Existence results for some Kirchhoff–Carrier problems

Nonlinear Analysis 75 (2012) 2904–2921

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Existence results for some Kirchhoff–Carrier problems Senoussi Guesmia ∗ Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, CH-8057, Zurich, Switzerland Mathematics Department, College of Sciences, Qassim University, Al-Qassim, Saudi Arabia

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Article history: Received 1 December 2010 Accepted 28 November 2011 Communicated by S. Carl MSC: 35A01 35L80 35L72 47H10 74K05 76M45

abstract Analyzing the viscoelastic problem for small vibrations of elastic strings, Kirchhoff and Carrier proposed two different models of nonlinear partial differential equations. By combining these two models, we deal here with some nonlocal hyperbolic problems that cover a large class of Kirchhoff and Carrier type problems. The existence of local solutions of degenerate problems as well as local and nonlocal solutions of nondegenerate problems is established. The proofs are based on the combination of the Schauder fixed point theorem with some asymptotic method. © 2011 Elsevier Ltd. All rights reserved.

Keywords: Quasilinear hyperbolic equations Degenerate problems Local and non-local existence Asymptotic methods Schauder fixed-point theorem Strings

1. Introduction Let us consider the nonlinear viscoelastic problem for small vibrations of elastic string of length L, fixed at the ends. The string in the rest position is on x axis. The vertical displacement u (t , x) of the point x ∈ (0, L) at the time t is a solution of the following model proposed by Kirchhoff [1]

     2 1 ∂2 σ (t , x) L ∂ u 2 ∂ ∂ τ0 + u− dx u + k (t , x) u = 0, 2 2 ∂t Ld (t , x) 2L ∂x ∂x ∂t 0

(1)

where d (t , x) is the mass density, σ (t , x) is the area of the cross section of the string, τ0 represents the initial tension and ∂ d(t ,x)

k (t , x) = ∂dt (t ,x) + η, η ≥ 0. Analyzing the same physical problem, Carrier [2] proposed another model depending on the square integral of u, i.e. we replace the nonlocal term in (1) by L



u2 dx. 0

∗

Correspondence to: Institute of Mathematics, University of Zurich, Winterthurerstrasse 190, CH-8057, Zurich, Switzerland. E-mail addresses: [email protected], [email protected].

0362-546X/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2011.11.033

S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

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Combining these two models, we may consider the following nonlocal hyperbolic equation

  L  L  2  2 ∂u ∂ ∂ ∂2 2 u − m0 (t , x) + m1 (t , x) u dx + m2 (t , x) dx u + m3 (t , x) u = 0, 2 ∂t2 ∂ x ∂ x ∂ t 0 0

(2)

coupled with some natural initial and boundary conditions u (0, x) = u0 (x)

∂ u (0, x) = u1 (x) ∂t

in (0, L) ,

in (0, L) ,

(3)

u (t , 0) = u (t , L) = 0,

(4)

where m0 , . . . , m3 are real-valued functions in t and x. Let z0 be a real valued function solution to the linear hyperbolic problem

 2 ∂2 ∂     ∂ t 2 z0 − ∂ x2 z0 = 0, z0 (t , 0) = z0 (t , L) = 0,   ∂  z0 (0, x) = u0 (x) , z0 (0, x) = u1 (x) . ∂t By setting z (t , x) = u (t , x) − z0 (t , x) ,

t ≥ 0, x ∈ (0, L) ,

the problem (2)–(4) leads to

 2 ∂2 ∂ ∂2 ∂  ′  z − ℓ z z = − m t , x z + m t , x z z0 , + 1 + ℓ z ( ) ( ) ( ) ( ( ))  3 3 0  ∂t2 ∂ x2 ∂t ∂ x2 z (t , 0) = z (t , L) = 0,    z (0, x) = ∂ z (0, x) = 0, ∂t

(5)

where L



ℓ ( z ) = m0 + m1

z02 dx + m2 0

L

 + 2m2 0

∂ z0 ∂x  L

 L 0

∂ ∂ z0 zdx + m1 ∂x ∂x

2

L

 dx + 2m1

z0 zdx 0

z 2 dx + m2

0

 L 0

∂z ∂x

2

dx.

(6)

Note that ℓ (z ) contains different types of nonlocal terms. Motivated by this model let us now introduce the natural generalized problem. Let Ω be a bounded open subset of RN where N > 0 is an integer. For a positive constant T , we set QT = (0, T ) × Ω , where the index T will be dropped when there is no ambiguity. We denote by x the point in RN and (t , x) in Rt × RN . With this notation we set

 ⊥ ∇x u = ∇ u = ∂x1 u, . . . , ∂xN u ,

u′ = ∂t u.

Let us denote by a = a(t , x, s) (resp. f = f (t , x, r , s)) a real-valued function depending on a nonlocal term s = l (u) (resp. on a nonlocal term s = l (u) and the velocity r = u′ ) and consider the nonlocal evolution problem defined as

   u′′ − ∇ · (a (·, l (u)) ∇ u) = f ·, u′ , l (u) u = 0 on (0, T ) × ∂ Ω , u (0) = 0 and u′ (0) = 0 in Ω .

in QT , (7)

We say that the problem (7) has a local solution if there exist T0 ≤ T and a function u : (0, T0 ) × Ω → R satisfying (7) in the weak sense. If T0 = T we say that u is a nonlocal solution. Then in order to cover the model (5) with (6) we assume that

p0  p1  2 l (u) (t , x) = , ,..., }0 (t , x, y) u (y) dy }1 (t , x, y) ∂x1 u (y) dy Ω Ω  pN  q0  2 , h0 (t , x, y) u (y) dy , }N (t , x, y) ∂xN u (y) dy 

2

Ω

Ω

q1

 Ω

h1 (t , x, y) ∂x1 u (y) dy

,...,

qN 

 Ω

hN (t , x, y) ∂xN u (y) dy

,

(8)

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where pi , qi for i = 0, . . . , N are nonnegative integers and }i , hi for i = 0, . . . , N are real-valued functions. The existence of local solutions, in the degenerate as well as in the nondegenerate case (in more general and natural situation motivated by the problem (5)) is shown by employing an asymptotic method together with the Schauder fixed point theorem. Then we establish the existence of the nonlocal solution under some smallness variations of the data. There are many contributions about the mathematical aspects of the Kirchhoff model

    u′′ − a |∇ u|2 dy ∆u = f (t , x) Ω

in Q , (9)

 u = 0 on (0, T ) ×′ ∂ Ω , u (0) = u0 and u (0) = u1 in Ω , when a : R → R is independent of t and x. Assume that u0 , u1 and f are analytic functions, Pohozaev [3] obtained the global existence solutions (T = +∞) in the nondegenerate case i.e. a ≥ m for some constant m > 0. The following reformulate abstract problem of (9)





 

2 u′′ − a A1/2 uV Au = f (t ) ,

(10)

and u′ (0) = u1 ,

u (0) = u0

was established by Lions [4] to generalize and improve Pohozaev’s results. In (10), A : V → V ′ is a self-adjoint positive unbounded operator, A1/2 its square root and V is a Hilbert space. They used, in particular, the following useful identity    1/2 2 A u 

  2    d Au, u′ = a A1/2 u V

dt

V

a (s) ds.

