On a Kirchhoff wave model with nonlocal nonlinear damping

This paper is concerned with the well-posedness as well as the asymptotic behavior of solutions for a quasi-linear Kirchhoff wave model with nonlocal nonlinear damping term \begin{document}$ \sigma\left(\int_{\Omega}|\nabla u|^2\,dx\right )g(u_t), $\end{document} where \begin{document}$ \sigma $\end{document} and \begin{document}$ g $\end{document} are nonlinear functions under proper conditions. The analysis of such a damping term is presented for this kind of Kirchhoff models and consists the main novelty in the present work.

1. Introduction. In this paper we address well-posedness and long-time behavior to the following quasi-linear Kirchhoff wave model with nonlocal nonlinear damping u tt − φ( ∇u(t) 2 2 )∆u + σ ∇u(t) 2 2 where Ω ⊂ R n is a bounded domain with smooth boundary Γ, φ and σ are scalar functions defined on R + = [0, +∞), f and g are nonlinear functions on R corresponding to source and damping terms, and h is a external force. Here, · 2 stands for L 2 -norm. The precise assumptions on φ, σ, f and g shall be given later. The following initial-boundary conditions are considered: u(·, 0) = u 0 (·), u t (·, 0) = u 1 (·) in Ω and u = 0 on Γ × R + .
The main object of study of this paper is to analyze the existence and asymptotic behavior of solutions for (1)- (2) under the influence of the nonlocal nonlinear damping term σ ∇u(t) 2 2 g(u t ).
(3) Since it consists in a product of two nonlinear terms, which is unusual of seeing for these kind of models, then it deserves some attention from mathematical viewpoint as we shall explain later. Before, let us consider some existing literature on the subject dealing with Kirchhoff (wave) models with nonlocal damping term.
In this direction, a first paper we found is due to Lazo [12] which considers the following abstract model

