Non-uniqueness for an energy-critical heat equation on R2
Non-uniqueness for an energy-critical heat equation on R2
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R2 上能量临界热方程的非唯一性
DOI：
10.1007/s00208-020-01961-2
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发表时间：
2020
影响因子：
1.4
通讯作者：
Wei Juncheng
中科院分区：
文献类型：
--
作者：
Ibrahim Slim;Kikuchi Hiroaki;Nakanishi Kenji;Wei Juncheng
我们构造了一个定常非线性薛定谔方程的奇异解，其中平方指数非线性项在零附近具有线性行为。根据Trudinger-Moser不等式，这类非线性具有能量临界增长。我们利用这个奇异解证明了相应的半线性热方程的柯西问题的温和解的非唯一性。证明依赖于显式计算显示的热方程在适当的功能空间的正则化效果。
We construct a singular solution of a stationary nonlinear Schrödinger equation onwith square-exponential nonlinearity having linear behavior around zero. In view of Trudinger-Moser inequality, this type of nonlinearity has an energy-critical growth. We use this singular solution to prove non-uniqueness of mild solutions for the Cauchy problem of the corresponding semilinear heat equation. The proof relies on explicit computation showing a regularizing effect of the heat equation in an appropriate functional space.