On a Beam Equation in Banach Spaces

This paper is concerned with the existence and asymptotic behavior of solutions of the Cauchy problem for an abstract model for vertical vibrations of a viscous beam in Banach spaces. First is obtained a local solution of the problem by using the method of successive approximations, a characterization of the derivative of the non-linear term of the equation defined in a Banach space and the Ascoli–Arzelà theorem. Then the global solution is found by the method of prolongation of solutions. The exponential decay of solutions is derived by considering a Lyapunov functional.


Introduction
The small transverse vibrations due to flexion of an extensible beam, of length L, whose ends are held at fixed distance apart can be described by the following equation where 0 < x < L and t > 0.Here u(x, t) denotes the displacement of the point x of the beam at the instant t and σ, m 0 and m 1 are positive constants.The nonlinear term indicates the change in the tension of the beam due to its extensibility.Equation (1.1) was introduced by Woinowsky-Krieger [28].Equation (1.1) with σ = 0 describes the small transverse vibrations of an elastic stretched string of length L. This equation was introduced by Kirchhoff [16].Analyzing the same phenomenon, Carrier [7] obtained the following model: Let Ω ⊂ R n be a bounded open set of R n .A generalization of (1.1) and (1.2) is the following equation: u (x, t) + σ(−∆) 2 u(x, t) + m 0 + m 1 Ω |(−∆) α u(x, t)| 2 dx (−∆u(x, t)) = 0, (1.3) where x ∈ Ω, t > 0 and 0 ≤ α ≤ 1.
An abstract formulation for a mixed problem of equation (1.3) is the following: where M(ξ) is a smooth function satisfying M(ξ) ≥ m 0 > 0, A is an unbounded self-adjoint operator of a real separable Hilbert space H with A coercive and A −1 compact.Here σ and α are real numbers such that σ ≥ 0 and 0 ≤ α ≤ 1.
The existence of a global solution of (1.4) was obtained by Medeiros [22].The decay of solution with a dissipation in the equation of (1.4) was studied by [3-5, 9, 25].
In the above papers the Faedo-Galerkin method is used.The study of hyperbolic problems using the theory of semigroups can be seen in J. A. Goldstein [13] and [12] for the linear and nonlinear case, respectively.
In Izaguirre et al. [14] is formulated problem (1.4), with σ = 0, in the context of Banach space.More precisely, they consider the problem where V is a real separable Hilbert space with dual V ; A, B : V → V are two positive linear symmetric operators with A −1 and B −1 not neccessarily compact; W is a real Banach space such that V is continuously embedded in W and β is a real number with β > 1.They obtain a local solution for (1.5).Also, with the introduction of the damping δBu (t), δ > 0, in the equation of (1.5), Izaguirre et al. [15] obtain a global solution and exponential decay of the energy for (1.5).
Considering B ≡ I and introducing the expression F(u) + (1 + α u β )Au in the problem (1.5), where F is an operator and α > 0, β ≥ 2, Araruna and Carvalho [1] studied the existence of the global solution, uniqueness and exponential decay.
Motivated by (1.4) and (1.5), we formulate the following problem: Note that the nonlinear term M( u 2 D(A α ) )Au of (1.4) is a particular case of the nonlinear term M( u β W )Au of (1.6) since the Hilbert space D(A α ) is a particular case of the Banach space W. Thus (1.6) generalizes (1.4).
The results of [22] are obtained in the framework of Hilbert spaces and under the hypothesis A −1 a compact operator.We want to work in the framework of Banach spaces and where A −1 is not necessarily compact, therefore the results of [22] do not apply in our case.
In our approach, we need to obtain two a priori estimates but we cannot differentiate two times with respect to t the term u(t) β W , β > 1.To overcome this difficulty we introduce a strong dissipation in equation (1.6), more precisely, we consider where M and K are functions satisfying suitable conditions.It is possible to solve problem (1.6) with a weak internal dissipation δu , δ > 0, but in this case we obtain only solutions under the condition that the initial data belong to a ball whose radius depends on δ.We are interested in obtaining global solutions of (1.6) without restrictions on the norms of the initial data.For this purpose, we consider the dissipation of (1.7).
The objective of this paper is to investigate the existence and asymptotic behavior of solutions of problem (1.7).The plan is as follows: first, with general functions M(ξ) and K(t), we obtain a local solution of (1.7).Then for particular M(ξ) and K(t) increasing in t, we get a global solution of (1.7).Finally, with K(t) = K positive constant and particular M(t, ξ) with M(t, ξ) decreasing in t, we derive a global solution of (1.7).This last solution decays exponentially in t.
To obtain a solution of (1.7), we proceed in the following way.First, by the successive approximation method, the characterization of the derivative of the nonlinear term M( u(t) β W ) and the Ascoli-Arzelà theorem, we obtain a local solution of (1.7).Then by the method of prolongation of solutions, we deduce the existence of a global solutions of (1.7).The exponential decay of the energy is derived by considering a Lyapunov functional (see V. Komornik and E. Zuazua [17] and V. Komornik [18]).In the last section, we give some examples.