(11)

0

To cover Carrier’s problem we may replace A1/2 by Aθ in (10) where 0 ≤ θ ≤ 1. In this context Cousin et al. [5] showed analogues results of those obtained by Pohozaev. In the degenerate case (a ≥ 0) with θ = 21 , Hozoya and Yamada [6], Yamada [7] and Yamazaki [8] established the existence of local solutions for nonsmooth data. For more general problem when 0 < θ < 1 we refer the reader to Izaguirre et al. [9] and its bibliography. As proposed in Lions [4] and more close to the physical model (5) it is assumed that a depends on t and x. Nondegenerate problems of this type are established in Frota [10] to get nonlocal solutions and in Limaco et al. [11] to obtain global solutions with restrictions on the initial data in both papers. In Gourdin and Mechab [12] they proved the existence of global solutions for the same model with analytic data nonlinear terms or small data. It is interesting to cite that in [11] the  and small  viscoelastic model (i.e. when f = f t , x, u′ ) is investigated. Note that almost all authors in the previous papers they employed the approximation Galerkin method. In the same framework, using the Schauder fixed point theorem as here, the corresponding nonlocal elliptic and parabolic problems to (9) are considered in [13–16]. The questions of existence, uniqueness and asymptotic behavior of the solution are established therein. Here we propose a model that covers all Kirchhoff–Carrier problems considered in the previous papers and investigate the degenerate case to show the existence of the local solutions. Then considering nondegenerate problems, we study the existence of the local as well as the nonlocal solution. Note that the model proposed in this paper cannot be reformulate in the form of an abstract problem as in a lot of previous papers and by consequence to estimate the energy we cannot use a particular identity as (11). Moreover, here, the nonlocal terms also appear in the right hand side of the hyperbolic equation and we consider the divergence structure which is, in general, more complicate than the nondivergence one as in Limaco et al. [11]. Finally, for more details about the mathematical results of the Kirchhoff–Carrier models we refer the reader to the famous summarize in the paper by Medeiros et al. [17] and the references therein. In the following, for a given domain ω, we will denote by (·, ·)ω the L2 (ω)-scalar product or for simplicity in the notation by (·, ·) if there is not ambiguity, by ⟨·, ·⟩ω the duality between H01 (ω) and H −1 (ω) or simply ⟨·, ·⟩ and by |·|2 the norm of L2 (ω) . Next let us make the following hypotheses on a in order to define the degenerate and nondegenerate cases. More assumptions on a will be added in the next sections. We assume that a is a continuous function i.e. a ∈ C ([0, T ] × Ω × R2N +2 )

(12)

and we distinguish two cases according to the values of the function a. (I) Degenerate case. a (0, x, 0) > 0 ∀x ∈ Ω .

(13)

(II) Nondegenerate case. a≫0

on [0, T ] × Ω × R2N +2 .

(14)

Remark 1. Assuming only the hypothesis (13) on a, we may say that we have a degenerate nonlocal problems (see [7]). But as we will see and noted in [8], since the existence of solutions is given in an interval where a (t , x, l (u)) ̸= 0, we say that we have a mildly degenerate problem as in [18].

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In addition we assume, for the real-valued functions }i , hi for i = 0, . . . , N introduced in (8), that

}i , hi ∈ C ([0, T ] × Ω × Ω ) ∀i = 0, . . . , N , and there exists a constant δ > 0 such that

    δ |}i | , }′i  , |∇ }i | , ∇ }′i  ≤

2

a.e. (t , x, y) ∈ (0, T ) × Ω × Ω ∀i = 0, . . . , N ,

    δ |hi |2 , h′i 2 , |∇ h|2 , ∇ h′i 2 ≤

a.e. (t , x) ∈ (0, T ) × Ω ∀i = 0, . . . , N . 2 In general the function f is supposed to be continuous i.e. f ∈ C [0, T ] × Ω × R × R2N +2 ,





(15) (16)

(17)

but more hypotheses will be assumed in the next sections according to the degenerate and nondegenerate cases. In the next section we deal with the existence of local weak solutions of (7) in the degenerate case. This is done by solving a sequence of the corresponding smooth problems using the Schauder fixed point theorem, so that the solutions of these smoothing problems converge toward solutions of our problem. The third section is devoted to study the existence of local and nonlocal weak solutions of the nondegenerate problem assuming some smallness variations of the data. In the last section we will see that some improvements are possible when we assume more smoothness hypotheses on f . Before ending this section let us recall the following slightly modified Schauder fixed point theorem which will be used in this paper. Lemma 1. Let Φ be a relatively compact convex set in a Banach space B and let H : Φ → Φ be a continuous mapping on any finite dimensional subset of Φ . Then there exists a converging sequence un ∈ Φ , such that the sequence H (un ) converges to the limit of un i.e. lim H (un ) = lim un .

n→∞

n→∞

The proof is the same as that of the Schauder theorem (see [19]). 2. Local solutions of degenerate problems In this section we will investigate the existence of local solutions of the degenerate quasilinear hyperbolic problems given by (7) with a nonlocal term l defined by (8) i.e. we suppose that a satisfies (13). To this end, let us make some hypotheses on the data. Then by (12) and (13) there exist ρ, λ, γ1 > 0 and a constant ≤ T (still labeled T ), such that

|a (t , x, s)| ≤

γ1 2

,

a (t , x, s) ≥ 2λ,

∀ (t , x, s) ∈ (0, T ) × Ω × B¯ ρ ,

(18)

where B¯ ρ ⊂ R2N +2 is the closed ball of radius ρ centered at the origin. In addition we assume that there exist constants γ , γ0 , b > 0 such that

 ′     a  , |∇ a| , |∇s a| , |∇s (∇ a)| , ∇ a′  , ∇s a′  , |∇s (∇s a)| ≤ γ , a.e. (t , x, s) ∈ (0, T ) × Ω × B¯ ρ , 2   |∇x f | ≤ γ0 , |∇s f | , f ′  ≤ γ , |∂r f | ≤ b a.e. (t , x, r , s) ∈ (0, T ) × Ω × R × B¯ ρ ,

(19) (20)

where ∇s is the Gradient in s. As a consequence of the last inequality and (17) we derive

|f (t , x, r , s)| ≤ b0 + b |r |

∀ (t , x, r , s) ∈ (0, T ) × Ω × R × B¯ ρ ,

(21)

where b0 > 0. Note that the model problem (5) is covered by the assumption (21) since we will see that the nonlocal term ℓ (z ) is bounded. Let us start by introducing the same problem with smooth data. 2.1. Existence of solutions to the smooth problems Our method to solve Problem (7) is based on the idea of solving a sequence of the corresponding smooth problems using the Schauder fixed point theorem given in Lemma 1. So the solutions of these smoothing problems converge toward solutions of our problem. Let us introduce families of smooth functions an ∈ C ∞ ([0, T ] × Ω × B¯ ρ ) and }ni , hni ∈ C ∞ ([0, T ] × Ω × Ω ) for every i = 0, . . . , N , such that

 an → a, a′n → a′ , ∇ an → ∇ a, ∇ a′n → ∇ a′ , ∇s ∇ an → ∇s ∇ a,   ∇ a → ∇ a, ∇ a′ → ∇ a′ , ∇ ∇ a → ∇ ∇ a in L∞ ((0, T ) × Ω × B¯ ) s n s s s n ρ  s n ′ s n ′  n s′ s n n ′ → ∇ in L∞ ((0, T ) × Ω × Ω ), }i → }i , }i  → }i , ∇ }i → ∇ }i , ∇ } }  i i   ′  n ′ hi → hi , hni → h′i , ∇ hni → ∇ hi ∇ hni → ∇ h′i in L∞ ((0, T ) × Ω ; L2 (Ω )).

(22)

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Since we have (15), (16), (18), (19) and (22) we can suppose that, for any n > 0, an (t , x, s) ≥ λ,

∀ (t , x, s) ∈ (0, T ) × Ω × B¯ ρ ,  ′     |an | ≤ γ1 , an  , |∇s an | , |∇s (∇x an )| , ∇ a′n  , |∇ an | , ∇s a′n  , |∇s (∇s an )| ≤ γ , ∀ (t , x, s) ∈ (0, T ) × Ω × B¯ ρ ,  n   n ′       }  ,  }  , ∇x }n  , ∇x }n ′  ≤ δ, ∀ (t , x, y) ∈ (0, T ) × Ω × Ω ∀i = 0, . . . , N , i i i i     n   n ′     h  ,  h  , ∇x hn  , ∇x hn ′  ≤ δ, ∀ (t , x) ∈ (0, T ) × Ω , ∀i = 0, . . . , N . i 2 i i 2 i 2

(23) (24) (25) (26)

2

The smooth sequence converging toward f is the subject of the following fundamental lemma which can be shown as a consequence of the classical density results. Lemma 2. Let f be a function satisfying the hypotheses (17), (20) then there exists a sequence of functions fn ∈ C ∞ ([0, T ] × Ω × R × B¯ ρ ) such that

|∂t fn | , |∇s fn | ≤ γ ,

|∂r fn | ≤ b,

|fn (t , x, r , s)| ≤ b0 + b |r | ,

∀ (t , x, r , s) ∈ (0, T ) × Ω × R × B¯ ρ ,

∀ (t , x, r , s) ∈ (0, T ) × Ω × R × B¯ ρ ,

fn (t , ·, r , s) = 0 on a neighborhood of ∂ Ω , ∀ (t , r , s) ∈ (0, T ) × R × B¯ ρ .

(27) (28) (29)

Moreover for every sequence zn converges toward z in L2 (Q ) we have fn (·, zn , ·) → f (·, z , ·)

in L∞ B¯ ρ ; L2 (Q ) .





