On a Nonlinear Degenerate Evolution Equation with Nonlinear Boundary Damping

This paper deals essentially with a nonlinear degenerate evolution equation of the form supplemented with nonlinear boundary conditions of Neumann type given by . Under suitable conditions the existence and uniqueness of solutions are shown and that the boundary damping produces a uniform global stability of the corresponding solutions.


Introduction
Let Ω be a smooth bounded open set of R  , with  ≥ 1, and its boundary Ω = Γ of class  2 .Assume that Γ is constituted by two disjoint closed parts Γ 0 and Γ 1 both with positive Lebesgue measure.
The parabolic-hyperbolic equation   − Δ( −   ) + () = 0 when  = 1 or  = 2; this equation governs the motion of a nonlinear Kelvin solid.That is, a bar for  = 1 and a plate for  = 2, subject to no nonlinear elastic constraints.The function  represents the mass density of the solid.
The existence of solutions of the linear problem associated with () ( = 1,   = 0, and without the function () = ||   and with ℎ(, ) = ()) was established by Komornik and Zuazua in [1], via semigroup theory and by Milla Miranda and Medeiros in [2], applying the Galerkin's method, with a special basis.The advantage of this second method is to define the Sobolev space where /] is lying.In the same context, applying this second method for a wave equation with a nonlinear term, Araruna and Maciel [3], derive similar results.In Cavalcanti et al. [4] the existence of solution and an exponential decay rate is established supposing  = 0 and ℎ being a particular function considered in our work; see also Cavalcanti et al. [5].
For the wave equation with  = 1 and  = 0 there is a vast literature on this problem.We cite the papers Cavalcanti et al. [6], Lasiecka and Tataru [7], and references contained therein for the reader.
From spectral theory it follows that (−Δ) is dense in ; see [15].Moreover, it will be denoted Theorem 1. Assume hypotheses (H1)-(H6); there exists at least a function satisfying In addition, if and further Theorem 1 remains valid.Indeed, from (10), we obtain where (  ) is the Strauss' approximations (see [16]) of the function . for all  ∈ R (see [10]).
For use later, note that where  1 is the first eigenvalue of the spectral problem ((, V)) = (, V) for all V ∈  (see [15]).
Theorem 6.Under the hypothesis of Theorem 1 and (H7)-(H11), with ℎ 1 () satisfying ( 9) and (10), the energy associated with the solution, , obtained in Corollary 2 is uniformly stable; that is, there exists a positive constant such that where  and  are positive constants.

Journal of Applied Mathematics
Here,  →  indicates that the subspace  is continuously embedded in the space .
Next, following the ideas contained in Strauss [16], we approximate the function ℎ by Lipschitz-continuous ones ℎ  .
From ( 25) to (28) we have the results of this Lemma.
Comparing ( 54) and (56) and using the Lipschitz property of ℎ  , we obtain 3.1.5.Passage to the Limit in  → 0 and  → ∞.As estimates (36) and (49) are independent of , , and  we obtain a subsequence of (  ), which still denoted by (  ), and a function   such that all convergences (49) and (52) are valid.These convergences will be denoted by (49)  , (51)  , and (52)  , respectively.These results imply that there exists a function   belonging to class (49) and it is a solution of equation integrating the inequality above from 0 to , and using the hypothesis (H5) and (H6), we find where  > 0 is a constant independent of  ≥  0 and  ∈ [0, ].
The verification of the initial conditions follows by convergence (49)  .
The proof of Corollary 2 follows from Remark 10 and from regularity of elliptic problems (see Lions and Magenes [22]).

Asymptotic Behavior
In this section, by applying Nakao's method (see [23]), we will prove the uniform stabilization of the energy associated with the solution of the Problem ().
Proof of Theorem 6.First, we prove the inequality (15) for the approximate energy   () given by and Theorem 6 will follow by taking the lim inf in .