:
C. Fernández et al. / Applied Mathematics and Computation 217 (2011) 8522–8533
8531
pffiffiffi Since Au0 2 D(A1/2), by Theorem 2.4 (ii), the second derivative of r(t) belongs to Dð AÞ and
pffiffiffi k Ar 00 ðtÞk 6 CðkA3=2 u0 k þ kAu0 kÞ: Using (2.9) and Corollary 2.5 (c), we finally obtain that r(t) satisfies problem (3.25). Let u0 = u2 = 0 and f = 0. Similarly as in the above cases, applying Theorem 2.4 and Corollary 2.5, we obtain that p(t) is the strong solution of problem (3.25) and
pffiffiffi pffiffiffi kp000 ðtÞk þ kp00 ðtÞk þ kp0 ðtÞk þ kpðtÞk þ k Ap00 ðtÞk þ kAp0 ðtÞk þ kApðtÞk 6 Cðku1 k þ k Au1 k þ kAu1 kÞ: pffiffiffi Analogously for q(t), since u2 2 Dð AÞ; u0 ¼ u1 ¼ 0 and f = 0, using repeatedly Theorem 2.4 and Corollary 2.5, we obtain that q(t) is strong solution of problem (3.25) and
pffiffiffi pffiffiffi kq000 ðtÞk þ kq00 ðtÞk þ kq0 ðtÞk þ kqðtÞk þ k Aq00 ðtÞk þ kAq0 ðtÞk þ kAqðtÞk 6 Cðku2 k þ k Au2 kÞ: Finally, concerning the regularity of u00 , note that
1 c 1 1 u00 ðtÞ ¼ S00 ðtÞu0 þ S0 ðtÞu0 þ SðtÞAu0 þ S0 ðtÞu1 þ SðtÞu1 þ SðtÞu2 þ ðS f ÞðtÞ;
a
a
a
a
pffiffiffi for t 2 [0, 1), we have from Theorem 2.4 and Corollary 2.5 that Au00 ðtÞ is continuous.
h
4. Applications Physical motivation for studying the problem (1.1) arises from the problem of vibrations of an elastic structure with internal material damping, however small, always present in real materials, see [7]. Torsional and longitudinal vibrations of a linear uniform structure lead to problem (1.1) in the case A = D and problem (1.1) is its generalization, which includes several other cases of interest. 1. We consider a linear model of vibrations governed by the standard linear model of viscoelasticity. In this model, a linear spring is connected in series with a combination of another linear spring and a dashpot in paralell (see [11] and the references therein). The stress r and the strain e are then related by the differential equation
r þ krt ¼ Eðe þ let Þ; where the constants k, l are very small satisfying 0 < k < l and E is the Young’s modulus of the structure. Consequently, the vibrations of flexible structures with external forces are governed by the linear differential problem
8 kuttt ðt; xÞ þ utt ðt; xÞ ¼ c2 ðDuðt; xÞ þ lDut ðt; xÞÞ þ f ðt; xÞ in 0; T X; > > > > > > < uðt; xÞ ¼ 0 on 0; T @ X; uð0; xÞ ¼ u0 ðxÞ in X; > > > > u in X; t ð0; xÞ ¼ u1 ðxÞ > > : utt ð0; xÞ ¼ u2 ðxÞ in X
ð4:28Þ
in a smooth bounded region X Rn . In [3], the authors study the problem (4.28) in the case f = 0. Now, we assume that f 2 L1 ð½0; 1Þ; L2 ðXÞÞ. In what follows, we will apply the previous results to the operator A D, with Dirichlet boundary conditions on oX. The properties of the corresponding regular resolvent {S(t)}tP0 and the fact that the solutions have bounded energy will allow us to obtain compactness of the trajectories for the problem (4.28). Note that the solution of the Cauchy problem (4.28) is given by the formula (3.26), that is, u = v + w, where v is the solution of the non-homogeneous problem with zero data, and w is the solution of the homogeneous equation
8 kw ðt; xÞ þ wtt ðt; xÞ ¼ c2 ðDwðt; xÞ þ lDwt ðt; xÞÞ in 0; T X; > > > ttt > > > < wðt; xÞ ¼ 0 on 0; T @ X; wð0; xÞ ¼ u0 ðxÞ in X; > > > > wt ð0; xÞ ¼ u1 ðxÞ in X; > > : wtt ð0; xÞ ¼ u2 ðxÞ in X: Moreover, with the notation of previous sections, we have
1 c 1 wðtÞ ¼ SðtÞu0 þ ð1 SÞðtÞu0 þ ðt SÞðtÞAu0 þ ð1 SÞðtÞu1 þ ðt SÞðtÞu1 þ ðt SÞðtÞu2
a
a
a
and
v ðtÞ ¼
1
a
ðt S f ÞðtÞ:
The following is our main result of L2 well-posedness of problem (4.28).
ð4:29Þ
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Theorem 4.13. Let X be a bounded domain with smooth boundary oX in Rn and 0 < k < l. Suppose ðu0 ; u1 ; u2 Þ 2 ðH2 ðXÞ \ H10 ðXÞÞ H10 ðXÞ L2 ðXÞ. Then, for every f 2 L1 ð½0; 1Þ; L2 ðXÞÞ, the initial value problem (4.28) has a unique mild solution. This solution satisfies
ku00 ðtÞk þ ku0 ðtÞk þ kru0 ðtÞk þ kuðtÞk þ kruðtÞk 6 Cðku0 k þ ku1 k þ ku2 k þ kDu0 k þ kru1 k þ kf k1 Þ:
ð4:30Þ
Proof. Recall that by Poincare’s inequality, A = D is positive and elliptic. Since 0 < k < l, the numbers a = k, b = c2 and c = c2l 2 1 satisfy pffiffiffi the1 condition ab < c. Moreover, with Dirichlet boundary conditions, we have DðAÞ ¼ H ðXÞ \ H0 ðXÞ and Dð AÞ ¼ H0 ðXÞ: The result follows directly from Theorem 3.11. h Hence, the homogeneous Eq. (4.29) is well-posed (i.e. admits a regular resolvent) if the initial data (u0, u1, u2) belongs to X :¼ ðH2 ðXÞ \ H10 ðXÞÞ H10 ðXÞ L2 ðXÞ. In this case, the energy functional takes the explicit form
EðtÞ ¼
1 2
Z
X
ðu2t þ c2 jruj2 þ kc2 ðl kÞjrxt j2 Þdx:
By Theorem 2.7,
EðtÞ 6 Eð0Þ: Moreover,
E0 ðtÞ ¼ c2 ðl kÞ
Z
jrxt j2 dx;
X
where the integral shows that the energy of the system is dissipating throughout the domain due to the presence of the internal damping of the system. We next consider the dynamics of the system defined by T0(t)(u0, u1, u2) = (w, wt, wtt). Corollary 4.14. Given q > 0, there exists R > 0 such that if the initial data (u0, u1, u2) satisfy kðu0 ; u1 ; u2 ÞkX 6 q; then the following a priori estimate holds
kwk þ kwt k þ kwtt k þ krwt k 6 q: Let P : X ! X be the projection given by P(u0, u1, u2) = (0, u1, u2). Our next result shows that the projected dynamics T0(t)P is absorbing in H10 ðXÞ L2 ðXÞ. Corollary 4.15. Given q > 0, there exists R > 0 such that if the initial data (u0, u1, u2) satisfy kðu0 ; u1 ; u2 ÞkX 6 q; then kT 0 ðtÞPðu0 ; u1 ; u2 ÞkX 6 R; for all t 2 R: 2. Considering the border case k = 0 in the problem (4.28), we obtain
8 utt ðt; xÞ ¼ Duðt; xÞ þ lDut ðt; xÞÞ þ f ðt; xÞin0; T X; > > > < uðt; xÞ ¼ 0 on 0; T @ X; ; > > uð0; xÞ ¼ u0 ðxÞ in X; > : ut ð0; xÞ ¼ u1 ðxÞ in X
ð4:31Þ
the so-called Voigt model of viscoelasticity. This problem is treated for f = 0 in [10]. The energy functional is given by
EðtÞ ¼
1 2
Z
ðu02 þ jruj2 Þdx:
X
Applying Corollary 2.8 with a = 0, b = 1 and c = l, we obtain
E0 ðtÞ ¼ l
Z
jru0 j2 dx:
X
Again, the energy of the system is dissipating due to the presence of the internal damping of the system. According to Theorem 2.7, we have E(t) 6 E(0) for all t P 0. 3. Consider the Klein–Gordon equation
utt ðt; xÞ Duðt; xÞ þ m2 ðxÞuðt; xÞ ¼ 0 with initial conditions in a smooth bounded region X Rn : Suppose that m2(x) P q > 0 and define A = D + m2I in the Hilbert space L2(X). Note that A is self-adjoint, positive, and elliptic. Applying Corollary 2.8 with a = 0, b = 1 and c = l, we obtain conservation of energy, or in other words, t ? E(t) is constant, recovering a well known result (see for example [6, p.660]). Introducing the term auttt, that is the damped Klein–Gordon equation
auttt ðt; xÞ þ utt ðt; xÞ Duðt; xÞ þ m2 ðxÞuðt; xÞ ¼ 0; we conclude that E(t) is decreasing and, hence, the problem changes from conservative to dissipative.
