No abstract

No abstract

Abstract. The paper deals with local existence, blow-up and global existence for the solutions of a wave equation with an internal nonlinear source and a nonlinear boundary damping. The typical problem studied is; 1Þ Â À 1 ; uð0; xÞ ¼ u 0 ðxÞ; u t ð0; xÞ ¼ u 1 ðxÞ on ;where & R n (n ! 1) is a regular and bounded domain,1 Þ, ! 0, and the initial data are in the energy space. The results proved extend the potential well theory, which is well known when the nonlinear damping acts in the interior of , to this problem.

Abstract. The aim of this paper is to study the initial and boundary value problemwhere Ω is a bounded regular open domain in R N (N ≥ 1), Γ = ∂Ω, ν is the outward normal to Ω, and k < 0. In particular we prove that the problem is ill-posed when N ≥ 2, while is well-posed in dimension N = 1. Moreover we carefully study the case when Ω is a ball in R N . As a byproduct we give several results on the elliptic eigenvalue problem −∆u = λu, in Ω, ∆u = ku ν on Γ.

This paper deals with the heat equation posed in a bounded regular domain Ω of R N (N 2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problemLaplacian operator with respect to the space variable, while Γ denotes the Laplace-Beltrami operator on Γ , ν is the outward normalto Ω, and k and l are given real constants, l > 0. Well-posedness is proved for data u 0 ∈ H 1 (Ω) such that u 0|Γ ∈ H 1 (Γ ). We also study higher regularity of the solution.