Notations and results
Let V and H be two real Hilbert spaces whose scalar product and norm are represented, respectively, by ((u, v)), u and (u, v), |u|.Here H is separable.
Let us represent by A the unbounded self-adjoint operator of H defined by the triplet {V, H; ((u, v))}.We have (Au, u) ≥ γ 0 |u| 2 , ∀u ∈ D(A), where γ 0 is a positive constant (see Lions [20]).We consider the following hypotheses: V is densely and continuously embedded in H, W is a real Banach space with dual W strictly convex, (2.2) Consider the functions M(ξ) and K(t) satisfying and K ∈ L ∞ loc (0, ∞) and K(t) ≥ 0, a.e. in (0, ∞). (2.5) Under the above considerations, we have the following result.
In what follows, we introduce the real number T 0 > 0 mentioned in Theorem 2.1.In fact, consider u 0 and u 1 satisfying hypothesis (2.6).Take a real number N 2 > 0 such that
Then there exists a unique function u in the class (2.19), u solution of (P2) with M(t, ξ) given by (2.20) and K(t) is the function constant K. Furthermore, there exists a positive constant τ 0 such that Note that k 1 was defined in (2.9) and k 7 denotes the immersion constant of D(A

Proof of the results
In order to prove the results we need some previous propositions.
where J : W → W is the duality application defined by For the proof of the Proposition 3.1, see [14] and [15].Now consider the real functions µ 1 and µ 2 satisfying the following: and Proof.We apply the Faedo-Galerkin method.Let {w 1 , w 2 , . . .} be a Hilbert basis of H. Consider the basis PA) has a solution on a certain interval [0, t m ), which can be extended by the next priori estimates, over the interval [0, T] for all real number T > 0.

Estimates
Taking z j = 2u m (t) in (PA) 1 , we obtain Integrating the above equality from 0 to t, t ≤ t m and using (3.1) and (3.2), we get where C > 0 is a constant independent of m and t.Applying Gronwall's inequality in (3.3) and using (3.1), we obtain ∀t ∈ [0, t m ), t m ≤ T, where As a consequence of estimates (3.4), we deduce, respectively, the existence of a subsequence of (u m ) m∈N , still denoted by (u m ) m∈N , such that 2 )). (3.5) Now, multiplying the approximate equation (PA) 1 by θ ∈ D(0, T), integrating the result of 0 to T and using the convergences (3.5), we get Finally, using the diagonal process we obtain equality . By standard arguments, we verify the initial conditions and the uniqueness of the solutions.This concludes the proof of Proposition 3.2.