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Acknowledgement The authors are supported by Laboratorio de Analisis Estocástico, Proyecto Anillo ACT-13. The first author is partially financed by Project FONDECYT 1100304. The second author is partially supported by Grant FONDECYT number 1100485. The third author is also partially financed by Proyecto Fondecyt de Iniciación 11075046. References [1] W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, Vol. 96, Birkhäuser, Basel, 2001. [2] S.K. Bose, G.C. Gorain, Exact controllability and boundary stabilization of flexural vibrations of an internally damped flexible space structure, Appl. Math. Comput. 126 (2–3) (2002) 341–360. [3] S.K. Bose, G.C. Gorain, Stability of the boundary stabilised damped wave equation y00 + ky000 = c2(Dy + lDy0 ) in a bounded domain in Rn , Indian J. Math. 40 (1) (1998) 1–15. [4] S.K. Bose, G.C. Gorain, Exact controllability and boundary stabilization of torsional vibrations of an internally damped flexible space structure, J. Optim. Theory Appl. 99 (2) (1998) 423–442. [5] P. Cannarsa, D. Sforza, Semilinear integrodifferential equations of hyperbolic type: existence in the large, Mediterr. J. Math. 1 (2004) 151–174. [6] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, Rhode Island, 2010. [7] R. Christensen, Theory of Viscoelasticity, Academic Pres, New York, 1971. [8] H.O. Fattorini, Second Order Differential Equations in Banach Spaces, North Holland, Amsterdam, 1985. [9] C. Fernandez, C. Lizama, V. Poblete, Maximal regularity for flexible structural systems in Lebesgue spaces, Mathematical Problems in Engineering, Vol. 2010, Article ID196956, 2010, p. 15. [10] G. Gorain, Exponential energy decay estimate for the solutions of internally damped wave equation in a bounded domain, J. Math. Anal. Appl. 216 (1997) 510–520. [11] G. Gorain, Stabilization for the vibrations modeled by the ‘‘standard linear model’’ of viscoelasticity, Proc. Indian Acad. Sci. (Math. Sci.) 120 (4) (2010) 495–506. [13] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. 87, Birkhäuser Verlag, 1993. [14] T.J. Xiao, J. Liang, The Cauchy Problem for Higher-order Abstract Differential Equations, Lecture Notes in Mathematics, Vol. 1701, Springer-Verlag, Berlin, 1998.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Regularity of solutions for a third order differential equation in Hilbert spaces Claudio Fernández a, Carlos Lizama b,⇑, Verónica Poblete c a
Pontificia Universidad Católica de Chile, Facultad de Matemáticas, Avenida Vicuña Mackenna 4860, Santiago, Chile Universidad de Santiago de Chile, Departamento de Matemática, Facultad de Ciencias, Casilla 307-Correo 2, Santiago, Chile c Universidad de Chile, Departamento de Matemática, Facultad de Ciencias, Las Palmeras 3425 Ñuñoa, Santiago, Chile b
a r t i c l e
i n f o
Keywords: Third order Cauchy problems Regularity of solutions Mild and strong solutions Energy functional
a b s t r a c t We study regularity of mild and strong solutions of an abstract mathematical model of a flexible space structure under appropriate initial conditions. We apply our results showing qualitative properties of the trajectories in the case of the negative Laplacian operator. Ó 2011 Elsevier Inc. All rights reserved.
1. Introduction Let H be a Hilbert space and A an unbounded, self-adjoint, elliptic operator in H. Consider the problem
8 000 au ðtÞ þ u00 ðtÞ þ bAuðtÞ þ cAu0 ðtÞ ¼ f ðtÞ; > > > < uð0Þ ¼ u ; 0 0 > u ð0Þ ¼ u > 1; > : 00 u ð0Þ ¼ u2 :
t P 0; ð1:1Þ
Here f : Rþ ! H; a; b; c 2 Rþ ; and u0, u1, u2 2 H are given data. In the case A is the negative Laplacian in H = L2(X) equipped with suitable boundary conditions, problem (1.1) was first derived by Bose and Gorain [4] to model flexible structural systems possessing internal damping. For further information and a complete description of the model, see [2,3]. Maximal regularity properties for problem (1.1) in Lebesgue spaces has been recently studied in [9]. In this paper, we will study existence, uniqueness and regularity properties for strong and mild solutions of problem (1.1). A popular approach is to reduce the third order problem to a first order system in a suitable phase space and use operator semigroup theory. The disadvantage of this approach is that finding an ideal space is generally difficult and the structure of the phase space (if any) may be complicated, so that it is inconvenient for computational applications; also, it is well known that some inherent properties of problems of order greater than or equal to two cannot always be reflected precisely from the corresponding first order systems (see for example [8,14]). Our idea is to establish a more inclusive theory about third order problems and therefore we give a direct treatment of problem (1.1). It is also important to note that abstract differential equations of order greater than two are, in general, ill-posed (see [14]). Consequently, in our case we necessarily have to assume ab < c. pffiffiffi In Section 3, it is shown that under these assumption, for any u0 2 DðAÞ; u1 2 Dð AÞ; u2 2 H and f 2 L1 ð0; þ1; HÞ, the solution u satisfies
⇑ Corresponding author. E-mail addresses: [email protected] (C. Fernández), [email protected] (C. Lizama), [email protected] (V. Poblete). 0096-3003/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2011.03.056
C. Fernández et al. / Applied Mathematics and Computation 217 (2011) 8522–8533
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pffiffiffi u 2 C 1 ð½0; 1Þ; Dð AÞÞ \ Cð½0; 1Þ; HÞ: pffiffiffi pffiffiffi For smoother data, u0 2 DðA3=2 Þ; u1 2 DðAÞ; u2 2 Dð AÞ; and f 2 L1 ð½0; þ1Þ; Dð AÞÞ; we show that the solution u is more regular, that is,
pffiffiffi u 2 C 3 ð½0; 1Þ; HÞ \ C 2 ð½0; 1Þ; Dð AÞÞ \ C 1 ð½0; 1Þ; DðAÞÞ \ Cð½0; 1Þ; DðAÞÞ: The paper is organized as follows. Section 2 is mainly devoted to the construction of a resolvent operator family for problem (1.1) with f 0. In passing, we define an energy functional naturally associated to third order problems, by means of the expression
