On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources
On the wave equation with hyperbolic dynamical boundary conditions, interior and boundary damping and supercritical sources
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DOI：
10.1016/j.jde.2018.06.022
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发表时间：
2016-01
影响因子：
2.4
通讯作者：
Enzo Vitillaro
中科院分区：
文献类型：
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作者：
Enzo Vitillaro
本文研究了问题{ut t− Δ u+ P（x，ut）= f（x，u）in（0，∞）× Ω，u= 0 on（0，∞）× Γ 0，ut t+ Δ v u− Δ Γ u+ Q（x，ut）= g（x，u）on（0，∞）× Γ 1，u（0，x）= u0（x），ut（0，x）= u1（x）in Ω ∞，其中Ω是RN中具有C1边界（N≥ 2）的有界开子集，Γ= π Ω，Γ 1在Γ上相对开，Δ Γ表示Γ上的Laplace-Beltrami算子，ν是Ω的外法线，P和Q表示非线性阻尼项，f和g表示非线性扰动项.在本文中，我们建立了局部和整体的存在性，唯一性和Hadamard适定性结果时，源项可以是超临界或超超临界。
The aim of this paper is to study the problem {u t t− Δ u+ P (x, u t)= f (x, u) in (0,∞)× Ω, u= 0 on (0,∞)× Γ 0, u t t+∂ ν u− Δ Γ u+ Q (x, u t)= g (x, u) on (0,∞)× Γ 1, u (0, x)= u 0 (x), u t (0, x)= u 1 (x) in Ω‾, where Ω is a bounded open subset of R N with C 1 boundary (N≥ 2), Γ=∂ Ω, Γ 1 is relatively open on Γ, Δ Γ denotes the Laplace–Beltrami operator on Γ, ν is the outward normal to Ω, and the terms P and Q represent nonlinear damping terms, while f and g are nonlinear perturbations. In the paper we establish local and global existence, uniqueness and Hadamard well-posedness results when source terms can be supercritical or super-supercritical.