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In this paper, we establish the existence of spatially inhomogeneousclassical self-similar solutions to a non-Lipschitz semi-linear parabolicCauchy problem with trivial initial data. Specifically we consider boundedsolutions to an associated two-dimensional non-Lipschitz non-autonomousdynamical system, for which, we establish the existence of a two-parameterfamily of homoclinic connections on the origin, and a heteroclinic connectionbetween two equilibrium points. Additionally, we obtain bounds and estimates onthe rate of convergence of the homoclinic connections to the origin.

The evolution to localized and front solutions in a non-Lipschitz reaction-diffusion Cauchy problem with trivial initial data