EðtÞ ¼
i pffiffiffi pffiffiffi pffiffiffi 1h kaS00 ðtÞx þ S0 ðtÞxk2 þ bka AS0 ðtÞx þ ASðtÞxk2 þ aðc abÞk AS0 ðtÞxk2 : 2
In Section 3, we state our main results on regularity of mild and strong solutions of problem (1.1), as well as important a priori-estimates, necessary for applications in nonlinear problems. In Section 4, we give some practical examples of application of our results, including the Voigt model of viscoelasticity and the Klein–Gordon equation. 2. The homogeneous problem Let H be a real Hilbert space with scalar product h, i. In this section, we consider the homogeneous equation
au000 ðtÞ þ u00 ðtÞ þ bAuðtÞ þ cAu0 ðtÞ ¼ 0; for t P 0;
ð2:1Þ
where A is an unbounded, self-adjoint linear operator with domain D(A) dense in H. The domain D(A) will be regarded as a Hilbert space endowed with the graph-norm. We further assume that A is elliptic, or in other words there exists a constant M > 0 such that
hAx; xi P Mkxk2 ;
for all x 2 DðAÞ:
ð2:2Þ
pffiffiffi Since D(A) is dense in Dð AÞ, we easily obtain the abstract version of Poincare’s inequality with Dirichlet boundary conditions
pffiffiffi k Axk P Mkxk;
pffiffiffi for all x 2 Dð AÞ:
ð2:3Þ
In this section, we will denote the solution of Eq. (2.1) by means of a strongly continuous family of bounded, linear operators, that we denote by {S(t)}tP0. We define the abstract energy functional associated to Eq. (2.1) and get a priori estimates. Recall that a family fCðtÞgt2R of continuous linear operators on H is called a strongly continuous operator cosine family if C(0) = I, 2C(t)C(s) = C(t + s) + C(t s) and limt?0 C(t)x = x for all x 2 H. For further information on cosine families, see [1,8], and the references given there. The following observation, stated as a Lemma, will be the key to associate to Eq. (2.1) a family of operators which will then provide the solution of the non-homogeneous problem (1.1) by means of a kind of variation of parameters formula. Lemma 2.1. If A is a self-adjoint, strictly positive operator, then – A generates a strongly continuous operator cosine family. pffiffiffi Proof. By Stone’s theorem, there exists a strongly continuous group of unitary operators fUðtÞgt2R with generator i A: Define for t 2 R and x 2 H
CðtÞx ¼
UðtÞx þ UðtÞx : 2
Then we obtain a strongly continuous operator cosine family with generator C00 (0)x = Ax, x 2 D(A). h Now we define an appropriate concept of resolvent family for the homogeneous problem. Definition 2.2. A family {S(t)}tP0 of bounded linear operators in H is called a regular resolvent for Eq. (2.1) if the following conditions are satisfied: (S1) (S2) (S3) (S4)
S(t) is strongly continuous on [0, 1) and S(0) = I. For any x 2 D(A) and t P 0, we have that S(t)x 2 D(A) and AS(t)x = S(t)Ax. For any x 2 D(A), we have that S()x is twice continuously differentiable in H on [0, 1). For any x 2 D(A2) and t P 0, we have that S0 (t)x 2 D(A), AS0 ()x is continuous and S()x is three times continuously differentiable in H on [0, 1). Moreover, the equation
aS000 ðtÞx þ S00 ðtÞx þ cAS0 ðtÞx þ bASðtÞx ¼ 0 holds.
ð2:4Þ
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In order to establish existence of a regular resolvent for Eq. (2.1), we will use the theory of integral evolutionary equations [13]. For this, note that by integrating three times Eq. (2.1), we find that it is equivalent to the perturbed integral Volterra equation
uðtÞ ¼ gðtÞ þ ða ðAÞuÞðtÞ þ ðb uÞðtÞ;
ð2:5Þ
where
aðtÞ ¼
b 2 c t þ t a 2a
gðtÞ ¼
t c 2 t2 t2 u0 ð0Þ þ u00 ð0Þ: 1þ þ t A uð0Þ þ t þ a 2a 2a 2
and bðtÞ ¼
1
ð2:6Þ
a
and
ð2:7Þ
Our main result of this section is the following. Theorem 2.3. Eq. (2.1) admits a regular resolvent {S(t)}tP0 which satisfies the following:
aS0 ðtÞx þ SðtÞx þ cAð1 SÞðtÞx þ bAðt SÞðtÞx ¼ 0
ð2:8Þ
aS00 ðtÞx þ S0 ðtÞx þ cASðtÞx þ bAð1 SÞðtÞx ¼ 0 for all x 2 DðAÞ:
ð2:9Þ
and
Proof. It follows from Lemma 2.1 that A is the generator of a cosine family, say C(t). Hence the equation
v ðtÞ ¼ gðtÞ þ
c a
Z
t
ðt sÞðAÞv ðsÞds
ð2:10Þ
0
is well-posed and (t ⁄ C)(t) 2 D(A) for all t P 0 (see [13, Definition 1.3 and Proposition 1.1]). Consider Eq. (2.5) where a(t) is given in (2.6). Define cðtÞ ¼ ac t, kðtÞ ¼ bc : Note that Eq. (2.5) can be rewritten as
uðtÞ ¼ gðtÞ þ
Z
t
aðt sÞðAÞuðsÞds þ 0
¼ gðtÞ þ
b 2a
Z
t
bðt sÞuðsÞds
0
Z
t
ðt sÞ2 ðAÞuðsÞds þ
0
c a
Z
t
ðt sÞðAÞuðsÞds þ ðb uÞðtÞ
0
¼ gðtÞ þ ðc k ðAÞuÞðtÞ þ ðc ðAÞuÞðtÞ þ ðb uÞðtÞ: Hence
uðtÞ ¼ gðtÞ þ ½ðc þ c kÞ ðAÞuðtÞ þ ðb uÞðtÞ:
ð2:11Þ
Since it is a perturbed form of (2.10), it follows from [13, Theorem 1.2] that Eq. (2.5) admits a resolvent family. Furthermore, c u 2 Cð½0; þ1Þ; DðAÞÞ if and only if ðc þ c kÞ u 2 Cð½0; þ1Þ; DðAÞÞ (see the remark after formula (2.16) of [13, p.39]). Then there exists a family {S(t)}tP0 of bounded linear operators in H such that (i) S(0) = I and S(t) is strongly continuous on [0, 1); (ii) S(t) commutes with A, which means that S(t)D(A) D(A) and AS(t)x = S(t)Ax, for all x 2 D(A) and t P 0; (iii) for t P 0 and x 2 D(A), the resolvent equation
SðtÞx ¼ x þ ða ðAÞSÞðtÞx þ ðb SÞðtÞx
ð2:12Þ
holds. By [13, Proposition 1.1], for any x 2 H, the function (a ⁄ S)()x belongs to Cð½0; 1Þ; DðAÞÞ; Aða SÞ is strongly continuous in H and
SðtÞx ¼ x Aða SÞðtÞx þ ðb SÞðtÞx;
for x 2 H:
ð2:13Þ
Note that, for all x 2 D(A), from (2.12) and the fact that S(t) commutes with A, we conclude that (b ⁄ S)(t) x = S(t)x x + (a ⁄ S)(t)Ax and hence (1 ⁄ S)(t) is leaving D(A) invariant and A(1 ⁄ S)() is strongly continuous. Moreover, from the proof of [13, Theorem 1.2] we deduce that (t ⁄ S)(t) maps H into D(A) and A(t ⁄ S)() is strongly continuous. By (i) and (ii), it is clear that (S1) and (S2) are satisfied. For x 2 D(A), differentiating (2.12) we obtain (2.8). Again, differentiating (2.12) we obtain (2.9), for all x 2 D(A). We conclude that (S3) holds.
