Existence and nonexistence of solutions for a model of gravitational interaction of particles, I
Existence and nonexistence of solutions for a model of gravitational interaction of particles, I
复制标题
粒子引力相互作用模型解的存在性和不存在性，I
DOI：
10.4064/cm-66-2-319-334
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发表时间：
1993
影响因子：
0.4
通讯作者：
T. Nadzieja
中科院分区：
文献类型：
--
作者：
P. Biler;T. Nadzieja
其中，ν表示到Ω Ω的向外单位法向量。对于势φ，我们假设（4.1）φ = 0在Ω上，或者代替（4.2）中的狄利克雷边界条件，φ = En <$u，其中En是n维拉普拉斯算子的基本解。对于初边值问题，我们增加了条件（5）u（x，0）= u 0（x）≥ 0。对于（1）-（5）的解释，我们请读者参考[9]的介绍。在一维情况下，解在时间上是全局的（参见[7]中的推理，第二节）。3][14]，[18]，[19]，[19]。对于n维球（n ≥ 2）中的径向问题，[7，定理3]首次证明了（1）-（4）在时间上整体定义的解的不存在性。初始质量的临界值
where ν denotes the outward unit normal vector to ∂Ω. For the potential φ we assume either (4.1) φ = 0 on ∂Ω, or instead of the Dirichlet boundary condition above (4.2) φ = En ∗ u with En the fundamental solution of the n-dimensional Laplacian. For the initial-boundary problem we add the condition (5) u(x, 0) = u0(x) ≥ 0. For the interpretation of (1)–(5) we refer the reader to the introduction of [9]. In the one-dimensional case solutions are global in time (see the reasoning in [7, Sec. 3] based on an idea from [14], and [18]). The first proof of nonexistence of solutions to (1)–(4) defined globally in time has been discovered for the radial problem in an n-dimensional ball, n ≥ 2, in [7, Theorem 3]. The critical value of mass of the initial