1,720,977 research outputs found
Local Hadamard well-posedness and blow-up for reaction-diffusion equations with non-linear dynamical boundary conditions
No full textAbstract on my talk on special session 5
Global existence for the heat equation with nonlinear dynamical boundary conditions
No full textThis paper deals with local and global existence for the solutions of the heat equation in bounded domains with nonlinear boundary damping and source terms. The typical problem studied is
u_t − Δu = 0 in (0, ∞) × Ω,
u = 0 on [0, ∞) × Λ_0,
∂u/∂v = −|u_t|^{m−2}u_t + |u|^{p−2}u on [0, ∞) × Λ1,
u(0, x) = u_0 (x) on Ω,
where Ω ⊂ R^n (n ≥ 1) is a regular and bounded domain, ∂Ω = Λ_ ∪ Λ_1, m > 1, 2 ≤ p r/(r + 1−p) or n = 1, 2 and global existence when p ≤ m or the initial datum is inside the potential well associated to the stationary problem
On the heat and Laplace equations with dynamical boundary conditions of reactive – diffusive type
No full textSlides of my tal
On the wave equation with Second Order Dynamical Boundary Conditions
No full textAbstract of my talk on Special Session 1
The damped wave equation with acoustic boundary conditions and non-locally reacting surfaces
No full textThe aim of the paper is to study the problem
⎧⎩⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪utt+dut−c2Δu=0μvtt−divΓ(σ∇Γv)+δvt+κv+ρut=0vt=∂νu∂νu=0u(0,x)=u0(x),ut(0,x)=u1(x)v(0,x)=v0(x),vt(0,x)=v1(x)inR×Ω,onR×Γ1,onR×Γ1,onR×Γ0,inΩ,onΓ1,
where Ω
is a open domain of RN with uniformly Cr boundary (N≥2, r≥1), Γ=∂Ω, (Γ0,Γ1) is a relatively open partition of Γ with Γ0 (but not Γ1) possibly empty. Here divΓ and ∇Γ denote the Riemannian divergence and gradient operators on Γ, ν is the outward normal to Ω, the coefficients μ,σ,δ,κ,ρ are suitably regular functions on Γ1 with ρ,σ and μ uniformly positive, d is a suitably regular function in Ω and c is a positive constant. In this paper we first study well-posedness in the natural energy space and give regularity results. Hence we study asymptotic stability for solutions when Ω is bounded, Γ1 is connected, r=2, ρ is constant and κ,δ,d≥0
Global existence for the wave equation with nonlinear boundary damping and source terms
No full textAbstractThe 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 isutt−Δu=0in(0,∞)×Ω,u=0on[0,∞)×Γ0,∂u∂ν=−|ut|m−2ut+|u|p−2uon[0,∞)×Γ1,u(0,x)=u0(x),ut(0,x)=u1(x)onΩ,where Ω⊂Rn(n⩾1) is a regular and bounded domain, ∂Ω=Γ0∪Γ1, 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
Blow-up for the wave equation with nonlinear source and boundary damping terms
No full textWe present some sufficient conditions to obtain compactness properties for the Euler–Lagrange functional of an elliptic equation. As an application, we extend some existence and multiplicity results for superlinear problems
Decay and stability for some nonlinear quasi-variational systems
No full textthe paper deals with the decay and stability problem for a nonlinear quasi-variational syste
Wave equation with second-order non-standard dynamical boundary conditions
No full textThe paper deals with the well-posedness of the problem
u_tt-Δu=0 in RxΩ,
u_tt=k u_ν on RxΓ,
u(0,x)= u_0(x), u_t(0,x)= v_0(x) in Ω,
where u = u(t, x), t ∈ R, x ∈ Ω, Δ = Δx denotes the Laplacian operator with respect to the space variable, Ω is a bounded regular (C^∞) open domain of R^N (N ≥ 1), Γ = ∂Ω, ν is the outward normal to Ω, k is a constant. We prove that it is ill-posed if N ≥ 2, while it is well-posed when N = 1. In the one-dimensional case, we give a complete existence, uniqueness and regularity theory. We also give some existence result for regular initial data when N ≥ 2 and Ω is a ball
Strong solutions for the wave equation with a kinetic boundary condition
No full textThe paper deals with local existence and uniqueness for wave equation with a kinetic boudary conditio
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