C. Fernández et al. / Applied Mathematics and Computation 217 (2011) 8522–8533
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Now, for x 2 D(A2), from (2.8) it follows that S0 (t)x 2 D(A) and AS0 () is strongly continuous. Finally, differentiating (2.9) we obtain that the resolvent equation in (S4) holds. h The next Theorem gives a number of bounds and inequalities that are the key pieces for the main results obtained later in this paper. Theorem 2.4. Suppose ab < c. Then the regular resolvent {S(t)}tP0 for Eq. (2.1) satisfies the following properties: (i) For any x 2 H, the functions
t ! ð1 SÞðtÞx ¼
Z
t
SðsÞxds and t ! ðt SÞðtÞx ¼
pffiffiffi 0 belong to Cð½0; 1Þ; Dð AÞÞ: In addition, for t P 0, we have
Z
t
ðt sÞSðsÞxds;
0
pffiffiffi k Að1 SÞðtÞxk 6 Ckxk
ð2:14Þ
pffiffiffi k Aðt SÞðtÞxk 6 Ckxk:
ð2:15Þ
and pffiffiffi pffiffiffi (ii) For any x 2 Dð AÞ; SðÞx belongs to Cð½0; 1Þ; Dð AÞÞ: Moreover, for t P 0, we have
pffiffiffi pffiffiffi k ASðtÞxk 6 Ck Axk:
ð2:16Þ
pffiffiffi Rt Rt (iii) For any x 2 Dð AÞ; the functions t ! ð1 SÞðtÞx ¼ 0 SðsÞxds and t ! ðt SÞðtÞx ¼ 0 ðt sÞSðsÞxds belong to Cð½0; 1Þ; DðAÞÞ. In addition, for t P 0, we have
pffiffiffi kAð1 SÞðtÞxk 6 Ck Axk
ð2:17Þ
pffiffiffi kAðt SÞðtÞxk 6 Ck Axk:
ð2:18Þ
and pffiffiffi (iv) For any x 2 Dð AÞ; SðÞx is continuously differentiable in H on [0, 1). Moreover, for t P 0, we have
pffiffiffi kS0 ðtÞxk 6 Cðkxk þ k AxkÞ: (v) For any x 2 D(A), S()x belongs to Cð½0; 1Þ; DðAÞÞ. Moreover, for t P 0, we have
kASðtÞxk 6 CkAxk: pffiffiffi (vi) For any x 2 D(A), S0 ()x belongs to Cð½0; 1Þ; Dð AÞÞ: Moreover, for t P 0, we have
pffiffiffi k AS0 ðtÞxk 6 CkAxk: Here C denotes a suitable positive constant which depends on a, b and c. Proof. (i) Let x 2 D(A) and ð1 SÞðtÞx ¼
Rt 0
SðsÞxds: Multiplying (2.8) by aS(t)x + (1 ⁄ S)(t)x we obtain
2
a hS ðtÞx; SðtÞxi þ akSðtÞxk þ abhAðt SÞðtÞx; SðtÞxi þ achAð1 SÞðtÞx; SðtÞxi þ ahS0 ðtÞx; ð1 SÞðtÞxi þ hSðtÞx; ð1 SÞðtÞxi þ bhAðt SÞðtÞx; ð1 SÞðtÞxi þ chAð1 SÞðtÞx; ð1 SÞðtÞxi ¼ 0 2
0
or equivalently
2 ac d pffiffiffi 1 d 0 kSðtÞxk2 þ akSðtÞxk2 þ abhAðt SÞðtÞx; SðtÞxi þ kð1 SÞðtÞxk2 Að1 SÞðtÞx þ ahS ðtÞx; ð1 SÞðtÞxi þ 2 dt 2 dt 2 dt 2 b d pffiffiffi þ Aðt SÞðtÞx þ chAð1 SÞðtÞx; ð1 SÞðtÞxi ¼ 0: 2 dt
a2 d
Integrating the last identity on [0, t], we have
Z t Z t 2 a2 ac pffiffiffi kSðsÞxk2 ds þ ab hAðt SÞðsÞx; SðsÞxids þ Að1 SÞðtÞx kSðtÞxk2 kSð0Þxk2 þ a 2 2 2 0 0 Z t Z t 2 1 b pffiffiffi þ a hS0 ðsÞx; ð1 SÞðsÞxids þ kð1 SÞðtÞxk2 þ Aðt SÞðtÞx þ c hAð1 SÞðsÞx; ð1 SÞðsÞxids ¼ 0: 2 2 0 0
a2
Integrating by parts the fourth and sixth terms on the left-hand side of the preceding equality, we obtain
Z t Z t pffiffiffi 2 2 ac pffiffiffi kSð0Þxk2 þ a kSðsÞxk2 ds þ abhAðt SÞðtÞx;ð1 SÞðtÞxi ab Að1 SÞðsÞx ds þ Að1 SÞðtÞx 2 2 2 0 0 Z t Z t pffiffiffi 2 2 1 b pffiffiffi þ ahSðtÞx;ð1 SÞðtÞxi a kSðsÞxk2 ds þ kð1 SÞðtÞxk2 þ Aðt SÞðtÞx þ c Að1 SÞðsÞx ds ¼ 0: 2 2 0 0
a2
kSðtÞxk2
a2
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Taking into account the Hilbert space structure, we can reorganize the above equality as follows:
1 kaSðtÞx þ ð1 SÞðtÞxk2 þ abhAðt SÞðtÞx; ð1 SÞðtÞxi þ ðc abÞ 2 2 a2 b pffiffiffi þ Aðt SÞðtÞx ¼ kSð0Þxk2 ; 2 2
Z t pffiffiffi 2 2 ac pffiffiffi Að1 SÞðsÞx ds þ Að1 SÞðtÞx 2 0
or equivalently
Z t pffiffiffi 2 2 pffiffiffi 1 b pffiffiffi kaSðtÞx þ ð1 SÞðtÞxk2 þ Aðt SÞðtÞx þ a Að1 SÞðtÞx þ ðc abÞ Að1 SÞðsÞx ds 2 2 0 2 a2 ac pffiffiffi a2 b pffiffiffi 2 þ k Að1 SÞðtÞxk2 Að1 SÞðtÞx ¼ kSð0Þxk : 2 2 2 Hence, we obtain the identity
Z t pffiffiffi pffiffiffi pffiffiffi kaSðtÞx þ ð1 SÞðtÞxk2 þ bka Að1 SÞðtÞx þ Aðt SÞðtÞxk2 þ 2ðc abÞ k Að1 SÞðsÞxk2 ds 0 pffiffiffi þ aðc abÞk Að1 SÞðtÞxk2 ¼ a2 kxk2 :
ð2:19Þ
Since ab < c, we have that the sum on the left-hand side of the equality preceding is positive for all x non-zero. Then
pffiffiffi k Að1 SÞðtÞxk 6 Ckxk;
for any t P 0;
x 2 DðAÞ:
Again, for t P 0, from (2.19) we obtain
pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi pffiffiffi Aðt SÞðtÞx ¼ Að1 SÞðtÞx þ Aðt SÞðtÞx Að1 SÞðtÞx 6 Að1 SÞðtÞx þ Aðt SÞðtÞx þ Að1 SÞðtÞx 6 Ckxk; where C is a generic constant which depends on a, b and c. The proof of (i) follows using the density of D(A) in H. (ii) Fix x 2 D(A). As in the proof of (i), multiplying (2.8) by aS0 (t)x + S(t)x and integrating on [0, t],t P 0, we obtain the identity
pffiffiffi 2 pffiffiffi 2 pffiffiffi 2 pffiffiffi kaS0 ðtÞx þ SðtÞxk2 þ ba ASðtÞx þ Að1 SÞðtÞx þ aðc abÞ ASðtÞx ¼ ac Ax :
ð2:20Þ
pffiffiffi pffiffiffi pffiffiffi Since ab < c, we have ASðtÞx 6 C Ax; for any t P 0 and x 2 D(A). Using the density of D(A) in Dð AÞ; the result follows. (iii) Fix x 2 D(A). According to (2.8), we have
cAð1 SÞðtÞx ¼ aS0 ðtÞx þ SðtÞx þ bAðt SÞðtÞx: Then
kAð1 SÞðtÞxk 6
1
c
b kaS0 ðtÞx þ SðtÞxk þ kAðt SÞðtÞxk:
c
pffiffiffi From (2.20) we have kaS ðtÞx þ SðtÞxk 6 Ck Axk and by (2.15) we obtain that 0
pffiffiffi pffiffiffi pffiffiffi kAðt SÞðtÞxk ¼ k Aðt SÞðtÞ Axk 6 Ck Axk: We conclude that
pffiffiffi kAð1 SÞðtÞxk 6 Ck Axk: pffiffiffi pffiffiffi Finally, note that the previous inequality holds for any x 2 Dð AÞ thanks to the density of D(A) in Dð AÞ: This proves (2.17) and (2.18). (iv) Fix x 2 D(A). It follows from (2.8) that, for t P 0, we have
kS0 ðtÞxk 6
1
a
pffiffiffi b c kSðtÞxk þ kAðt SÞðtÞxk þ kAð1 SÞðtÞxk 6 Cðkxk þ k AxkÞ:
a
a
pffiffiffi Using the density of D(A) in Dð AÞ again, the conclusion follows. (v) Let x 2 D(A). It follows from (ii) that