(30)

Remark 2. Note that the estimate

|∇x f | ≤ γ0 a.e. (t , x, r , s) ∈ [0, T ] × Ω × R × B¯ ρ

(31)

is only used here and is not really needed. Instead of (31), for instance, we can only assume that f is uniformly continuous in x. As mentioned above let us now establish the existence of local solutions of the following smooth nonlocal problems

   u′′ − ∇ (an (·, ln (u)) ∇ u) = fn ·, u′ , ln (u) u = 0 on (0, T ) × ∂ Ω , u (0) = 0 and u′ (0) = 0 in Ω ,

in Q , (32)

where ln is defined as l in (8) replacing }i , hi by }ni , hni respectively for i = 0, . . . , N. This is the subject of the following theorem. Theorem 1. Let Ω be a smooth bounded open subset of Rn , fn be a sequence of smooth functions given by Lemma 2. Assume that (23)–(26) hold and we have



4 (b0 + b) b0 meas (Ω ) + bCΩ M 2 < min 1,





λ 2



λM 2 ,

(33)

where M is given below by (43). Then there exists 0 < T0 ≤ T such that for every n there is at least one solution un to

         un ∈ C 0, T0 ; H01 (Ω ) ∩L∞ 0, T0 ; H 2 (Ω ) , ∂t un ∈ L∞ 0, T0 ; H01 (Ω ) ∩ C 0, T0 ; L2 (Ω ) ,    2 ∂t un ∈ L∞ 0, T0 ; L2 (Ω ) , ′  un′′ (0) = 0 and un (0) = 0 in Ω  ,  un − ∇ · (an (·, ln (un )) ∇ un ) = fn ·, u′n , ln (un ) .

(34)

Moreover for every n the estimates

  |∆un |L∞ (0,T0 ;L2 (Ω )) , ∇ u′n L∞ (0,T ;L2 (Ω )) ≤ M , 0  ′′  u  ∞ n L (0,T0 ;L2 (Ω )) ≤ R,

(35) (36)

are true for some R > 0 independent of n. Remark 3. The hypothesis (33) does not imply necessarily the smallness of b, b0 . For instance we may consider that b0 is bigger and bigger if M is. Of course in this case according (18) and (43), λ is small and b behaves as λ. Note that the geometry of the domain Ω also plays a role in the choice of b, b0 .

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2.2. Proof of Theorem 1 In order to apply the Schauder fixed point theorem (Lemma 1), we know (see [20,21]) that for every w ∈ C ∞ Q vanishing on [0, T ] × ∂ Ω there exists a unique un solution to the following linear hyperbolic problem

 

   u′′n − ∇ (an (·, ln (w)) ∇ un ) = fn ·, w′ , ln (w) u = 0 on (0, T ) × ∂ Ω , un (0) = 0 and ∂ u (0) = 0 in Ω , n t n

in QT , (37)

in the weak sense i.e.

     un ∈ C 0, T ; H01 (Ω ) , u′n ∈ C 0, T ; L2 (Ω ) ,    un (0) = 0 and u′n (0) = 0 in Ω ,      d2   an (·, ln (w)) ∇ un · ∇v dx = fn ·, w ′ , ln (w) , v  2 (un , v) + dt

Ω

(38) in D ′ (0, T ) , ∀ v ∈ H01 (Ω ).

In particular thanks to (29) and the vanishing initial condition we deduce that the mth-order compatibility conditions, for hyperbolic problems, hold for m ∈ N (see [22]). Then since all the data of the problem are smooth and if we assume that ∂ Ω is smooth enough, for instance ∂ Ω is C ∞ , it follows that un is very smooth i.e. un ∈ C ∞ Q T .





(39)

Combining this with the fact that un = 0 on (0, T ) × ∂ Ω leads to

∂tk un = 0 on (0, T ) × ∂ Ω , ∀k ≥ 0.

(40)

We also deduce, using the vanishing initial conditions, that

∂xki un (0) = ∂xki u′n (0) = 0 in Ω for i = 1, . . . , N , ∀k ≥ 0.

(41)

We are now ready to define a sequence Hn of mappings defined as

w → Hn (w) = un and make the following hypotheses sup ∂t2 w 2 ≤ R,



sup |∂t ∇w|22 + |∆w (t )|22





t ≤T 0

1/2

≤ M,

(42)

t ≤T0

where R, T0 > 0 are constants independent of n that we will choose later. Of course we have T0 ≤ T . M is a constant satisfying the second inequality of (42) such that ln (w) belongs to B¯ ρ i.e.





M = sup M ′ | ∀w ∈ C0∞ (Ω ), |∆w|2 ≤ M ⇒ l (w) ∈ B¯ ρ



2

∀w ∈ C0∞ (Ω ),

,

(43)

|∆w|2 ≤ M ⇒ ln (w) ∈ B¯ ρ ,   where C0∞ (Ω ) is the class of functions in C ∞ Ω vanishing on ∂ Ω . Note that since we have (22) the second line in (43) is deduced from the definition of M in the first line. Remark 4. In fact we do not need more than un ∈ C 4 Q T .





(44)

Then several assumptions can be weakened. In particular the constants pi , qi ≥ 0 for i = 0, . . . , N can be taken in R, large enough in order to get (44). We next apply the Laplace operator to the first equation of (37) i.e.

  ∂t2 ∆un − ∆∇ (an (·, ln (w)) ∇ un ) = ∆fn ·, w′ , ln (w) , then since the assertions (39) and (40) allows to test the above identity with v = −∂t un we get, integrating on Ω ,

  − ∂t2 ∆un , ∂t un +

 Ω

[∆∇ (an (·, ln (w)) ∇ un )] ∂t un dx = −

 Ω

  ∆fn ·, w ′ , ln (w) · ∂t un dx.

Integrating by part we derive

 2  ∂t ∇ un , ∂t ∇ un −

 Ω

∇ [∇ (an (·, ln (w)) ∇∂ un )] · ∂t ∇ un dx =

 Ω

  ∇ fn ·, w′ , ln (w) · ∂t ∇ un dx,

(45)

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S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

since ∂t un = 0 on (0, T ) × ∂ Ω . As a consequence of (40) and Lemma 2 with this identity

  ∇ (an (·, ln (w)) ∇ un ) = ∂t2 un − fn ·, w′ , ln (w) , we have, for every t ∈ [0, T ] ,

∇ (an (·, ln (w)) ∇ un ) ∈ H01 (Ω ). Going back to (45), it follows that



1 d  2 dt

 ∇ u′ 22 + n L (Ω )

Ω

[∇ (an (·, ln (w)) ∇ un )] ∆u′n dx = −



fn ·, w ′ , ln (w) ∆u′n dx,



Ω



(46)

since we also have fn = 0 on (0, T ) × ∂ Ω (by Lemma 2). Expending the first integral we get



1 d  2 dt

 ∇ u′ 22 + n L (Ω )

Ω

an (·, ln (w)) ∆un ∆u′n dx = −



fn ·, w ′ , ln (w) ∆u′n dx





Ω

 − Ω

{∇ [an (·, ln (w))] · ∇ un } ∆u′n dx.

(47)

Since all the quantities are smooth, one rewrites the first integral in the above identity as

 Ω



1

an (·, ln (w)) ∂t |∆un |2 dx 2 Ω   1 1 d an (·, ln (w)) |∆un |2 dx − ∂t [an (·, ln (w))] |∆un |2 dx. = 2 dt Ω 2 Ω

an (·, ln (w)) ∆un ∆u′n dx =

Replacing in (47) and integrating over (0, t ) for t ≤ T0 lead to

 ∇ u′ 22 + 1 n L (Ω )

1 2



2

 t

Ω

fn ·, w ′ , ln (w) ∆u′n dxdσ −



− 0

an (·, ln (w)) |∆un | dx = 2

Ω



 t Ω

0

 t

1 2

Ω

0

∂t [an (·, ln (w))] |∆un |2 dxdσ

{∇ [an (·, ln (w))] · ∇ un } ∆u′n dxdσ .