Proof of Theorem 2.1
A sketch of the proof of Theorem 2.1 is as follows.First, we approximate u 0 and u 1 by functions u 0 l and u 1 l belonging to D(A 4 ) and D(A 3 ), respectively.Then by Proposition 3.1 and 3.2 and the method of successive approximations, we determine the solution u l of the problem Estimates obtained for the solution u l allow us to pass to the limit in the equation in (P l ).The limit of the nonlinear terms follows by applying Proposition 3.1 and the Ascoli-Arzelà theorem for real functions.
In the sequel we will use the method of successive approximations to obtain the solution of problem (P l ).Thus, we consider the following problem: It follows from hypotheses (2.4), (2.5) and Proposition 3.2 that u l,1 belongs to class (3.13).Now taking the scalar product of H of both sides of the equation in (P l,1 ) with 2A 3 u l,1 , integrating this result on [0, t], 0 < t ≤ T 0 , using (3.12) and the hypothesis (2.4), we obtain that u l,1 satisfies (3.14).
Define the sequence (u l,ν ) ν≥2 , where u l,ν is the solution of the problem Using induction we shall prove that u l,ν satisfies the (3.13) and (3.14).In fact, assume that u l,ν−1 satisfies (3.13) and (3.14).Then, by Lemma 3.3, we have Also by Proposition 3.2, we derive that u l,ν belongs to class (3.13).
Taking the scalar product of H of both sides of equation (P l,ν ) 1 with 2A 3 u l,ν (t), applying similar arguments used to prove that u l,1 satisfies (3.14) and using the last inequality, we obtain Then by (3.12) we find Then thanks to the choice of T 0 , this inequality provides Thus u l,ν satisfies (3.13) and (3.14).
(3.16) Convergences (3.16) are not sufficient to pass to the limit in problem (P l,ν ) due to the nonlinear terms.Next we will prove that and Let us begin considering the sequence (ϕ l,ν ) ν∈N , where ϕ l,ν (t) = u l,ν−1 (t) β W .As a consequence of (2.12) and (3.14) it follows that Now using the mean value theorem, Proposition 3.1, (2.11), (2.12) and (3.14), we have Therefore from (3.19), (3.20) and the Ascoli-Arzelà theorem it follows that there exists Consequently we obtain from (3.21) and (2.4) the convergence Now let us consider the sequence (ψ l,ν ) ν∈N , where ψ l,ν (t In a similar way as in (3.22), we conclude that there exists a sequence For that, one proceeds as follows.Let u l,ν and u l,σ be the solutions of problems (P l,ν ) and (P l,σ ), respectively.Consider w σν = u l,σ − u l,ν .So w σν is the solution of the problem Taking the scalar product of H of both sides of the equation (P σν ) with 2A 2 w σν (t), we obtain t ∈ [0, T 0 ] .By Lemma 3.3, the first term of the second member of (3.24) can be bounded by As (M( u l,ν−1 (t) This inequality and (3.14) imply that the second term of the second member of (3.24) can be bounded by Aw σν (t) , ∀σ, ν ≥ ν 0 .In a similar way, the third term of the second member of (3.24)can be bounded by Integrating both members of (3.24) on [0, t] , 0 < t ≤ T 0, and taking into account the last four results, we obtain ∀σ, ν ≥ ν 0 , where C > 0 is a generic constant which is independent of σ and ν.
The last inequality and the Gronwall inequality imply that (A which provides convergence (3.18).Using (2.11) and the convergence (3.25) it follows that (u l,ν ) ν∈N is a Cauchy sequence in which implies the convergence  3.18), we can pass to the limit in (P l,ν ).The limit u l is a solution of problem (P l ).
Our next goal is to take the limit in problem (P l ).Write (3.14) with u l,ν and take the limit inf of both sides of this inequality.Then convergences (3.16) provide ess sup ∀l ∈ N.This implies that there exists a subsequence of (u l ), still denoted by (u l ), such that 2 )) 2 )) u l → u weak in L 2 (0, T 0 ; D(A 2 )). (3.27) In the sequel we will prove that and Let us consider two sequences (ϕ l ) l∈N and (ψ l ) l∈N , such that ϕ l (t) = u l (t) β W and ψ l (t) = A Thus hypothesis (2.4) and convergence (3.30) provide In the sequel, we will show that ϕ = u β W and ψ = A Taking the scalar product of H of both sides of the equation (P lk ) with 2A 3 w lk (t) and integrating the result into [0, t] , 0 < t ≤ T 0 , we obtain (3.33) By similar arguments used to bound the terms of the second member of (3.24) and using convergences (3.30) and (3.31), we obtain a.e. in (0, T 0 ), and for ε > 0, and ∀k, l ≥ l 0 .
Taking into account the last three inequalities in (3.33) , noting that in (0, T 0 ) and that the first three terms of the second member of (3.33) can be bounded by ε 2 , we find The last inequality and Gronwall's inequality imply that t ∈ [0, T 0 ] , ∀k, l ≥ l 0 , where C > 0 denotes a generic constant which is independent of l and k.
The last inequality implies that In view of Remark 2.3, it follows from (3.34) the following convergence: which implies convergence  3.29), we can pass to the limit in (P l ).The limit u is a solution of problem (P1) 1 and u verifies (2.7).Using a standard argument, we can verify the initial conditions (P1) 2 .
The uniqueness of the solution is proved by the energy method.In fact, we consider u and v in the conditions of the Theorem 2.1.Then w = u − v satisfies (P) Taking the scalar product in H of both sides of equation of (P) with 2Aw (t), we obtain By the mean value theorem, we get where ξ * is between the real numbers v(t) where k 5 denotes the immersion constant of D(A In similar way we obtain that Combining (3.37) with (3.38) and (3.39) and using the Lemma 3.3, we obtain t ∈ [0, T 0 ] , where C is a generic constant which is independent of u and v. Integrating (3.40) from 0 to t ≤ T 0 and noting that M(ξ) ≥ m 0 > 0 and w(0) = w (0) = 0, we have t ∈ [0, T 0 ] .Finally, applying Gronwall's inequality in (3.41), we obtain that |Aw(t)| = 0, for all t ∈ [0, T 0 ] , that is, u(t) = v(t), for all t ∈ [0, T 0 ].This concludes the proof of Theorem 2.1.