VANDO NARCISO
where A is an operator defined in a real Hilbert space (H, · ). Only existence of weak solutions for (4) is obtained by the author in [12] under proper conditions on φ and σ, and considering α = θ. The damping term in (4) is just nonlinear in one of its components. Besides, one of the main goals in [12] is to generalize the results of Medeiros and Milla Miranda [16] which studied equation (4) in the particular case σ ≡ 1. In such a case the dissipation becomes to the linear one and it is proved in [16] that uniqueness holds only when θ varies the range 1 2 ≤ θ ≤ 1. Global existence and exponential decay of the energy are also addressed by the authors in [16].
Chueshov [3] studied the well-posedness and long-time dynamics of solutions for (4) by introducing a source term f (u) and taking α = 1 2 and 1 2 ≤ θ < 1. Such a choice for θ allowed the author in [3] to consider existence and uniqueness of weak solutions as well as existence of finite-dimensional compact global and exponential attractors. Later, in Chueshov [4] is considered the strong case θ = 1. More precisely, it is studied in [4] the following Kirchhoff wave model with nonlocal and nonlinear strong damping The strength of the nonlocal and nonlinear strong damping in (5) is capable to produce sufficient regularity in order to show the existence and uniqueness of weak solution as well as its regularizing property for a wide class of nonlinear source terms f (u) with supercritical growth and a (possible) degenerate stiffness coefficient φ. Moreover, global attractor and its properties are considered for strictly positive stiffness factors. See e.g. Chueshov [4]. See also the papers [8,23,24,26,27] where equation (5) is considered with linear strong damping −σ∆u t , σ ≡ σ 0 > 0. From the above mentioned works, e.g. [3,4,12,16], one can see that it is unlikely to consider a nonlocal weak damping term σ ∇u(t) 2 2 u t (6) or a nonlocal nonlinear damping term such as (3) in the equations (4) and (5) in order to conclude global well-posedness of (weak and strong) solutions. Indeed, as remarked in [3,4], uniqueness for (4) with 0 ≤ θ < 1 2 is still an open problem. We also note that the damping given in (6) corresponds to the case θ = 0 in (4) and the (standard) case of (3) with g(s) = s. Motivated by these problems our main goal in this paper is to analyze the well-posedness and long-time behavior of (strong) solutions for (1)-(2) by keeping both nonlinearities σ and g in (3). However, for the global existence of solutions we must pay a tribute of taking small initial data. Our conditions on g (which shall be given later) allow us to consider damping terms like What differentiates this work from others is that (3) consists in a product of the two nonlinearities σ ∇u(t) 2 2 and g(u t ). In this case, it is not possible to apply same arguments as used in [3,4] in order to prove uniqueness of solutions for initial data in the natural weak energy space H = H 1 0 (Ω) × L 2 (Ω). Indeed, a uniqueness result is only known for Kirchhoff wave models with nonlocal (strong) damping g(u t ) := (−∆) θ u t with 1 2 ≤ θ ≤ 1. But this is not our situation and then it seems to be a hard task to get the existence of global attractors for (1)-(2) on H. This damping given by the product of two nonlinearities was first used by Jorge and Narciso in [7] for a class of extensible beams.
However, in spite of all difficulties produced by the quasi-linear model (1) under the presence of the nonlocal nonlinear damping (3), we prove in this work that (1)-(2) is globally well posed on the strong phase space for small initial data. In addition, we study the asymptotic behavior of its solution on H 1 . To our best knowledge there are only a few works in this direction. For instance, one of them we found is due to Nakao [22] which considers the existence of a local attractor to the following Kirchhoff wave model with linear weak damping It is worth noting that model (7) is a particular case of (1) with φ(s) = 1 + s, σ ≡ 1, and g(s) = s. For this particular model (7) subject to initial-boundary conditions in (2) it is proved in [22] the existence of an attractor A for the corresponding semigroup S(t) in a neighborhood V of (0, 0) in H 1 , namely, A ⊂ V is compact, invariant and attracts any bounded set B ⊂ V . A related study on local attractor is also given by Nakao [21].
The main results of this paper are Theorem 2.1, Theorem 3.1 and Theorem 4.2. To their proofs new estimates and arguments are required in view of the damping term (3). When compared to existing results on the subject we note that (3) allows us to address the case θ = 0 in (4) that complements the case not approached in [3,4,12,16]. Moreover, in view of our conditions on the functions φ, σ and g we also extend all results provided by Nakao [22] with respect to nonlocal damping and stiffness factor.
We finish this section by noting that (1) is a generalized equation of the classical model of vibrating strings given by equation which was introduced by Kirchhoff [9] in 1883, where u = u(x, t) is the lateral displacement, m is the mass density, τ 0 is the initial tension, L is the length of the string in the rest position, E is the Young's modulus of the material of the string and I the area of the cross section. After that, all kinds of generalizations of the Kirchhoff wave model (8) have been studied by many authors. Existing literature is truly long, see e.g. [1,2,5,6,10,11,15,17,18,19,24,25,28,29] and references therein. A suitable survey on precise references dealing with Kirchhoff wave models like (1) with σ ≡ 1 (and also φ ≡ 1) can be found in Chueshov [3,4] and Nakao [20,21,22]. The remaining paper is organized as follows. In section 2 we introduce some initial assumptions and local existence of solution for (1)- (2). In section 3 we show the global well-posedness of solution under suitable additional assumptions. Finally, in Section 4 we establish our result on asymptotic behavior.
2. Local existence. We start this section by fixing some notations and assumptions that shall be used throughout this paper. Spaces L p (Ω) stand for p-Lebesgue integrable functions with norm and W m,p (Ω) or W m,p 0 (Ω) denote well-known Sobolev spaces. In particular, if p = 2, then L 2 (Ω) is a Hilbert space with inner-product and norm (Ω) := H m 0 (Ω). In the special case H 1 0 (Ω) we have, in view of the Poincaré inequality, the following inner-product and norm . We also set the following Hilbert phase spaces In order to consider the Hadamard well-posedness to the problem (1)-(2) in the strong topology of H 1 , we assume the following hypotheses on φ, σ, f and g. (H1) The stiffness factor φ ∈ C 1 (R + ) satisfies for some constant φ 0 > 0, where we denote φ(s) : where we consider C f > 0 and the growth ρ satisfying In addition, for some η ∈ (0, λ 1 ) and C f ≥ 0, where λ 1 > 0 is the first eigenvalue of −∆, we assume that where f (s) := s 0 f (τ )dτ. (H4) The damping g ∈ C 1 (R) satisfies for some κ 1 , κ 2 > 0, and γ ≥ 0.
Remark 1. Condition (11) implies that W m+1,2 (Ω) → W m,2(ρ+1) (Ω). So we denote by C m+1,ρ > 0 the embedding constant for We also observe the choice for η implies that Theorem 2.1. Let us assume that assumptions (H1)-(H4) hold and take h ∈ H 1 0 (Ω). If (u 0 , u 1 ) ∈ H 1 , then there exists a T > 0 such that the problem (1)-(2) has a unique solution u = u(x, t) in the class Proof. The proof on existence relies on Faedo-Galerkin method which was extensively applied in the study of wave equations, see for instance Lions et al. [13,14]. Let (ω j ) j∈N be eigenfunctions of the following problem From elliptic regularity we have ω j ∈ H 2 (Ω) ∩ H 1 0 (Ω), j ∈ N. Let V m be a subspace generated by the first m vectors ω 1 , · · · , ω m . For each m ∈ N, we can find a function which is a solution to the approximate problem by using standard methods in ODE. In what follows, the first estimate shall allow us to extend the local solution to the interval [0, T ], for any given T > 0. The remaining estimates shall allow us to conclude the proof of Theorem 2.1.
A Priori Estimate I. Taking ω j = u m t (t) in the approximate problem (15) yields where From assumption (9) we have Using assumption (12), Hölder inequality and u m 2 Combining (17)- (19) and using assumption (14), we obtain On the other hand, using assumption (13) we have
Therefore, as a consequence of (53), we conclude that problem (1)-(2) has a unique "local" strong solution (u, u t ). The proof of Theorem 2.1 is now complete.
Remark 2. The same conclusion of Theorem 2.1 can be obtained if we replace condition (H3) by the following: (H5) f ∈ C 2 (R) is a monotonous nondecreasing function such that f (0) = 0, where C f > 0 and ρ satisfies It is worth noting that assumption (H5) is stronger than (H3). Here assumption (55) implies that H 1 0 (Ω) → L ρ+2 (Ω). It allows us to work with proper estimates on the second derivative of f and Hölder inequality, and recover assumption (12) with C f = 0. The proof relies on similar arguments which shall be given later in Theorem 3.1.

