Enzo Vitillaro

GLOBAL EXISTENCE FOR THE WAVE EQUATION WITH NONLINEAR BOUNDARY DAMPING AND SOURCE TERMS

Abstract The paper deals with local and global existence for the solutions of the wave equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is u tt − Δ u=0 in (0,∞)×Ω, u=0 on [0,∞)×Γ 0 , ∂u ∂ν =−|u t | m−2 u t +|u| p−2 u on [0,∞)×Γ 1 , u(0,x)=u 0 (x), u t (0,x)=u 1 (x) on Ω, where Ω⊂ R n (n⩾1) is a regular and bounded domain, ∂Ω=Γ 0 ∪Γ 1 , m >1, 2⩽ p r , where r =2( n −1)/( n −2) when n ⩾3, r =∞ when n =1,2, and the initial data are in the energy space. We prove local existence of the solutions in the energy space when m > r /( r +1− p ) or n =1,2, and global existence when p ⩽ m or the initial data are inside the potential well associated to the stationary problem.

A potential well theory for the wave equation with nonlinear source and boundary damping terms

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 \cases{u_{tt}-\Delta u=|u|^{p-2}u \hfill & \rm{~}OPEN~7~in [0,\rm{inf}ty )\times \Omega ,}\hfill \cr u=0 \hfill & \rm{~}OPEN~6~on [0,\rm{inf}ty )\times \Gamma _0,}\hfill \cr \frac {\partial u}{\partial \nu }=-\alpha (x)|u_t|^{m-2}u_t \hfill & \rm{~}OPEN~2~on [0,\rm{inf}ty )\times \Gamma _1,}\cr u(0,x)=u_0(x),u_t(0,x)=u_1(x) & \rm{~}OPEN~1~on \Omega ,}\hfill } where \Omega \subset R^n ( n\ge 1 ) is a regular and bounded domain, \partial \Omega =\Gamma _0\cup \Gamma _1 , \lambda _{n-1}(\Gamma _0)>>;0 , 2<>;p\le 2(n-1)/(n-2) (when n\ge 3 ), m>>;1 , \alpha \in L^\rm{inf}ty (\Gamma _1) , \alpha \ge 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 \Omega , to this problem.

Heat Equation with Dynamical Boundary Conditions of Reactive Type

The aim of this paper is to study the initial boundary problem where Ω is a bounded regular open domain in ℝN (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 it is well-posed in dimension N = 1. Moreover we carefully study the case when Ω is a ball in ℝN. As a byproduct we give several results on the elliptic eigenvalue problem

Heat equation with dynamical boundary conditions of reactive–diffusive type

This paper deals with the heat equation posed in a bounded reg- ular domain of R N (N � 2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem 8 utu = 0 in (0,1) × , ut = ku� + lu on (0,1) × , u(0,x) = u0(x) on , where u = u(t,x), t � 0, x 2 , = @, � = �x denotes the Laplacian operator with respect to the space variable, whiledenotes the Laplace- Beltrami operator on , � is the outward normal to , and k and l are given real constants, l > 0. Well-posedness is proved for data u0 2 H 1 () such that u0| 2 H 1 (). We also study higher regularity of the solution.

On the laplace equation with non-linear dynamical boundary conditions

The main part of the paper deals with local existence and global existence versus blow-up for solutions of the Laplace equation in bounded domains with a non-linear dynamical boundary condition. More precisely, we study the problem consisting in: (1) the Laplace equation in

Global existence for the heat equation with nonlinear dynamical boundary conditions

(0, \infty) \times \Omega

WAVE EQUATION WITH SECOND-ORDER NON-STANDARD DYNAMICAL BOUNDARY CONDITIONS

; (2) a homogeneous Dirichlet condition

On the Wave Equation with Hyperbolic Dynamical Boundary Conditions, Interior and Boundary Damping and Source

(0, \infty) \times \Gamma_0

Asymptotic stability for abstract evolution equations and applications to partial differential systems

; (3) the dynamical boundary condition

Periodic solutions with prescribed energy for some Keplerian N-body problems

\frac {\partial u}{\partial \nu} = - |u_t|^{m-2} u_t + |u|^{p - 2} u