Quantitative unique continuation for the semilinear heat equation in a convex domain
Quantitative unique continuation for the semilinear heat equation in a convex domain
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凸域半线性热方程的定量唯一延拓
DOI：
10.1016/j.jfa.2010.04.015
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发表时间：
2010-09-01
影响因子：
1.7
通讯作者：
Wang, Gengsheng
中科院分区：
文献类型：
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作者：
Phung, Kim Dang;Wang, Gengsheng
在本文中，我们研究了具有齐次狄利克雷边界条件的半线性热方程偏导数 (t)u - Delta u = g(u) 的解在 Omega x (0, T(*)) 上的某些独特的连续性质。 Omega 是 Rd 的有界凸开子集，子集具有平滑边界。函数g R -> R 满足一定的条件。我们对 (u-v) 建立了一些观测估计，其中 u 和 v 是上述方程的两个解。观察是在 omega x {T} 上进行的，其中 omega 是 Omega 的任何非空开子集，T 是一个正数，使得 u 和 v 都存在于区间 [0, TI.从这些估计中至少可以得出两个结果：(i) 如果 11(u v)(., T)II L2(0 = 3，则 11(/ v)(:, T)11 0(Q) C8，其中常数 C > 0 和 alpha epsilon (0, 1) 在某些情况下可以独立于 u 和 v；(ii) 如果上述方程的两个解在 w x {T} 上保持相同的值，则它们在 Omega x 上一致 [0，T(m))。 T(m)表示[0,T,)上存在这两个解的最大数量。 (C) 2010 Elsevier Inc. 保留所有权利。
In this paper, we study certain unique continuation properties for solutions of the semilinear heat equation partial derivative(t)u - Delta u = g(u), with the homogeneous Dirichlet boundary condition, over Omega x (0, T(*)). Omega is a bounded, convex open subset of Rd, with a smooth boundary for the subset. The function g R -> R satisfies certain conditions. We establish some observation estimates for (u-v), where u and v are two solutions to the above-mentioned equation. The observation is made over omega x {T}, where omega is any non-empty open subset of Omega, and T is a positive number such that both u and v exist on the interval [0, TI. At least two results can be derived from these estimates: (i) if 11(u v)(., T)II L2(0 = 3, then 11(/ v)(:, T)11 0(Q) C8 where constants C > 0 and alpha epsilon (0, 1) can be independent of u and v in certain cases; (ii) if two solutions of the above equation hold the same value over w x {T}, then they coincide over Omega x [0, T(m)). T(m) indicates the maximum number such that these two solutions exist on [0, T,). (C) 2010 Elsevier Inc. All rights reserved.