  In this paper, we establish the existence of spatially inhomogeneous
classical self-similar solutions to a non-Lipschitz semi-linear parabolic
Cauchy problem with trivial initial data. Specifically we consider bounded
solutions to an associated two-dimensional non-Lipschitz non-autonomous
dynamical system, for which, we establish the existence of a two-parameter
family of homoclinic connections on the origin, and a heteroclinic connection
between two equilibrium points. Additionally, we obtain bounds and estimates on
the rate of convergence of the homoclinic connections to the origin.