pffiffiffi pffiffiffi kASðtÞxk ¼ k ASðtÞ Axk 6 CkAxk: (vi) Fix x 2 D(A2). Multiplying (2.4) by a S00 (t)x + S0 (t)x, we obtain
C. Fernández et al. / Applied Mathematics and Computation 217 (2011) 8522–8533
8527
ac d pffiffiffi 0 2 1 d 0 kS00 ðtÞxk2 þ abhASðtÞx; S00 ðtÞxi þ kS ðtÞxk2 þ ahS000 ðtÞx; S0 ðtÞxi AS ðtÞx þ 2 dt 2 dt 2 dt 2 b d pffiffiffi 0 0 þ ASðtÞx þ chAS ðtÞx; S ðtÞxi ¼ 0: 2 dt
akS00 ðtÞxk2 þ
a2 d
Integrating on [0, t], we obtain
a
Z
Z t Z t 2 1 ac pffiffiffi kS00 ðtÞxk2 þ ab hASðsÞx; S00 ðsÞxids þ AS0 ðtÞx þ kS0 ðtÞxk2 þ a hS000 ðsÞx; S0 ðsÞxids 2 2 2 0 0 0 Z t 2 b pffiffiffi b pffiffiffi a2 00 1 0 ac pffiffiffi 0 2 0 2 2 0 þ k ASðtÞxk þ c hAS ðsÞx; S ðsÞxids ¼ kS ð0Þxk þ kS ð0Þxk þ AS ð0Þx þ k ASð0Þxk2 : 2 2 2 2 2 0 t
kS00 ðsÞxk2 ds þ
a2
Integrating by parts the third and sixth terms on the left-hand side of the above equality, we obtain
Z t pffiffiffi pffiffiffi 2 pffiffiffi 2 2 pffiffiffi 1 0 kaS00 ðtÞx þ S0 ðtÞxk2 þ ba AS0 ðtÞx þ ASðtÞx þ aðc abÞ AS0 ðtÞx þ ðc abÞ AS ðsÞx ds 2 0 2 pffiffiffi pffiffiffi a2 1 ac pffiffiffi b pffiffiffi ¼ kS00 ð0Þxk2 þ kS0 ð0Þxk2 þ k AS0 ð0Þxk2 þ ASð0Þx þ ahS0 ð0Þx; S00 ð0Þxi þ abh AS0 ð0Þx; ASð0Þxi: 2 2 2 2 From (2.8) and (2.9), we have that S0 ð0Þx ¼ a1 x; S00 ð0Þx ¼ a12 x ac Ax: Replacing in preceding identity, we obtain
Z t pffiffiffi pffiffiffi 2 pffiffiffi 2 2 pffiffiffi 0 kaS00 ðtÞx þ S0 ðtÞxk2 þ ba AS0 ðtÞx þ ASðtÞx þ aðc abÞ AS0 ðtÞx þ 2ðc abÞ AS ðsÞx ds
c ab pffiffiffi 2 2 2 ¼ Ax þ c kAxk : a
0
ð2:21Þ
pffiffiffi Since ab < c, and since A is elliptic, we obtain AS0 ðtÞx 6 CkAxk; for any t P 0 and x 2 D(A2). Using the density of D(A2) in D(A) completes the proof. h Corollary 2.5. Under the assumptions of Theorem 2.4, we have: (a) For x 2 H, there exists C > 0 such that, for t P 0 we have
kðt SÞðtÞxk 6 Ckxk and kð1 SÞðtÞxk 6 Ckxk: (b) For x 2 H, there exists C > 0 such that, for t P 0, we have
kSðtÞxk 6 Ckxk: pffiffiffi (c) For x 2 Dð AÞ; there exists K > 0 such that, for t P 0, we have
pffiffiffi kSðtÞxk 6 Kk Axk: pffiffiffi (d) Equality (2.8) holds for all x 2 Dð AÞ: Proof pffiffiffi (a) For x 2 H, according to Theorem 2.4 (i), we deduce that (t ⁄ S)(t)x and (1 ⁄ S)(t)x belong to Dð AÞ: Assertion (a) now follows from (2.3), (2.14) and (2.15). (b) For x 2 H and t P 0, note that
akSðtÞxk 6 kaSðtÞx þ ð1 SÞðtÞxk þ kð1 SÞðtÞxk: Assertion (b) now pffiffiffifollows from (2.19) and assertion (a). pffiffiffi (c) For x 2 Dð AÞ; according to Theorem 2.4 (ii), we have that S(t)x belongs to Dð AÞ: Assertion (c) now follows from (2.3) and (2.16). pffiffiffi Assertion (d) follows from Theorem 2.4 (iii), (2.8), and the density of D(A) in Dð AÞ:
h
Based on the proof of the previous theorem, we define the following energy functional associated to our problem. We consider a P 0. Definition 2.6. For x 2 D(A) and ab 6 c, we define the abstract energy functional associated to Eq. (2.1) by
EðtÞ ¼
2 2 pffiffiffi pffiffiffi pffiffiffi 1 kaS00 ðtÞx þ S0 ðtÞxk2 þ ba AS0 ðtÞx þ ASðtÞx þ aðc abÞ AS0 ðtÞx ; 2
for t P 0:
ð2:22Þ
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Note that by the previous Theorem E : ½0; 1Þ ! R is a well defined, positive and continuous function. Moreover, note that in the case a = 0, c = 0 and b = 1 we obtain
EðtÞ ¼
2 1 0 1 pffiffiffi kS ðtÞxk2 þ ASðtÞx ; 2 2
which is the usual expression of the energy functional for the second order equation
u00 ðtÞ þ AuðtÞ ¼ f ðtÞ: Compare also with Sforza [5, p. 160]. Moreover, note that (2.22) is the abstract version of the energy functional defined by Bose-Gorain [3] in the case A = D, the Laplacian. We conclude this section with the following result on an important property of the energy functional E(t). Theorem 2.7. Suppose ab 6 c. For any x 2 D(A), E(t) is non-increasing. In particular, E(t) 6 E(0). Proof. Let x 2 D(A). If a > 0, note that
Eð0Þ ¼
pffiffiffi 2 pffiffiffi pffiffiffi 1 c ab pffiffiffi 2 c2 2 kaS00 ð0Þx þ S0 ð0Þxk2 þ ba AS0 ð0Þx þ ASð0Þx þ aðc abÞk AS0 ð0Þxk2 ¼ Ax þ kAxk : 2 2a 2
From (2.21) we have that
EðtÞ þ ðc abÞ
Z t pffiffiffi 2 0 AS ðsÞx ds ¼ Eð0Þ:
ð2:23Þ
0
For ab < c we obtain that hðtÞ ¼ ðc abÞ
2 Rt pffiffiffi 0 AS ðsÞx ds is a positive and increasing function, we conclude that E(t) is non0
increasing and E(t) 6 E(0) for any t P 0. for any x 2 D(A) and t P 0, hence E(t) is non-increasing, in particular E(t) 6 E(0). h From (2.23) we directly obtain the following Corollary. Corollary 2.8. Suppose ab 6 c. For any x 2 D(A), we have
pffiffiffi 2 E0 ðtÞ ¼ ðab cÞ AS0 ðtÞx :
ð2:24Þ
3. Regularity of strong and mild solutions In this section we will assume that A is a self-adjoint linear operator which satisfies the ellipticity condition (2.2). The existence of a regular resolvent {S(t)}tP0 proved in the preceding section allows us to solve the non-homogeneous problem
8 000 au ðtÞ þ u00 ðtÞ þ bAuðtÞ þ cAu0 ðtÞ ¼ f ðtÞ; > > > < uð0Þ ¼ u ; 0 0 > ð0Þ ¼ u u > 1; > : 00 u ð0Þ ¼ u2
t P 0; ð3:25Þ
for which we will now state the notions of mild and strong solutions. Definition 3.9. Let f 2 Cð½0; 1Þ; HÞ. We say that u is a strong solution of problem (3.25) in ½0; 1Þ u 2 C 3 ð½0; 1Þ; HÞ \ C 1 ð½0; 1Þ; DðAÞÞ, and u satisfies problem (3.25). Let f 2 L1 ð0; 1; HÞ; u0 2 DðAÞ and u1, u2 2 H. The mild solution of problem (3.25) in [0, 1) with initial conditions
