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

On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearity