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

Abstract

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 wellposedness 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 η → ∞.2 Self-similar solution structure to [CP] In this section, we establish a priori bounds for solutions to [CP]. Consequently, we consider a radial self-similar solution structure of solutions to [CP] which yields (P). To begin, we have, Lemma 2.1. Let u :D T → R be a solution to [CP]. Then,Proof. Since [CP] has spatially homogeneous initial data, the maximal solution u + :D T → R and minimal solution u − :D T → R to [CP] are spatially homogeneous for t ∈ [0, T ] (see, for example, [21, Proposition 8.31] for n = 1). We note that u ± :D T → R given byare the maximal and minimal solutions to [CP], and hence any solution u :D T → R to [CP] satisfies u − ≤ u ≤ u + onD T , as required.