uð0Þ ¼ u0 ;
u0 ð0Þ ¼ u1 ;
if
u00 ð0Þ ¼ u2
is the function u 2 Cð½0; 1Þ; HÞ defined by
1 c 1 1 uðtÞ ¼ SðtÞu0 þ ð1 SÞðtÞu0 þ ðt SÞðtÞAu0 þ ð1 SÞðtÞu1 þ ðt SÞðtÞu1 þ ðt SÞðtÞu2 þ ðt S f ÞðtÞ:
a
a
a
a
We begin with the following result. Theorem 3.10. If u is a strong solution of problem (3.25), then u is a mild solution of problem (3.25). Proof. A strong solution u of problem (3.25) is also a strong solution of the integral equation
Z t t c 2 t2 t2 1 u1 þ u2 þ ð1 t f ÞðtÞ þ uðtÞ ¼ 1 þ þ t A u0 þ t þ Aðt sÞuðsÞds; a 2a 2a 2 a 0
ð3:26Þ
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where AðtÞ ¼ ð2ba t2 þ ac tÞA a1 I: According to [13, Proposition 6.3], we have
uðtÞ ¼ ¼
d dt Z t
# 1 c ðt sÞ2 ðt sÞ2 1 1 þ ðt sÞ þ ðt sÞ2 A u0 þ ðt sÞu1 þ u1 þ u2 þ ð1 t f Þðt sÞ ds a 2a 2a 2 a 0 1 c 1 1 SðsÞ þ ðt sÞA u0 þ u1 þ ðt sÞu1 þ ðt sÞu2 þ ðt f Þðt sÞ ds þ SðtÞu0
Z
t
"
SðsÞ
a
a
a a c 1 1 ¼ SðtÞu0 þ ð1 SÞðtÞu0 þ ðt SÞðtÞAu0 þ ð1 SÞðtÞu1 þ ðt SÞðtÞu1 þ ðt SÞðtÞu2 þ ðt S f ÞðtÞ: a a a a 0
1
In what follows, we will assume the condition ab < c. Our next regularity result establishes conditions on the initial data under which mild solutions are obtained. We also state a priori bounds that will be useful later. pffiffiffi Theorem 3.11. If f 2 L1 ð½0; 1Þ; HÞ, u0 2 DðAÞ; u1 2 Dð AÞ; and u2 2 H, then the mild solution of problem (3.25) belongs to p ffiffiffi C 1 ð½0; 1Þ; Dð AÞÞ \ C 1 ð½0; 1Þ; HÞ: Moreover, for any t 2 [0, 1), the following estimate holds:
pffiffiffi pffiffiffi pffiffiffi ku00 ðtÞk þ ku0 ðtÞk þ k Au0 ðtÞk þ kuðtÞk þ k AuðtÞk 6 Cðku0 k þ ku1 k þ ku2 k þ kAu0 k þ k Au1 k þ kf k1 Þ:
ð3:27Þ
pffiffiffi Proof. It follows from (i) and (ii) of Theorem 2.4 that u 2 Cð½0; 1Þ; Dð AÞ \ Cð½0; 1Þ; HÞ: According to 2.4 and Remark 2.5 we have
1 c 1 kuðtÞk 6kSðtÞu0 k þ kð1 SÞðtÞu0 k þ kðt SÞðtÞAu0 k þ kð1 SÞðtÞu1 k þ kðt SÞðtÞu1 k þ kðt SÞðtÞu2 k
a
a
a
1
þ kðt S f ÞðtÞk a pffiffiffi 6 Cðk Au0 k þ ku0 k þ kAu0 k þ ku1 k þ ku2 k þ kf k1 Þ 6 Cðku0 k þ kAu0 k þ ku1 k þ ku2 k þ kf k1 Þ with in the last inequality by Definition (2.2). On the other hand, according to Theorem 2.4, we also have
pffiffiffi pffiffiffi 1 pffiffiffi c pffiffiffi pffiffiffi 1 pffiffiffi AuðtÞ 6 ASðtÞu0 þ Að1 SÞðtÞu0 þ Aðt SÞðtÞAu0 þ Að1 SÞðtÞu1 þ Aðt SÞðtÞu1 a a a pffiffiffi 1 pffiffiffi þ Aðt SÞðtÞu2 þ Aðt S f ÞðtÞ
a
6 Cðku0 k þ kAu0 k þ ku1 k þ ku2 k þ kf k1 Þ: Furthermore, note that
1 c 1 1 u0 ðtÞ ¼ S0 ðtÞu0 þ SðtÞu0 þ ð1 SÞðtÞAu0 þ SðtÞu1 þ ð1 SÞðtÞu1 þ ð1 SÞðtÞu2 þ ð1 S f ÞðtÞ:
a
a
a
a
pffiffiffi pffiffiffi Since u0 2 D(A) and u1 2 Dð AÞ; Theorem 2.4 implies that u 2 C 1 ð½0; 1Þ; Dð AÞ \ Cð½0; 1Þ; HÞ: The second derivative of u is
1 c 1 1 u00 ðtÞ ¼ S00 ðtÞu0 þ S0 ðtÞu0 þ SðtÞAu0 þ S0 ðtÞu1 þ SðtÞu1 þ SðtÞu2 þ ðS f ÞðtÞ:
a
a
a
a
Combining (2.21), (2.20), Corollary 2.5 (c), and Theorem 2.4, we have u 2 C 2 ð½0; 1Þ; HÞ. Applying Theorem 2.4, we obtain estimate (3.27). h The main result of this section, Theorem 3.12, deals with the regularity of strong solutions of the non-homogeneous problem (3.25). pffiffiffi pffiffiffi 3=2 Theorem 3.12. Let f 2 L1 ð½0; 1Þ; pffiffiffiDð AÞÞ; u0 2 DðA Þ; u1 2 DðAÞ, and u2 2 Dð AÞ: Then the mild solution of problem (3.25) is 2 strong, belongs to C ð½0; 1Þ; Dð AÞÞ, and for any t P 0,the estimate
pffiffiffi ku000 ðtÞk þ ku00 ðtÞk þ ku0 ðtÞk þ kuðtÞk þ Au00 ðtÞ þ kAu0 ðtÞk þ kAuðtÞk pffiffiffi 6 Cðku0 k þ ku1 k þ ku2 k þ kA3=2 u0 k þ kAu1 k þ k Au2 k þ kf k1 Þ
holds, where C is a positive constant depending on a, band c. Proof. We consider the functions
v ðtÞ ¼
1
a
ðt S f ÞðtÞ;
wðtÞ ¼ SðtÞu0 ;
1 pðtÞ ¼ ð1 SÞðtÞu1 þ ðt SÞðtÞu1
a
rðtÞ ¼
1
a
c a
ð1 SÞðtÞu0 ; þ ðt SÞðtÞAu0 ;
and qðtÞ ¼ ðt SÞðtÞu2 :
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We will show that each one of the functions v, w, r, p and q is a strong solution of the problem (3.25) and that the corresponding estimates holds. The claim pffiffiffiof Theorem will follows by the superposition principle. Let u0 = u1 = u2 = 0. Since f ðtÞ 2 Dð AÞ for all t P 0, using Theorem 2.4 (iv) we obtain that v is three times differentiable, moreover, it follows from the closedness of A and Theorem 2.4 (iii) that v and v0 belong to Cð½0; þ1Þ; DðAÞÞ. We differentiate v(t), and obtain
1 b c av 000 ðtÞ þ v 00 ðtÞ þ bAv ðtÞ þ cAv 0 ðtÞ ¼ f ðtÞ þ ðS0 f ÞðtÞ þ ðS f ÞðtÞ þ Aðt S f ÞðtÞ þ Að1 S f ÞðtÞ a a a Z Z Z
t 1 t 1 t S0 ðt sÞf ðsÞds þ Sðt sÞf ðsÞds þ bAt Sðt sÞf ðsÞds ¼ f ðtÞ þ a 0 a 0 0 Z t 1 þ cAð1 SÞðt sÞf ðsÞds a 0 Z 1 t 0 ¼ f ðtÞ þ ½aS ðt sÞ þ Sðt sÞ þ bAðt SÞðt sÞ þ cAð1 SÞðt sÞf ðsÞds:
a
0
pffiffiffi Since f ðtÞ 2 Dð AÞ; Corollary 2.5 (c) implies that [aS0 (t s) + S(t s) + bA(t ⁄ S)(t s) + cA(1 ⁄ S)(t s)]f(s) = 0, and hence problem (3.25) holds. Now we show that the estimative of Theorem holds for v. According to Corollary 2.5(a) and (b), we have kv ðtÞk 6 Ckf kL1 ; kv 0 ðtÞk 6 Ckf kL1 ; and kv 00 ðtÞk 6 Ckf kL1 : It follows from Theorem 2.4 (iv) that kv 000 ðtÞk 6 Ckf kL1 : Applying pffiffiffi Theorem 2.4 (iii) we obtain kAv ðtÞk þ kAv 0 ðtÞk 6 Ckf kL1 : Again, according to Theorem 2.4 (ii), we have v 00 ðtÞ 2 Dð AÞ and pffiffiffi 00 k Av ðtÞk 6 Ckf kL1 : Hence
pffiffiffi kv 000 ðtÞk þ kv 00 ðtÞk þ kv 0 ðtÞk þ kv ðtÞk þ kAv ðtÞj þ kAv 0 ðtÞk þ k Av 00 ðtÞk 6 Ckf kL1
