In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. Specifically we establish well-posedness for an associated initial value problem for a singular two-dimensional non-autonomous dynamical system with non-Lipschitz nonlinearity. Additionally, we establish that solutions to the initial value problem converge algebraically to the origin and oscillate as $\eta\to \infty$.

中文翻译：

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关于具有非Lipschitz非线性的二阶半线性抛物线偏微分方程的双符号解

在本文中，我们针对具有非 Lipschitz 非线性和平凡初始数据的半线性抛物线偏微分方程建立了柯西问题的空间非均匀径向对称经典自相似解的 1 参数族的存在性。具体来说，我们为具有非 Lipschitz 非线性的奇异二维非自治动力系统的相关初始值问题建立适定性。此外，我们确定初始值问题的解决方案代数收敛到原点并振荡为 $\eta\to\infty$。