Proof of Theorem 2.5
Initially we consider the following problem: where (u 0 l ) l∈N and (u 1 l ) l∈N are sequences of D(A 4 ) and D(A 3 ), respectively.Consequently we have and 2 ).
(3.43)By Theorem 2.1, there exists a unique solution u l of (P l ) belonging to class (2.7), but Proposition 3.2 with µ 1 Fix l ∈ N. Let M l be the set constituted by the real numbers T > 0 such that there exists a unique solution u l of (P l ) belongs to class (3.44) (changing T 0l with T).By the preceding arguments it follows that M l = ∅.We denote by T max,l the supremum of the T ∈ M l .
Next we obtain estimates for the solution u l .Taking the scalar product of the H of both sides of equation in (P l ) with 2A 3 u l (t), t ∈ [0, T max,l ), and using the Proposition 3.1, we obtain t ∈ [0, T max,l ).Here we assume that u l (t) = 0.
Consequently, we obtain t ∈ [0, T max,l ), where C 1 is a constant independent of l and t.
Remark 3.7.Due the Proposition 3.1 is possible to obtain the inequality (3.50) even when u l (t) = 0. Now we will prove that T max,l is infinite ∀l ∈ N. Let us suppose that T max,l < ∞.Now consider a sequence of the real numbers (t ν ) such that 0 < t ν < T max,l with t ν → T max,l .By (3.51) and (3.52) we obtain, respectively, that there exists ξ ∈ D(A 2 ) and u l (t ν ) → η weak in D(A ).
With ξ and η we determine, by Theorem 2.1, the local solution of the problem We note that the function . This is a contradiction with the definition of T max,l .So T max,l is infinite.Consequently we obtain from (3.50) that ∀t ∈ [0, ∞[ , where C 1 is a constant independent of l and t.By arguments similar to those employed in the proof of Theorem 2.1, we obtain the convergences  3.57), we can pass to the limit in (P l ).The limit u is a solution of problem (P2) 1 and u verifies (2.19).Using a standard argument, we can verify the initial conditions (P2) 2 .The uniqueness of solution is obtained as is the proof of Theorem 2.1.So Theorem 2.5 is proved.

Proof of Theorem 2.6 i) Existence of solutions
Let us begin by showing that problem (P2) possesses solution in the class (2.19) when we consider the hypothesis (2.20) instead of the hypothesis (2.17) and when we consider a function K(t) satisfying hypothesis K(t) = K.As the calculations are similar, we will obtain only the estimates.
Initially we consider the following problem: where (u 0 l ) l∈N ⊂ D(A 4 ) and (u 1 l ) l∈N ⊂ D(A 3 ).Consequently we obtain the convergences (3.42) and (3.43).With the same arguments as were used in the proof of Theorem 2.5, we have t ∈ [0, T max,l ).
It results from (2.11) and of the fact that β ≥ 2 the following:

.59)
Now let us note that where k 6 is the immersion constant of D(A 2 ) into D(A that is, (3.61) Integrating (3.61) from 0 to t, t < T max,l , using the hypothesis (2.20) and convergences (3.42) and (3.43), we derive where C > 0 is a generic constant independent of l and t.
As m 1 ≤ 0 and m 1 ∈ L 1 (0, ∞) (see (2.20)), it results from (3.62) and Gronwall's inequality t ∈ [0, T max,l ), where C > 0 is a generic constant independent of l and t.Note that with (3.63) we derive similar estimates to (3.51) and (3.52) for u l .With (3.63) and similar arguments used in the proof of Theorem 2.5 we obtain that T max,l is infinite and that u is the solution of (P2) in the class (2.19).

ii) Decay of solutions
Take the scalar product of the H of both sides of equation in (P2) with 2Au (t) and use (2.21), Proposition 3.1 and hypothesis (2.20).We obtain where E(t) was defined in (2.21).Here u(t) = 0.
It follows from (2.9), (2.20) and the inequality ab where k 7 is the immersion constant of D(A where M 1 (ξ) and M 2 (ξ) are similar to M(ξ) of problem (P2) and σ ≥ 0 is a real number.Also with our techniques it is possible to solve problem (P ) replacing M 1 ( u(t) 2 ) by M 1 (|u(t)| 2 ), where • and |•| denote the norms of V and H, respectively.Remark 3.9 also remains valid in these cases.
3 o ) As A −1 is not necessarily compact, we can consider problems defined on Ω × ]0, ∞[ with Ω an unbounded smooth open set of R n .

3 2
u l (t) β .Then by applying arguments similar to those used to obtain (3.21) and (3.23), we get two functions ϕ, ψ ∈ C 0 ([0, T 0 ]) such that us begin considering u l and u k two solutions of problems (P l ) and (P k ), respectively.Consider still w lk = u l − u k .So w lk is the solution of the problem

βW
and u(t) β W and s * is between the real numbers v(t) W and u(t) W .By R given in (2.15), inequality(3.14)and (2.9), we find