and thus the claim is proved for v. Let u1 = u2 = 0 and f = 0. By hypothesis, u0 2 D(A3/2), and hence w(t) 2 D(A). It follows fromp(2.8) that w0 ðtÞ ¼ a1 SðtÞu0 ab ðt SÞðtÞAu0 ac ð1 SÞðtÞAu0 . ffiffiffi Since Au0 2 Dð AÞ; Theorem 2.4 (iii), (v) implies that w0ðÞ 2 Cð½0; þ1Þ; DðAÞÞ. Using (2.9), it now follows that
1 b c w00 ðtÞ ¼ w0 ðtÞ ð1 SÞðtÞAu0 SðtÞAu0 :
a
a
a
pffiffiffi Due to Theorem 2.4 (i), (ii) and (vi), we have w00 ðÞ 2 Cð½0; þ1Þ; Dð AÞÞ: Note that
1 b c w000 ðtÞ ¼ w00 ðtÞ SðtÞAu0 S0 ðtÞAu0 :
a
a
a
It now follows from Theorem 2.4 (iv) and Corollary 2.5 (b) that w000 ðÞ 2 Cð½0; þ1Þ; HÞ. Finally, is easy to verify that w(t) satisfies problem (3.25). Now we show that the estimate holds for w. From Corollary 2.5 (b) and Theorem 2.4 (v), it follows that
pffiffiffi kwðtÞk 6 Ck Au0 k and kAwðtÞk 6 CkAu0 k: pffiffiffi By Corollary 2.5(a) and (b) we obtain kw0 ðtÞk 6 CðkAu0 k þ k Au0 kÞ: Using Theorem 2.4 (iii), (v) we have kAw0 (t)k 6 C(kA3/ 2 u0k + kAu0k). Considering again Theorem 2.4 and Corollary 2.5 we obtain
pffiffiffi pffiffiffi kw00 ðtÞk þ k Aw00 ðtÞk 6 Cðku0 k þ k Au0 k þ kA3=2 u0 k þ kAu0 kÞ and
pffiffiffi kw000 ðtÞk 6 Cðku0 k þ k Au0 k þ kA3=2 u0 k þ kAu0 kÞ: Let u1 = u2 = 0 and f = 0. Thanks to Theorem 2.4 (iii), (v), we obtain that r() pffiffiffiand their first derivative belong to Cð½0; þ1Þ; DðAÞÞ. Moreover, by (vi), their second derivative belongs to Cð½0; þ1Þ; Dð AÞÞ: The third derivative of r is given by
r000 ðtÞ ¼
1
a
c a
w00 ðtÞ þ S0 ðtÞAu0 ;
pffiffiffi and belongs to Cð½0; þ1Þ; HÞ because w00 2 Cð½0; þ1Þ; Dð AÞÞ and S0 ðÞAu0 2 Cð½0; þ1Þ; HÞ, by Theorem 2.4 (iv). It follows from Corollary 2.5(a) and (b) that
pffiffiffi krðtÞk þ kr 0 ðtÞk 6 Cðku0 k þ k Au0 k þ kAu0 kÞ: By Theorem 2.4 and Corollary 2.5,
pffiffiffi kr 00 ðtÞk þ kr 000 ðtÞk 6 CðkA3=2 u0 k þ k Au0 k þ kAu0 k þ ku0 kÞ and
pffiffiffi kAr 0 ðtÞk þ kArðtÞk 6 CðkA3=2 u0 k þ k Au0 k þ kAu0 kÞ:
C. Fernández et al. / Applied Mathematics and Computation 217 (2011) 8522–8533
8531
pffiffiffi Since Au0 2 D(A1/2), by Theorem 2.4 (ii), the second derivative of r(t) belongs to Dð AÞ and
pffiffiffi k Ar 00 ðtÞk 6 CðkA3=2 u0 k þ kAu0 kÞ: Using (2.9) and Corollary 2.5 (c), we finally obtain that r(t) satisfies problem (3.25). Let u0 = u2 = 0 and f = 0. Similarly as in the above cases, applying Theorem 2.4 and Corollary 2.5, we obtain that p(t) is the strong solution of problem (3.25) and
pffiffiffi pffiffiffi kp000 ðtÞk þ kp00 ðtÞk þ kp0 ðtÞk þ kpðtÞk þ k Ap00 ðtÞk þ kAp0 ðtÞk þ kApðtÞk 6 Cðku1 k þ k Au1 k þ kAu1 kÞ: pffiffiffi Analogously for q(t), since u2 2 Dð AÞ; u0 ¼ u1 ¼ 0 and f = 0, using repeatedly Theorem 2.4 and Corollary 2.5, we obtain that q(t) is strong solution of problem (3.25) and
pffiffiffi pffiffiffi kq000 ðtÞk þ kq00 ðtÞk þ kq0 ðtÞk þ kqðtÞk þ k Aq00 ðtÞk þ kAq0 ðtÞk þ kAqðtÞk 6 Cðku2 k þ k Au2 kÞ: Finally, concerning the regularity of u00 , note that
1 c 1 1 u00 ðtÞ ¼ S00 ðtÞu0 þ S0 ðtÞu0 þ SðtÞAu0 þ S0 ðtÞu1 þ SðtÞu1 þ SðtÞu2 þ ðS f ÞðtÞ;
a
a
a
a
pffiffiffi for t 2 [0, 1), we have from Theorem 2.4 and Corollary 2.5 that Au00 ðtÞ is continuous.
h
4. Applications Physical motivation for studying the problem (1.1) arises from the problem of vibrations of an elastic structure with internal material damping, however small, always present in real materials, see [7]. Torsional and longitudinal vibrations of a linear uniform structure lead to problem (1.1) in the case A = D and problem (1.1) is its generalization, which includes several other cases of interest. 1. We consider a linear model of vibrations governed by the standard linear model of viscoelasticity. In this model, a linear spring is connected in series with a combination of another linear spring and a dashpot in paralell (see [11] and the references therein). The stress r and the strain e are then related by the differential equation
r þ krt ¼ Eðe þ let Þ; where the constants k, l are very small satisfying 0 < k < l and E is the Young’s modulus of the structure. Consequently, the vibrations of flexible structures with external forces are governed by the linear differential problem
8 kuttt ðt; xÞ þ utt ðt; xÞ ¼ c2 ðDuðt; xÞ þ lDut ðt; xÞÞ þ f ðt; xÞ in 0; T X; > > > > > > < uðt; xÞ ¼ 0 on 0; T @ X; uð0; xÞ ¼ u0 ðxÞ in X; > > > > u in X; t ð0; xÞ ¼ u1 ðxÞ > > : utt ð0; xÞ ¼ u2 ðxÞ in X
ð4:28Þ
in a smooth bounded region X Rn . In [3], the authors study the problem (4.28) in the case f = 0. Now, we assume that f 2 L1 ð½0; 1Þ; L2 ðXÞÞ. In what follows, we will apply the previous results to the operator A D, with Dirichlet boundary conditions on oX. The properties of the corresponding regular resolvent {S(t)}tP0 and the fact that the solutions have bounded energy will allow us to obtain compactness of the trajectories for the problem (4.28). Note that the solution of the Cauchy problem (4.28) is given by the formula (3.26), that is, u = v + w, where v is the solution of the non-homogeneous problem with zero data, and w is the solution of the homogeneous equation
8 kw ðt; xÞ þ wtt ðt; xÞ ¼ c2 ðDwðt; xÞ þ lDwt ðt; xÞÞ in 0; T X; > > > ttt > > > < wðt; xÞ ¼ 0 on 0; T @ X; wð0; xÞ ¼ u0 ðxÞ in X; > > > > wt ð0; xÞ ¼ u1 ðxÞ in X; > > : wtt ð0; xÞ ¼ u2 ðxÞ in X: Moreover, with the notation of previous sections, we have
1 c 1 wðtÞ ¼ SðtÞu0 þ ð1 SÞðtÞu0 þ ðt SÞðtÞAu0 þ ð1 SÞðtÞu1 þ ðt SÞðtÞu1 þ ðt SÞðtÞu2
a
a
a
and
v ðtÞ ¼
1
a
ðt S f ÞðtÞ:
The following is our main result of L2 well-posedness of problem (4.28).