(48)

(Of course the initial conditions (41) are taken into account.) 2.2.1. Estimate I Next we need to estimate the integrals in the right hand side of the above identity. To this end we first deal with some frequently arising quantities. (i) l′n (w) (t , x) is given by

 ln (w) (t , x) = ′

 p1

Ω

 pN

Ω

 q0

Ω

 qN

Ω

n }1

n }N

 p0

Ω

 2 ∂x1 w

Ω

2 }n0 ww ′ + ∂t }n0 w 2 ,

2 }n1 ∂x1 w∂x1 w ′ + ∂t }n1 ∂x1 w



pN −1 

2

,...,

2 }nN ∂xN w ′ ∂xN w + ∂t }nN ∂xN w



Ω

Ω

∂ w

p0 −1 

Ω

 q 0 −1 

hnN xN

w

2

p1 −1 

 2 ∂xN w

hn0 w

n }0

hn0 w ′ + ∂t hn0 w, q1

 Ω

hn1 ∂x1 w

2

q1 −1  Ω

, hn1 ∂x1 w ′ + ∂t hn1 ∂x1 w, . . . ,



qN −1  Ω

∂ w +∂

hnN xN

′

∂ w .

n t hN x N

(49)

(We dropped the measures of integration.) Thanks to (25) and (26) we obtain

  N   ′   ′ 2 p0   2 pi p 2 p 2 ′ i 0 l (w) ≤ 2 p0 δ |w| + w  |∇w|2 + ∇w 2 + pi δ n 2 2 i =1

√ +

 2 q0 δ

q0



N   2  q20   2  q2i |w|22 + w ′ 2 + qi δ pi |∇w|22 + ∇w ′ 2 i =1



S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

2911



N    2 pi  2 p0  p + pi δ pi |∇w|22 + ∇w ′ 2 ≤ 2 p0 δ p0 CΩ0 |∇w|22 + ∇w ′ 2



i =1

√



q0 2

 2   |∇w|22 + ∇w′ 2

2 qδ CΩ

+

q0

q0 2

+

N 

  2  q2i |∇w|22 + ∇w ′ 2 qi δ

 ,

pi

(50)

i=1

where CΩ is the Poincaré inequality constant. On the other hand for every v ∈ H01 (Ω ) such that ∆v ∈ L2 (Ω ) we have by Poincaré’s inequality





2

1/2 v ∆v dx ≤ |∆v|2 |v|2 ≤ CΩ |∆v|2 |∇v|2 ,

|∇v| dx = − Ω

Ω

whence

|∇v|2 ≤ CΩ1/2 |∆v|2 .

(51)

Applying this in (50) we derive

  N   ′   2 p0   2 pi p p 2 ′ p 2 ′ 0 l (w) ≤ 2 p0 δ 0 C   + pi δ i CΩ |∆w|2 + ∇w 2 Ω CΩ |∆w|2 + ∇w 2 n i =1

√ +



q0 2

2 q0 δ q0 CΩ



N   2  q20   2  q2i CΩ |∆w|22 + ∇w ′ 2 + qi δ pi CΩ |∆w|22 + ∇w ′ 2

 .

i =1

Then it follows, by (42), that

′  l (w) ≤ Υ (δ, M ) , n

(52)

where q0

p

q0

Υ (δ, M ) = 2p0 δ p0 CΩ0 (CΩ + 1)p0 M 2p0 + 2γ q0 δ q0 CΩ2 (CΩ + 1) 2 M q0   N N   qi q pi pi 2pi pi i . +2 pi δ (CΩ + 1) M + qi δ (CΩ + 1) 2 M i=1

(53)

i=1

(ii) ∂xi ln (w) (t , x) is given by

 ∂xi ln (w) (t , x) =

 p0

Ω

 pN

w

2

p0 −1  Ω

n }N ∂xN w



Ω

 q1

n }0

Ω

∂ w

hn1 x1

2

∂

n xi }0

pN −1  Ω

q1 −1  Ω

∂

w , p1 2

 Ω

n }1



∂x1 w

 2 ∂xi }nN ∂xN w , q0 ∂ w, . . . , qN

n xi h1 x1

2

Ω



 Ω

p1 −1 

Ω

hn0 w

∂ w

hnN xN

 2 ∂xi }n1 ∂x1 w , . . . ,

 q 0 −1  Ω

∂xi hn0 w, 

 q N −1  Ω

∂

∂ w .

n xi hN xN

Then using (25), (26) and (51) we obtain N    2p q ∂x ln (w) ≤ p0 δ p0 |w|2p0 + q0 δ q0 |w|q0 + pi δ pi |∇w|2 i + qi δ pi |∇w|2i 2 2 i i=1 2p

2p

q

q

≤ p0 δ p0 CΩ 0 |∆w|2 0 + q0 δ q0 CΩ0 |∆w|20 +

N 

p

2p

qi

q

pi δ pi CΩi |∆w|2 i + qi δ pi CΩ2 |∆w|2i .

i =1

Setting 2p q Υ¯ (δ, M ) = p0 δ p0 CΩ 0 M 2p0 + q0 δ q0 CΩ0 M q0 +

N 

p

qi

pi δ pi CΩi M 2pi + qi δ pi CΩ2 M qi ,

(54)

i =1

it is clear from the above inequality and (42) that

  ∂x ln (w) ≤ Υ¯ (δ, M ) . i (iii) Finally, ∂xi ln (w) (t , x) is given by ′

∂xi l′n (w) (t , x) = A + B,

(55)

2912

S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

where

 A =

 p0

Ω

 p1

Ω

n }1

 pN

p0 −1 

2  ∂ x1 w

Ω

hn0 w

 Ω

2∂xi }n0 ww ′ + ∂t ∂xi }n0 w 2 ,

Ω





qN

2

n }N ∂xN w

Ω

q0

w

n }0

p1 −1  pN −1 

Ω

∂ w

2

,...,

2∂xi }nN ∂xN w ′ ∂xN w + ∂xi ∂t }nN ∂xN w



Ω

 q 0 −1 

hnN xN



Ω

2

2∂xi }n1 ∂x1 w∂x1 w ′ + ∂xi ∂t }n1 ∂x1 w

∂xi hn0 w′ + ∂xi ∂t hn0 w, q1



hn1 ∂x1 w

Ω

2

q1 −1  Ω

, ∂xi hn1 ∂x1 w ′ + ∂xi ∂t hn1 ∂x1 w, . . . ,



 q N −1  Ω

∂ w ,

∂ w +∂ ∂

∂

′

n x i hN x N

n x i t hN x N

and

 B =

p0 (p0 − 1) p1 (p1 − 1)

q1 (q1 − 1)

Ω

2

p0 −2  Ω

n }1 ∂x1 w



Ω

 Ω

Ω

 Ω

hn0 w

∂ w

Ω

2

pN −2  Ω

Ω

∂

 q N −2  Ω



2 }n0 ww ′ + ∂t }n0 w 2 dy ,

Ω



 Ω

Ω

 Ω

2

dy, . . . ,

2 }nN ∂xN w ′ ∂xN w + ∂t }nN ∂xN w



2

dy,

hn0 ∂t w + hn0 w ′ dy,

∂ wdy

n x i h1 x 1

∂

2 }n1 ∂x1 w∂x1 w ′ + ∂t }n1 ∂x1 w



 2 ∂xi }nN ∂xN w dy

∂xi hn0 wdy

Ω



 2 ∂xi }n1 ∂x1 w dy

Ω

 q 1 −2 

∂ w

hnN xN

∂xi }n0 w 2 dy

p1 −2 

 q 0 −2 

hn1 x1



2

n }N ∂xN w





qN (qN − 1)

}0 w n



pN (pN − 1) q0 (q0 − 1)



 Ω

∂ wdy

n x i hN x N

hn1 ∂x1 w ′ + ∂t hn1 ∂x1 w dy, . . . ,



 Ω

∂ w +∂

hnN xN

′

∂ wdy .

n t hN x N

Arguing as we did in the points (i) and (ii) we get

 ′  ∂x l (w) ≤ c Υ (δ, M ) , i n

(56)

where c = 1 + max {pi − 1, qi − 1 for i = 0, . . . , N } . In the following we will drop (for simplicity) δ, M in Υ¯ (δ, M ) and Υ (δ, M ). Let us now estimate the three terms of the right hand side of (48) separately. We rewrite the first term as

 t 0

Ω

∂t [an (·, ln (w))] |∆un |2 dxdσ =

 t Ω

0

∂t an (·, ln (w)) |∆un |2 dxdσ

 t + Ω

0

∇s an (·, ln (w)) · l′n (w) |∆un |2 dxdσ .