Invariant set and global existence. The energy functional E(t) associated with problem (1)-(2) is given by
We also set the modified energy is the local in time solution in Theorem 2.1.
Proof. First of all, we note that E(t) is non-negative. Indeed, from assumptions (9), (12) and Hölder inequality, it is easy to see that where ϑ = min 1 2 , φ0η0

4
. Taking the multiplier u t with (1) and using (13) we obtain d dt Let R > 0, we define B R by If (u 0 , u 1 ) ∈ B R ∩ H 1 , using that d dt E(t) = d dt E(t), we have from (60) and (61) that Therefore, (u(t), u t (t)) ∈ B R for all t ∈ [0, T ). This completes the proof of Proposition 1.

Remark 4.
In what follows, under suitable additional assumptions, we are going to show that there exists a "small" invariant set B ⊂ H 1 such that the local solution given by Theorem 2.1 can be extended to a global one for initial data belonging to B. More precisely, one of the additional assumptions to be considered is of the form g (0) > 0. With this hypothesis on g, one sees that it is not superlinear at the origin, which rules out interesting cases such as those considered in [10,11] for wave models with nonlinear boundary dissipation. However, due to the difficulties caused by the nonlinear nonlocal damping, it seems to be a hard task to apply the same ideas as introduced in [10,11] on damping g at the origin in the present problem. Summarizing, the next condition assumed on g is much weaker than that in [10,11].
More precisely, we have: Proof. We first note that assumptions of Theorem 3.1 are enough to conclude all statements made in Theorem 2.1 and Proposition 2. In addition, we perform formally the computations below since they hold for Galerkin approximate solutions and so for their limits. Let us consider (u 0 , u 1 ) ∈ B R ∩ H 1 with B R given in (62). From Proposition 1 we have (u(t), u t (t)) ∈ B R , 0 ≤ t < T . Moreover, from (60) we also get Deriving equation (1) with respect to variable t we have Taking the multiplier u tt with (79) we obtain Since g is continuous and we are additionally assuming that g (0) > 0, then there exists δ > 0 such that g (s) > 0 for all |s| < δ, which implies in the existence of a constant κ 3 > 0 such that Moreover, using (13) in particular on R\I δ , we also have Thus, taking κ = min{κ 3 , κ 1 |δ| γ }, we conclude from the above that Now, using again (13), estimate (24) (with u instead u m ) and (81), we obtain dx.