AI 中文总结
该研究针对拟递减加权L₂空间中含奇异时间可测伪微分算子的柯西问题，证明符号超指数增长时解仍存在唯一，还求解负阶分数阶拉普拉斯驱动的演化方程，建立初始爆破下的强解存在性。
AI 中文摘要
本研究考察由高度奇异的时间可测伪微分算子（傅里叶乘子的奇异可测族）支配的柯西问题。我们证明这类算子的符号可呈现任意爆破行为，尤其在符号随时间和频率呈超指数增长时，仍能证明解的存在性与唯一性。作为具体应用，我们求解由任意负阶分数阶拉普拉斯算子驱动的演化方程。此外，仅在符号在频率上局部可积的唯一条件下，即使初始时刻存在严重爆破，我们也能建立唯一的强解。
英文摘要
This study examines Cauchy problems governed by highly singular, time-measurable pseudo-differential operators (singular measurable families of Fourier multipliers). We show that the symbols of these operators can exhibit arbitrary blow-up behavior. In particular, we prove the existence and uniqueness of solutions even when the symbols grow super-exponentially in time and frequency. As a concrete application, we solve evolutionary equations driven by fractional Laplacians of any negative order. Additionally, we establish unique strong solutions under the sole condition that the symbol is locally integrable in frequency, even in the presence of severe blow-up at the initial time.
Comments39 pages
