Asymptotic behavior of global entropy solutions for nonstrictly hyperbolic systems with linear damping

In this paper we investigate the large time behavior of the global weak entropy solutions to the symmetric Keyftiz-Kranzer system with linear damping. It is proved that as t tends to infinite the entropy solutions tend to zero in the L p norm

(1. 1) with initial data This system models of propagation of forward longitudinal and transverse waves of elatic string wich moves in a plane, see [1], [3]. General source term for the system (2.8) was considered in [6]. The damping in the system (2.8) represents external forces proportional to velocity, and this term can be produce lost of total energy of system. Consider the scalar case,by example From the integral representation of (1.3) it is easy to find the following solution u(x, t) = u0(x − at)e −bt . (1.4) The term bu produce a dissipative effect in the solutions, i.e, the solutions tends to zero when t → ∞. We are looking for condition under wich the terms a, b have a dissipative efect in the solutions of 2.8. Let r(x, t) = u(x, t) 2 + v(x, t) 2 be, we are going to show the following main theorem.
then the Cauchy problem (2.8), ( 1.2) has a weak entropy solutions satisfaying Moreover r(u, v) converges to zero in L p with exponential time decay, i.e.

Preliminars
We start with some preliminaries about the general systems of conservation laws, see [2] chapter 5. Let f : Ω → R n be a smooth vector field. Consider Cauchy problem for the system When g(u) = 0 the system (2.1) is called homogeneous system of conservation laws, if g(u) = 0 the system (2.1) is called inhomogeneous system or balance system of consevation laws. We shall work also with the parabolic perturbation to the system (2.1), namely Denote by A(u) = Df (u) the Jacobian matrix of partial derivates of f .
Let ri(u) the correspond eigenvetor to λi(u), then Definition 2.2. We say that the i-th characteristic field is genuinely we say that the i-th characteristic field is linearly degenerate.
For the following definitions see [5], [7] if η(u) is a convex function then the pair (η, q) is called convex entropyentropy flux pair.
Definition 2.5. A bounded measurable function u(x, t) is an entropy (or admisible) solution for the Cauchy problem (2.1), if it satisfies the following inequality in the distributional sense, where (η, q) is any convex entropy-entropy flux pair.
We consider the general system of Keyftiz-Kranzer system to get some general observations about this type of systems. Making F (u, v) = (uφ(u, v), vφ(u, v) in (2.8), we have that the eigenvalues and eigenvector of the Jacobian's matrix Df are given by (2.10) From (2.9),(2.10) we have that ∇φ · r1 = 0, and ∇Z(u, v) · r2 = 0, where Z(u, v) = u v , then the Riemann invariants are given by (2.12) Lemma 2.6. The system (2.8) is always linear degenerate in the first characteristic field. If (u, v)∇φ(u, v) = 0, then the system (2.8) is strictly hyperbolic and non linear degenerate in the second characteristic field, moreover where H represents the Hessian matrix.

Global existence of weak entropy solutions and asymptotic behavior
We consider the parabolic regularization of the system (2.8), namely ut + (uφ(r))x + au = ǫuxx, vt + (vφ(r))x + bv = ǫvxx, where jǫ is a mollifier. In this case φ(u, v) = φ(r), with r = √ u 2 + v 2 . By (2.9) the eigenvectors and eigenvalues are given by The following conditions will be nesessaries in our next discution C1 limr→o rφ(r) = 0, rφ The condition C1 garanties the strictly hyperbolicity to the system (3.2), while condition C2 ensure the existence of a positive invarian region. Now we consider the following subset of R We affirm that Σ is an invariant region. Let h(u, v) = (au, bv) be, if (u, v) ∈ γ1 where γ1 is the level curve of Z = φ(r) we have that with i = 1, 2., then by the Theorem 14.7 of [5], Σ is an invariant region for the system (3.1).Is easy to verify that (au, bv) satisfies the condition H1 · · · H5 in [6], thus we have the following Lemma.

Acknowledgments
We would like to thanks to professor Laurent Gosse by his suggestions and review. To the professor Juan Galvis by his many valuable observation, and to the professor Yun-Guang Lu by his suggestion this problem.