(57)

Taking into account (24) and (52) we derive

 t 0

Ω

   ∂t an ·, w′ , ln (w) |∆un |2 dx ≤ γ (1 + Υ )

t



|∆un |22 dσ . 0

Going back to (48) and we estimate the second term of the right hand side. Integrating by part we get

 t

 t     fn t , ·, w , ln (w) ∆un dx − ∂t fn ·, w′ , ln (w) ∆un dxdσ 0 Ω Ω 0 Ω   t     ′′ ′ ′ = fn t , ·, w , ln (w) ∆un dx − ∂r fn ·, w , ln (w) w ∆un dxdσ Ω 0 Ω  t  t       − ∂t fn ·, w′ , ln (w) ∆un dxdσ − ∇s fn ·, w′ , ln (w) · l′n (w) ∆un dxdσ ,   fn ·, w ′ , ln (w) ∆u′n dxdσ =

0

Ω



′



0

Ω

(58)

S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

2913

since we have (41). Applying Lemma 2 and (52) leads to

  t      ′   ∂ ∆ u dxd σ ·, w , l f (w) t n n n   0 Ω   t  t    ′′   w  |∆un | dxdσ + γ (Υ + 1) |∆un | dxdσ . ≤ b0 + b w ′  |∆un | dx + b Ω

0

Ω

0

Ω

Thanks to Young’s inequality we deduce

 t     b0 meas (Ω ) γ (Υ + 1) meas (Ω ) t   ′  fn ·, w , ln (w) ∂t ∆un dxdσ  ≤ +  2ε 2 Ω

0

+

   ∇w′ 2 + ε (b0 + b) |∆un |2 + bt sup w ′′ 2 + 1 (b + γ (1 + Υ )) 2 2 2

bCΩ  2ε

2

2 σ ≤T0

t



2

|∆un |22 dσ , 0

where ε is a constant that we will choose later. We used Poincaré’s inequality since w ′ vanishes on [0, T ] × ∂ Ω . By (42), it follows that

 t   2       ′   ≤ b0 meas (Ω ) + bCΩ M + γ (1 + Υ ) meas (Ω ) + bR2 t f ·, w , l ∂ ∆ u dxd σ (w) n n t n   2ε 2 0 Ω  t ε 1 |∆un |22 dσ . + (b0 + b) |∆un |22 + (b + γ (1 + Υ )) 2

2

(59)

0

Again going back to (48) and integrating by part in the last term of the right hand side we get

 t 0



{∇ [an (·, ln (w))] · ∇ un } ∆un dxdσ = {(∇ [an (t , ·, ln (w))] · ∇ un )} ∆un dx Ω  t  t   − ∇ [an (·, ln (w))] · ∇ u′n ∆un dxdσ − (∇ [∂t [an (·, ln (w))]] · ∇ un ) ∆un dxdσ ′

Ω

0

Ω

0

Ω

whence

 t

{∇ [an (·, ln (w))] · ∇ un } ∆u′n dxdσ 0 Ω      {(∇ an · ∇ un )} ∆un dx + = ∇s an · ∂xi ln (w) i=1,...,N · ∇ un ∆un dx Ω Ω  t        ′ − ∇ an · ∇ un ∆un dx − ∇s an · ∂xi ln (w) i=1,...,N · ∇ u′n ∆un dxdσ 0 Ω Ω  t      ′  − ∇ an + ∂xi ∇s an · l′n (w) i=1,...,N + ∇s a′n · ∂xi ln (w) i=1,...,N 0

Ω

+ ∇s an · ∂xi ln (w) 

′



i=1,...,N

   − ∂sj ∇s an · l′n (w) j=1,...,2N +2 · ∂xi ln (w)

 · ∇ un ∆un dxdσ .

i=1,...,N

Note that for simplicity we dropped (t , ·, ln (w)) as a variable of an . Then using (24)–(26), (52), (55) and (56) we derive

 t    √     ′   ¯ |∇ un | |∆un | dx {∇ [an (·, ln (w))] · ∇ un } ∆un dxdσ  ≤ γ N Υ + 1  Ω 0 Ω       √  ′   t   t   ¯ ¯ ¯ | | |∇ | | | +γ N Υ + 1 ∇ un ∆un dxdσ + Υ + (c + 1) Υ + Υ Υ + 1 un ∆un dxdσ . Ω

0

Ω

0

Applying Young’s inequality it comes

   1 γ√  {∇ [an (·, ln (w))] · ∇ un } ∆u′n dxdσ ≤ N Υ¯ + 1 ε ′ |∆un |22 + ′ |∇ un |22 2 ε Ω       t  ′ 2   t γ√ 2 2 ∇ u  + |∆un |2 dσ + Υ¯ + (c + 1) Υ + Υ¯ Υ + 1 |∇ | | | + N Υ¯ + 1 u + ∆ u d σ . n n n 2 2 2 2

 t 0

2

0

0

2914

S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

Taking into account (51) leads to

  t     γ√    ′ 1 ′ 2 2   ¯ } | | {∇ |∇ | ] [a · ∇ u ∆ u dxd σ l ≤ ε ∆ u + N Υ + 1 u (w)) (·, n n n 2 n n 2 n   2 ε′ 0 Ω         t   t  ′ 2 γ√ 2   ¯ ¯ ¯ ¯ |∆un |2 dσ . + N ∇ un 2 dσ + Υ + 1 + Υ + (c + 1) Υ + Υ Υ + 1 (CΩ + 1) Υ +1 2

(60)

0

0

Finally, combining (48) and (58)–(60) it follows that

 t λ b0 meas (Ω ) + bCΩ M 2 |∂t ∇ un |22 + |∆un |22 ≤ + γ (Υ + 1) meas (Ω ) + bR2 2 2 2ε 2  t  ′   √    ε   γε √  γ ∇ u′ 2 dσ + 1 |∇ un |2 + N Υ¯ + 1 + (b0 + b) |∆un |22 + N Υ¯ + 1 n 2 2 2 2 2 ε′ 0  t    √     b γ |∆un |22 dσ . 2 + 2Υ + N Υ¯ + 1 + Υ¯ + (c + 1) Υ + Υ¯ Υ + 1 (CΩ + 1) + + 1

2

2

0

Note that we used (23). Taking ε = 1 2

λ

λ 4(b0 +b)

and ε = ′

4γ

√ λ

(

we end up with

)

N Υ¯ +1

2 (b0 + b) b0 meas (Ω ) + bCΩ M 2





 t |∂t ∇ | + | | ≤ + γ (Υ + 1) meas (Ω ) + bR2 4 λ 2  t  √     b γ  |∆un |22 dσ + + 2 + 2Υ + N Υ¯ + 1 + Υ¯ + (c + 1) Υ + Υ¯ Υ + 1 (CΩ + 1) un 22

2

∆un 22

2

0

 2  γ√  2γ N Υ¯ + 1 N Υ¯ + 1 |∇ un |22 + + λ 2 2

t



 ′ 2 ∇ u  dσ . n 2

0

(61)

2.2.2. Estimate II In order to apply Gronwall’s inequality, the following lemma is devoted to estimate the term |∇ un |22 . Lemma 3. We have, for every t ≤ T0

|∇ un |22 ≤

 4 2 γ b0 meas (Ω ) + b2 CΩ M 2 t 2 + (1 + Υ ) CΩ

λ

t



λ

|∆un |22 dσ . 0

Proof. Testing the equation in (38) with v = u′n we derive 1 d 2 dt

 Ω

 ′  u (t )2 dx + 1 n

2

 Ω

an (·, ln (w))∂t |∇ un |2 dx =



fn ·, w ′ , ln (w) u′n dxdσ .



Ω



Integrating over (0, t ) leads to 1



2

Ω

 ′  u (t )2 dx + 1 n 2

 Ω

an (·, ln (w))∇ un · ∇ un dx =

 t

1 2

Ω

0

 t

fn ·, w ′ , ln (w) u′n dxdσ .



+ Ω

0

∂t [an (·, ln (w))] ∇ un · ∇ un dxdσ 

Computing the first integral of the right hand side we get

    ′  1 t u (t )2 dx + 1 a (·, l u · ∇ u dx = a′n (·, ln (w))∇ un · ∇ un dxdσ (w))∇ n n n n n 2 Ω 2 Ω 2 0 Ω    t     1 t + ∇s an (·, ln (w)) · l′n (w) ∇ un · ∇ un dxdσ + fn ·, w ′ , ln (w) u′n dxdσ . 1



2

0

Ω

0

Ω

Thus — due to (23), (24), (52) and Lemma 2 we deduce that

 1 ′ u (t )2 + λ |∇ un |2 ≤ γ (1 + Υ ) n 2 2 2 2 2

t



|∇ un |22 dσ + 0

 t 0

 Ω

b0 + b w ′  u′n  dxdσ .