ð4:29Þ
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C. Fernández et al. / Applied Mathematics and Computation 217 (2011) 8522–8533
Theorem 4.13. Let X be a bounded domain with smooth boundary oX in Rn and 0 < k < l. Suppose ðu0 ; u1 ; u2 Þ 2 ðH2 ðXÞ \ H10 ðXÞÞ H10 ðXÞ L2 ðXÞ. Then, for every f 2 L1 ð½0; 1Þ; L2 ðXÞÞ, the initial value problem (4.28) has a unique mild solution. This solution satisfies
ku00 ðtÞk þ ku0 ðtÞk þ kru0 ðtÞk þ kuðtÞk þ kruðtÞk 6 Cðku0 k þ ku1 k þ ku2 k þ kDu0 k þ kru1 k þ kf k1 Þ:
ð4:30Þ
Proof. Recall that by Poincare’s inequality, A = D is positive and elliptic. Since 0 < k < l, the numbers a = k, b = c2 and c = c2l 2 1 satisfy pffiffiffi the1 condition ab < c. Moreover, with Dirichlet boundary conditions, we have DðAÞ ¼ H ðXÞ \ H0 ðXÞ and Dð AÞ ¼ H0 ðXÞ: The result follows directly from Theorem 3.11. h Hence, the homogeneous Eq. (4.29) is well-posed (i.e. admits a regular resolvent) if the initial data (u0, u1, u2) belongs to X :¼ ðH2 ðXÞ \ H10 ðXÞÞ H10 ðXÞ L2 ðXÞ. In this case, the energy functional takes the explicit form
EðtÞ ¼
1 2
Z
X
ðu2t þ c2 jruj2 þ kc2 ðl kÞjrxt j2 Þdx:
By Theorem 2.7,
EðtÞ 6 Eð0Þ: Moreover,
E0 ðtÞ ¼ c2 ðl kÞ
Z
jrxt j2 dx;
X
where the integral shows that the energy of the system is dissipating throughout the domain due to the presence of the internal damping of the system. We next consider the dynamics of the system defined by T0(t)(u0, u1, u2) = (w, wt, wtt). Corollary 4.14. Given q > 0, there exists R > 0 such that if the initial data (u0, u1, u2) satisfy kðu0 ; u1 ; u2 ÞkX 6 q; then the following a priori estimate holds
kwk þ kwt k þ kwtt k þ krwt k 6 q: Let P : X ! X be the projection given by P(u0, u1, u2) = (0, u1, u2). Our next result shows that the projected dynamics T0(t)P is absorbing in H10 ðXÞ L2 ðXÞ. Corollary 4.15. Given q > 0, there exists R > 0 such that if the initial data (u0, u1, u2) satisfy kðu0 ; u1 ; u2 ÞkX 6 q; then kT 0 ðtÞPðu0 ; u1 ; u2 ÞkX 6 R; for all t 2 R: 2. Considering the border case k = 0 in the problem (4.28), we obtain
8 utt ðt; xÞ ¼ Duðt; xÞ þ lDut ðt; xÞÞ þ f ðt; xÞin0; T X; > > > < uðt; xÞ ¼ 0 on 0; T @ X; ; > > uð0; xÞ ¼ u0 ðxÞ in X; > : ut ð0; xÞ ¼ u1 ðxÞ in X
ð4:31Þ
the so-called Voigt model of viscoelasticity. This problem is treated for f = 0 in [10]. The energy functional is given by
EðtÞ ¼
1 2
Z
ðu02 þ jruj2 Þdx:
X
Applying Corollary 2.8 with a = 0, b = 1 and c = l, we obtain
E0 ðtÞ ¼ l
Z
jru0 j2 dx:
X
Again, the energy of the system is dissipating due to the presence of the internal damping of the system. According to Theorem 2.7, we have E(t) 6 E(0) for all t P 0. 3. Consider the Klein–Gordon equation
utt ðt; xÞ Duðt; xÞ þ m2 ðxÞuðt; xÞ ¼ 0 with initial conditions in a smooth bounded region X Rn : Suppose that m2(x) P q > 0 and define A = D + m2I in the Hilbert space L2(X). Note that A is self-adjoint, positive, and elliptic. Applying Corollary 2.8 with a = 0, b = 1 and c = l, we obtain conservation of energy, or in other words, t ? E(t) is constant, recovering a well known result (see for example [6, p.660]). Introducing the term auttt, that is the damped Klein–Gordon equation
auttt ðt; xÞ þ utt ðt; xÞ Duðt; xÞ þ m2 ðxÞuðt; xÞ ¼ 0; we conclude that E(t) is decreasing and, hence, the problem changes from conservative to dissipative.
C. Fernández et al. / Applied Mathematics and Computation 217 (2011) 8522–8533
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Acknowledgement The authors are supported by Laboratorio de Analisis Estocástico, Proyecto Anillo ACT-13. The first author is partially financed by Project FONDECYT 1100304. The second author is partially supported by Grant FONDECYT number 1100485. The third author is also partially financed by Proyecto Fondecyt de Iniciación 11075046. References [1] W. Arendt, C. Batty, M. Hieber, F. Neubrander, Vector-valued Laplace transforms and Cauchy problems, Monographs in Mathematics, Vol. 96, Birkhäuser, Basel, 2001. [2] S.K. Bose, G.C. Gorain, Exact controllability and boundary stabilization of flexural vibrations of an internally damped flexible space structure, Appl. Math. Comput. 126 (2–3) (2002) 341–360. [3] S.K. Bose, G.C. Gorain, Stability of the boundary stabilised damped wave equation y00 + ky000 = c2(Dy + lDy0 ) in a bounded domain in Rn , Indian J. Math. 40 (1) (1998) 1–15. [4] S.K. Bose, G.C. Gorain, Exact controllability and boundary stabilization of torsional vibrations of an internally damped flexible space structure, J. Optim. Theory Appl. 99 (2) (1998) 423–442. [5] P. Cannarsa, D. Sforza, Semilinear integrodifferential equations of hyperbolic type: existence in the large, Mediterr. J. Math. 1 (2004) 151–174. [6] L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, Rhode Island, 2010. [7] R. Christensen, Theory of Viscoelasticity, Academic Pres, New York, 1971. [8] H.O. Fattorini, Second Order Differential Equations in Banach Spaces, North Holland, Amsterdam, 1985. [9] C. Fernandez, C. Lizama, V. Poblete, Maximal regularity for flexible structural systems in Lebesgue spaces, Mathematical Problems in Engineering, Vol. 2010, Article ID196956, 2010, p. 15. [10] G. Gorain, Exponential energy decay estimate for the solutions of internally damped wave equation in a bounded domain, J. Math. Anal. Appl. 216 (1997) 510–520. [11] G. Gorain, Stabilization for the vibrations modeled by the ‘‘standard linear model’’ of viscoelasticity, Proc. Indian Acad. Sci. (Math. Sci.) 120 (4) (2010) 495–506. [13] J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, Vol. 87, Birkhäuser Verlag, 1993. [14] T.J. Xiao, J. Liang, The Cauchy Problem for Higher-order Abstract Differential Equations, Lecture Notes in Mathematics, Vol. 1701, Springer-Verlag, Berlin, 1998.