一维空间中抛物方程的奇异解
Singular solutions to parabolic equations in one space dimension
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- Universit \'e Paris-Saclay, CNRS, Laboratoire de Math\' e matiques d'Orsay, 91405 Orsay, France
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中文总结 AI 辅助
本文构造一维复系数抛物方程的奇异弱解，证明能量估计可积性最优，并揭示De Giorgi--Nash--Moser理论局部有界性在复系数下失效。
中文摘要 AI 辅助
我们构造了一维空间中具有复系数的标量、线性、一致抛物方程的奇异弱解。我们的例子表明，能量估计所提供的时空可积性是最优的：对于每个 $p>6$，存在这样的方程，其能量解在时空域的紧子集上不满足 $p$-可积性。特别地，De Giorgi--Nash--Moser 理论中弱解的局部有界性在复系数情况下即使在一维空间中也会失效。
英文摘要
We construct singular weak solutions to scalar, linear, uniformly parabolic equations with complex coefficients in one space dimension. Our examples show that the space-time integrability provided by the energy estimates is sharp: for every $p>6$, there exists such an equation admitting an energy solution that fails to be $p$-integrable on a compact subset of the space-time domain. In particular, the local boundedness of weak solutions from the De Giorgi--Nash--Moser theory fails for complex coefficients already in one space dimension.