  

By Young’s inequality we get, for ε > 0,

   t  t  t  ′ 2  ′ 2  ′  2 u (t )2 + λ |∇ un |2 ≤ 1 b2 meas (Ω ) t + b2 w  dσ + γ (1 + Υ ) u  dσ . |∇ | u d σ + 2 ε n 2 n 2 n 2 0 2 2 ε 0 0 0

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2915

Then it follows that



2 2 sup u′n (σ )2 + λ |∇ un (t )|22 ≤ sup u′n (σ )2 + λ |∇ un (σ )|22









σ ≤t

σ ≤t

≤

1



ε

b20 meas

(Ω ) t + b

t



2

0

 ′ 2 w  dσ 2



+ γ (1 + Υ )

t



|∇ un |22 dσ 0

2  + 2ε t sup u′n (σ )2 . σ ≤t

Choosing ε =

1 , 4t

we derive

 1 ′ u (t )2 + λ |∇ un (t )|2 ≤ 4t n 2 2 2



b20 meas

(Ω ) t + b

t



2

0

 ′ 2 w  dσ 2



+ γ (1 + Υ )



t

|∇ un |22 dσ .

0

Applying Poincaré’s inequality and (51) we get 1 2

   t  t   ∇w ′ 2 dσ + γ (1 + Υ ) CΩ |∆un |22 dσ . |∂t un (t )|22 + λ |∇ un (t )|22 ≤ 4t b20 meas (Ω ) t + b2 CΩ 2 0

0

Thus, by (42) it follows that 1 2

  |∂t un (t )|22 + λ |∇ un |22 ≤ 4 b20 meas (Ω ) + b2 CΩ M 2 t 2 + γ (1 + Υ ) CΩ



t

|∆un |22 dσ .

0

This completes the proof of the lemma.



We now go back to (61) and applying the above lemma we derive 1 2

un 22

|∂t ∇ | +

λ

2 (b0 + b) b0 meas (Ω ) + bCΩ M 2



∆un 22

|

| ≤



λ 2    2   t 8 γ + γ (Υ + 1) meas (Ω ) + bR2 + 2 N Υ¯ + 1 b20 meas (Ω ) + b2 CΩ M 2 t 2 2 λ    2  √    2   b γ 4γ ¯ + 1 (1 + Υ ) CΩ + 2 + 2Υ + N Υ¯ + 1 + Υ¯ + (c + 1) Υ + Υ¯ Υ + 1 (CΩ + 1) + Υ + N 2 λ2 2  t  t √   γ |∂t ∇ un |22 dσ . |∆un |22 dσ + × N Υ¯ + 1 4

2

0

0

Setting



Gγ ,δ,b (M ) = max 1,

2



b+γ



4γ 2

2

(1 + Υ ) CΩ + 2 + 2Υ     ¯ ¯ ¯ + N 2Υ + 2 + Υ + (c + 1) Υ + Υ Υ + 1 (CΩ + 1) ,     2   2 16γ  zγ ,δ,b (t , M , R) = max 1, × γ (Υ + 1) meas (Ω ) + bR2 + 2 N Υ¯ + 1 b20 meas (Ω ) + b2 CΩ M 2 t , λ λ √ 

λ

λ2

N Υ¯ + 1



we then derive, using Gronwall’s inequality, that

 un 22

|∂t ∇ | + |



| ≤ max 2,

∆un 22

4



λ

2 (b0 + b) b0 meas (Ω ) + bCΩ M 2



λ



 + zγ ,δ,b t exp Gγ ,δ,b t .

(62)

Note that the constants above are independents of n. We have now to deal with the second time derivative of Hn (w) to cover the first type inequality in (42). Thus, by (38) we have

  ∂t2 un = fn ·, w′ , ln (w) − an (·, ln (w)) ∆un − ∇ [an (·, ln (w))] ∇ un . Applying Lemma 2, (24) and (55) we derive

 2 2    ∂ un  ≤ 4 fn ·, w′ , ln (w) 2 + 4 |an (·, ln (w)) ∆un |2 + 4 |∇ an (·, ln (w)) · ∇ un |2 t 2 2 2 2  2    + 4  ∇s an (·, ln (w)) · ∂xi l (w) i=1,...,N · ∇ un  2  2   ≤ 8meas (Ω ) b20 + 8b2 w′ 2 + 4γ12 |∆un |22 + 4N γ 2 1 + Υ¯ 2 |∇ un |22 .

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S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

Due to Poincaré’s inequality and (51) we get

 2 2       ∂ un  ≤ 8meas (Ω ) b2 + 8b2 CΩ ∇w ′ 2 + 4 γ 2 + N γ 2 1 + Υ¯ 2 CΩ |∆un |2 . t 0 1 2 2 2 Then by (42) and (62), this implies that

 2 2 ∂ un  ≤ Iγ ,δ,b (t , M , R) , t 2

(63)

where Iγ ,δ,b (M , R) = 8meas (Ω ) b20 + 8b2 CΩ M 2 + 4 γ12 + N γ 2 1 + Υ¯ 2 CΩ



 ×



max 2,

4







2 (b0 + b) b0 meas (Ω ) + bCΩ M 2



λ

 



λ

+ zγ ,δ,b t exp Gγ ,δ,b t .

(64)

2.2.3. Definition of Φ In order to apply the Schauder fixed point theorem, we have to check that Hn (w) satisfies inequalities similar to those given in (42) for w i.e.

 2 2 ∂ un  ≤ R2 , |∂t ∇ un |22 + |∆un |22 ≤ M 2. t L (Ω ) L (Ω ) 2

(65)

This will happen if we have





max 2,

4



λ

   (b0 + b) b0 meas (Ω ) + bM 2 + zγ ,δ,b t exp Gγ ,δ,b t ≤ M 2 , λ

I (M , R) ≤ R2 .

(66) (67)

First thanks to (33) we have



max 2,

4

λ



2 (b0 + b) b0 meas (Ω ) + bCΩ M 2



λ

 < M 2.

Then we distinguish two cases according to the values of γ , δ, b and T0 .

(C1 ) If γ , δ and b are fixed, we choose R satisfying     R2 > 8meas (Ω ) b20 + 8b2 CΩ M 2 + 4 γ12 + N γ 2 1 + Υ¯ 2 CΩ     4 2 (b0 + b) b0 meas (Ω ) + bCΩ M 2 . × max 2, λ λ Then there exists 0 < T0 ≤ T independent of n such that (66) and (67) hold, for every t ∈ (0, T0 ). (C2 ) We now fix T0 = T and choose R such that   4 8b20 γ12 meas (Ω ) R2 > 8meas (Ω ) b20 + max 2, , λ λ then there exist γ , b > 0 or δ, b > 0 independents of n, such that (66) and (67) hold for every t ∈ (0, T0 ), since one can see that when γ , b > 0 become small and small, zγ ,δ,b and Gγ ,δ,b become small and small and tend to 0. On the other hand, thanks to (43), when δ, b > 0 become small and small, M can be chosen bigger and bigger and of course M, in (62), can be controlled by b. In fact here we only need the smallness of supΩ h and supΩ } to let M becomes relatively bigger. The second case (C2 ) is devoted to study the existence of nonlocal solutions in the next section and the first one (C1 ) characterizes the local solutions. Finally we are now ready to define Φ by

      Φ = u ∈ C ∞ [0, T0 ] × Ω , u (t , ·) = 0 on ∂ Ω , |∂t ∇ u|22 + |∆u|22 ≤ M 2 , ∂t2 un 2 ≤ R ∀t ∈ (0, T0 )   and we conclude that Hn is defined on Φ into itself. It is clear that Φ ⊂ W 1 0, T0 ; H 2 (Ω ), H01 (Ω ) since we have   H 2 (Ω ) ∩ H01 (Ω ) = u ∈ H01 (Ω ), ∆u ∈ L2 (Ω ) .   We then deduce that Φ is a precompact convex set in L2 0, T0 ; H01 (Ω ) .

