In this work we study the existence of global solutions and exponential decay for the following nonlinear problem:

No abstract

The present paper studies the existence and uniqueness of global solutions and decay rates to the nonlinear hyperbolic problemwhere M is a C 1 function; M (λ) ≥ 0; ∀λ ≥ 0.

The purpose of this article is to study the existence and uniqueness of quasi-Einstein structures on 3-dimensional homogeneous Riemannian manifolds. To this end, we use the eight model geometries for 3-dimensional manifolds identified by Thurston. First, we present here a complete description of quasi-Einstein metrics on 3-dimensional homogeneous manifolds with isometry group of dimension 4. In addition, we shall show the absence of such gradient structure on Sol 3 , which has 3-dimensional isometry group. Moreover, we prove that Berger's spheres carry a non-trivial quasi-Einstein structure with non gradient associated vector field, this shows that a theorem due to Perelman can not be extend to quasi-Einstein metrics. Finally, we prove that a 3-dimensional homogeneous manifold carrying a gradient quasi-Einstein structure is either Einstein or H 2 κ × R.

Existence and uniform decay of solutions of a mixed problem based on the degenerate equation #iOM)2/tt + K2(x,t)y t -A x y = 0 are studied. Under the assumptions that we have a nonlinear boundary damping (1 + a(t) \yt\ p )yt and a boundary source term of type a(£)|2/| 7 t/ 5 we establish the global existence theorem provided p > 7 and we obtain the uniform decay of strong and weak solutions considering p = 7 and the coefficient a(t) producing a damping effect.