Uniqueness of profiles for a fixed origin coefficient

Prove or disprove uniqueness of a self-similar profile satisfying the origin behavior lim_{\xi\to0}\xi^{-(\sigma+2)/(m-p)}f(\xi)=K and the far-field behavior lim_{\xi\to\infty}\xi^{\sigma/(p-1)}f(\xi)=c^* for each fixed K\in(0,K^*).

Background

Theorem 3.3 guarantees existence of at least one profile with the specified behavior at the origin and infinity for every K in the admissible interval (0,K*). Because the underlying profile equation is degenerate at f(0)=0 and (fm)'(0)=0, the associated dynamical system permits non-uniqueness phenomena.

The paper establishes neither uniqueness nor non-uniqueness when both endpoint conditions are imposed simultaneously. Thus, even after the coefficient K is fixed, it remains unresolved whether multiple profiles can connect the corresponding origin behavior to the distinguished tail at infinity.

References

In change, there is no proof of either uniqueness or non-uniqueness of the profiles satisfying simultaneously beh.zero and beh.inf, but the uniqueness of such profiles, once fixed $K\in(0,K*)$, is highly expected.

beh.zero:

lim⁡ξ→0ξ−(σ+2)/(m−p)f(ξ)=K\lim\limits_{\xi\to0}\xi^{-(\sigma+2)/(m-p)}f(\xi)=K

beh.inf:

lim⁡ξ→∞ξσ/(p−1)f(ξ)=c∗,\lim\limits_{\xi\to\infty}\xi^{\sigma/(p-1)}f(\xi)=c^*,

— A porous medium equation with dominating weighted absorption: three types of self-similar solutions  (2609.20397 - Iagar et al., 17 Sep 2026) in Section 1, final paragraph of the Introduction, immediately before Section 2