S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

2917

2.2.4. End of the proof The next step is to show that Hn : Φ → Φ is a continuous mapping. Note that  Φ is equipped  with the norm of L2 0, T0 ; H01 (Ω ) . Let us consider a sequence w m ∈ Φ converging toward w ∈ Φ in L2 0, T0 ; H01 (Ω ) , i.e. T0



 m  w − w  1 → 0. H (Ω )

0

(68)

m m m For n fixed, we set um n = Hn (w ) ∈ Φ . Since we also have un , w ∈ Φ we can extract from m a subsequence (still labeled by m) such that

in C 0, T0 ; H01 (Ω ) ,

um n → un





∂

(69)

→ ∂t un in C 0, T0 ; L (Ω ) ,   ∂t wm ⇀ ∂t w in L2 0, T0 ; H01 (Ω ) ,

(70)

in L ((0, T0 ) × Ω ) ,

(72)

m t un

∂

2 m t un

⇀∂  1

2 t un

where un ∈ W (38) as T0



2





(71)

2

0, T0 ; H (Ω ), 2

 2 m  ∂t un , v ϕ dσ +

T0



0

 (Ω ) . The third convergence holds since w m ∈ Φ . Next, we rewrite the last equation in

H01

 Ω

0

  ϕ an ·, ln w m ∇ um n · ∇v dxdσ = 

T0



       ′ fn ·, w m , ln w m , v ϕ dσ ,

0

for every ϕ ∈ D (0, T0 ) and v ∈ D (Ω ). Then passing to the limit when m → ∞ using (68)–(72) and the continuity of an , fn in s we obtain T0



 2  ∂t un , v ϕ dσ +

0

T0

 0

 Ω

ϕ an (·, ln (w)) ∇ un · ∇v dxdσ =

T0



    fn ·, (w)′ , ln (w) , v ϕ dσ .

0

Note that we used Lebesgue’s theorem, (28), the continuity of fn and the fact that ln w m → ln (w)



a.e. in (0, T0 ) × Ω .



We also derive the initial conditions from (69) and (70) un (0) = u′n (0) = 0. Thus, of (38) and Hn (w) = un . Since un is the unique solution of (38) it follows that um n → un strongly in  un is a1 solution  2 L 0, T0 ; H0 (Ω ) for the whole sequence. Of course this means the continuity of Hn .

¯m ¯m Finally, thanks to Lemma 1, there exists a converging sequence u¯ m ∈ Φ have the same n ∈ Φ such that u n and H u n 



¯m limit in L 0, T0 ; H01 (Ω ) . Then arguing as above we deduce that un (the limit of u¯ m n and, of course, of Hn u n ) is a solution m of the problem (34). Moreover since u¯ n ∈ Φ it follows that – up to a subsequence –  2

∆u¯ m n ⇀ ∆un ,



∂t ∇ u¯ m n ⇀ ∂t ∇ un ,





  2 ∂t2 u¯ m weakly in L∞ 0, T0 ; L2 (Ω ) . n ⇀ ∂t un

Since the norm is lower semi-continuous with respect to the weak topology it follows that

   |∂t ∇ un |∗ ≤ lim inf ∂t ∇ u¯ m n ∗ ≤ M,  m |∆un |∗ ≤ lim inf ∆u¯ n ∗ ≤ M ,  2    ∂ un  ≤ lim inf ∂ 2 u¯ m  ≤ R. t t n ∗ ∗ This completes the proof of the theorem. 2.3. Local existence theorem for degenerate problems We are now ready to state the main local existence result in the degenerate case. Theorem 2 (Degenerate Problem). Under the assumptions (12), (13), (15)–(17), (19) and (20), in addition if (33) holds there exists 0 < T0 ≤ T such that the nonlocal hyperbolic problem

     u ∈ C 0,T0 ; H01 (Ω ) ∩ L∞ 0, T0 ; H 2 (Ω ) ,     ′′ u ∈ L∞ 0, T0 ; L2 (Ω ) , u (0) = 0 and u′ (0) = 0 in Ω ,     ′′ u − ∇ · (a (·, l (u)) ∇ u) = f ·, u′ , l (u) , has at least one solution.

    u′ ∈ L∞ 0, T0 ; H01 (Ω ) ∩ C 0, T0 ; L2 (Ω ) , (73)

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S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

Proof. According to the estimates (35) and (36), there exists a function u satisfying the first line of (73) such that we have – up to a subsequence – in C 0, T0 ; H01 (Ω ) ,



un → u



(74)

 ∂t un → ∂t u in C 0, T0 ; L (Ω ) ,   ∂t2 un ⇀ ∂t2 u in L∞ 0, T0 ; L2 (Ω ) . 2



(75) (76)

Next we pass to the limit in (34) term by term. For the first term using (76) we get, for every ϕ ∈ D (0, T0 ) and v ∈ D (Ω ), T0





∂

2 t un

, v ϕ dσ → 

T0



 2  ∂t u, v ϕ dσ .

(77)

0

0

Then since we have an (·, ln (un )) − a (·, l (u)) = (an (·, ln (un )) − a (·, ln (un ))) + (a (·, ln (un )) − a (·, l (u))) , applying the first line in (22), we deduce that the first brackets in the right hand side of the above identity tends to 0 in L∞ ((0, T0 ) × Ω ). For the last brackets we first use the last two lines in (22) with (74) to show that ln (un ) → l (u)

a.e. in (0, T0 ) × Ω ,

(78)

then thanks to the continuity of a, this term tends to 0. Thus we get a.e. in (0, T0 ) × Ω .

a (·, ln (un )) → a (·, l (u))

(79)

This with (74) and Lebesgue’s theorem lead to T0



 Ω

0

T0



ϕ an (·, ln (un )) ∇ un · ∇v dxdσ →

 Ω

0

ϕ a (·, l (u)) ∇ u · ∇v dxdσ .

(80)

Finally rewriting the right hand side of the hyperbolic equation in (34) as T0



fn ·, (un )′ , ln (un ) , v ϕ dσ =

 





T0



0

      fn ·, (un )′ , ln (un ) − fn ·, u′ , ln (un ) , v ϕ dσ

0 T0

 +

  ′   fn ·, u , ln (un ) , v ϕ dσ ,

0

then applying (28), (75) to the first integral of the right hand side and (30), (78) to the second one we derive T0



fn ·, (un )′ , ln (un ) , v ϕ dσ →

 





T0



0

  ′   f ·, u , l (u) , v ϕ dσ .

(81)

0

Combining (77), (80) and (81), the equation in (34) becomes when n → +∞, for ϕ ∈ D (0, T0 ) and v ∈ D (Ω ), T0



 2  ∂t u, v ϕ dσ +

0

T0

 0

 Ω

ϕ a (·, l (u)) ∇ u · ∇v dxdσ =

T0



    f ·, (u)′ , ln (u) , v ϕ dσ .

0

We also derive the initial conditions by (74) and (75) u (0) = u′ (0) = 0. This completes the proof of the theorem.



3. Local and nonlocal solutions of nondegenerate problems As mentioned above we say that our problem is nondegenerate if (14) holds i.e. there exists λ > 0 such that a (t , x, s) ≥ 2λ,

∀ (t , x, s) ∈ [0, T ] × Ω × R2N +2 .

(82)

Under this assumption the following theorem shows the existence of local and nonlocal solutions of nondegenerate problem. Theorem 3 (Nondegenerate Problem). Assume that the hypotheses (12), (14)–(17) hold and (19), (20) hold for ρ large enough. Then we have

S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

2919

– Local solution. If moreover we assume that



4 (b0 + b) bCΩ < min 1,

λ



2

λ,

(83)

then there exists 0 < T0 ≤ T such that the problem (73) has at least one solution. – Nonlocal solution. If we assume that γ or δ , and b are small enough then we can take T0 = T . Remark 5. Note that the linear hyperbolic problems are considered here when b = 0,

δ = 0.

Proof. Here we deal with the nondegenerate problem i.e. the second inequality in (18) holds on an arbitrary ball B¯ ρ ⊂ R2N +2 of radius ρ centered at the origin. In this case since we have (83) we may choose M large enough in order to get (33). Then there exists ρ > 0 such that

∀w ∈ C0∞ (Ω ),

|∆w|2 ≤ M ⇒ l (w) ∈ B¯ ρ . 2

Since the sequences }n0 , . . . , }nN , h0 , . . . , hnN are supposed to be uniformly converge to }0 , . . . , }N , h0 , . . . , hN respectively, we may suppose that

∀w ∈ C0∞ (Ω ),

|∆w|2 ≤ M ⇒ ln (w) ∈ B¯ ρ .

Note that all the assumptions of the smooth problems given in Theorem 1 hold for this choice of ρ and M. Thus the remainder of the proof is similar to the proof of Theorem 1 and the first point of this theorem can be shown as in Theorem 2. The second point is given by the same argument taking into account the case (C2 ) in Section 2.2.3 above, since here the choice of M is arbitrary. Note that assuming the smallness of γ is equivalent to assuming the smallness of δ .  4. Some improvement results In this section we will show that, under some smoothness assumptions on f , we may drop the hypotheses (33) and (83). To do this it is enough, first, to assume that the sequence fn defined in Lemma 2 also satisfies, for every n > 0

|∇ fn | ≤ γ ,

∀ (t , x, r , s) ∈ [0, T ] × Ω × R × B¯ ρ .

(84)

Now regarding the function f , the above assumption can be reformulated otherwise. It is clear that if the sequence fn satisfies (84) and the assertions of Lemma 2 then fn (t , ·, r , s) is uniformly bounded in H 1 (Ω ). This also implies, using (29), that f (t , ·, r , s) ∈ H01 (Ω ) ,

∀ (t , r , s) ∈ [0, T ] × R × R2N +2 .

(85)

The following lemma shows that the hypothesis (85) with the boundedness of the gradient of f are sufficient conditions to construct a sequence fn satisfying (84). So we have Lemma 4. Let f be a function satisfying the hypotheses (17), (20) and (21). Assume that f (t , ·, r , s) ∈ H01 (Ω )

|∇ f | ≤ γ

∀ (t , r , s) ∈ [0, T ] × R × B¯ ρ ,

(86)

on (0, T ) × Ω × R × B¯ ρ ,

then there exists a sequence of functions fn ∈ C

(87) [0, T ] × Ω × R × B¯ ρ such that



 ∞

fn = 0 on [0, T ] × ∂ Ω × R × B¯ ρ ,

|∂t fn | , |∇s fn | , |∇ fn | ≤ C0 γ , |∂r fn | ≤ C0 b,

(88) on (0, T ) × Ω × R × B¯ ρ ,

(89)

where C0 is a positive constant independent of n. Moreover for every sequence zn converges toward z in L2 ([0, T ] × Ω ) we have fn (·, zn , ·) → f (·, z , ·)

in L∞ B¯ ρ ; L2 ((0, T ) × Ω ) .





(90)

The proof is classic, based on the standard density properties and is thus omitted. Now we are ready to state the following theorem. Theorem 4. Let Ω be a smooth bounded open subset of Rn . Assume that (12), (13), (15)–(17) and (19), (20) for ρ large enough, hold. fn be a sequence of smooth functions given in Lemma 4, satisfying (84), then we have – Local solution. For some 0 < T0 < T , there exists at least one solution of the problem (33).

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S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921

– Nonlocal solution. If we assume that (14) hold, then for some γ or δ , and b small enough there exists a solution of the problem (33) such that T0 = T . Proof. The proof is almost the same as for Theorem 2. Here we deal otherwise with the third integral in (48). Integrating by part in x we get

 t

fn ·, w ′ , ln (w) ∂t ∆un dxdσ = −

 t

   ∇ fn ·, w′ , ln (w) · ∂t ∇ un dxdσ 0 Ω 0 Ω  t  t     ∇ fn ·, w′ , ln (w) · ∂t ∇ un dxdσ − ∂r fn ·, w′ , ln (w) ∇w ′ · ∂t ∇ un dxdσ =− 



Ω

0

 t − Ω

0

Ω

0



   ∇s fn ·, w′ , ln (w) · ∂xi ln (w) i=1,...,N · ∂t ∇ un dxdσ .

(91)

Note that the function x −→ fn (·, x, ·) vanishes on ∂ Ω . Then applying (55), (84) and Lemma 4 we derive

 t

fn ·, w ′ , ln (w) ∂t ∆un dxdσ ≤ γ



Ω

0



√ 

N 1 + Υ¯



 t Ω

0

|∂t ∇ un | dxdσ + b

 t 0

Ω

  ∇w′  |∂t ∇ un | dxdσ .

Using Young’s inequality we get

 t 0

    b t γ√  ∇w ′ 2 dσ ¯ N 1 + Υ meas (Ω ) t + fn ·, w , ln (w) ∂t ∆un dxdσ ≤ 2 2 2 0 Ω   t   1 √  ∇ u′ 2 dσ . + γ N 1 + Υ¯ + b n 2 ′



2

0

By (42) we derive

 t 0

fn ·, w ′ , ln (w) ∂t ∆un dxdσ ≤



Ω



  1 √  γ N 1 + Υ¯ meas (Ω ) + bM 2 t 2  t    1 √  ∇ u′ 2 dσ . ¯ γ N 1+Υ +b + n 2 2 0

(92)

This means that (59), where (33) or (83) were required, is replaced by (92). Using this time (58), (60) and (92), we obtain the equivalent of (61) 1 2

|∂t ∇ un |22 +

λ 2

|∆un |22 ≤

     γ√  1 1 √  γ N 1 + Υ¯ meas (Ω ) + bM 2 t + N Υ¯ + 1 ε ′ |∆un |22 + ′ |∇ un |22 2 2 ε  t   γ √  |∆un |22 dσ N Υ¯ + 1 + Υ¯ + (c + 1) Υ + Υ¯ Υ + 1 (1 + CΩ ) + 1 + Υ 2 0  t    1 √  ∇ u′ 2 dσ . 2γ N 1 + Υ¯ + b + n 2 +

2

Taking ε = ′

1 2

2γ

√ λ

(

N Υ¯ +1

|∂t ∇ un |22 +

λ 4

)

0

and applying Lemma 3, we get

|∆un |22 ≤

   1 √ N γ 1 + Υ¯ meas (Ω ) + bM 2 t 2

 2  γ2  γ + 4 2 N Υ¯ + 1 b20 meas (Ω ) + b2 CΩ M 2 t 2 + λ 2   √     2 γ2 N Υ¯ + 1 + Υ¯ + (c + 1) Υ + Υ¯ Υ + 1 (1 + CΩ ) + 1 + Υ + 2N 2 CΩ (1 + Υ ) Υ¯ + 1 × λ  t  t   √     1 ∇ u′ 2 dσ . |∆un |22 dσ + × 2 N γ 1 + Υ¯ + b n 2 2

0

0

This is the equivalent of (62), i.e.

|∂t ∇ un |22 + |∆un |22 ≤ z exp Gt ,

(93)

with



zγ ,δ,b (t , M ) = max 1,

2

λ

 ×

 √



N γ 1 + Υ¯ meas (Ω ) + bM 2 + 4





    γ2  ¯ + 1 2 b20 meas (Ω ) + b2 CΩ M 2 t t , N Υ λ2

S. Guesmia / Nonlinear Analysis 75 (2012) 2904–2921



Gγ ,δ,b (t , M ) = max 1,

2

√



λ

b+

+ γ (1 + Υ ) + 2N

N γ 3Υ¯ + 3 + Υ¯ + (c + 1) Υ + Υ¯ Υ + 1





2921



(1 + CΩ )

 2 γ3 CΩ (1 + Υ ) Υ¯ + 1 . 2 λ 

Note that (63) still holds with I (M , R) = 8meas (Ω ) b20 + 8b2 CΩ M 2 + 4 γ12 + N γ 2 1 + Υ¯ 2 CΩ z exp Gt .









Then we can now apply the Schauder fixed point theorem, without need of (33) in the degenerate case or (83) in the nondegenerate case. Note that in the nondegenerate case ρ can be chosen large enough. The remainder of the proof is similar to the proofs of Theorems 2 and 3.  Remark 6. The assumption (85) may be replaced by the following f (t , x, 0, s) = 0,

∀ (t , x, s) ∈ [0, T ] × Ω × B¯ ρ .

Of course the hypothesis (87) is also assumed. Indeed, thanks to (40) we have f t , ·, w ′ (t , ·) , s ∈ H01 (Ω ) ,





∀ (t , s) ∈ [0, T ] × R × R2N +2 ,

(94)

where w ∈ Φ . In fact, this is exactly what we need to integrate by part and get (91). Acknowledgments The author has been partially supported by the Swiss National Science Foundation under the contracts #20-113287/1 and #20-117614/1. He is very grateful to this institution, to Professor M. Chipot and the ICTP (Trieste, Italy) where a part of this work has been done. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

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