Finiteness of the maximal admissible coefficient K*

Determine whether the threshold K^* in Theorem 3.3 is finite by proving that no bounded 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^* exists for sufficiently large K.

Background

Theorem 3.3 establishes a value K*\in(0,\infty] such that, for every K\in(0,K*), at least one self-similar profile has the prescribed power-law behavior at the origin and the uniquely identified absorption-dominated tail at infinity. The proof also shows that K*>K_0, where K_0 is the coefficient of the explicit unbounded stationary solution.

The authors expect K* to be finite, as in the previously studied case \sigma=0 in one spatial dimension, but do not establish this. Resolving the problem requires ruling out bounded self-similar profiles with both prescribed endpoint behaviors when K is sufficiently large.

References

We expect $K*<\infty$, as it occurs for $\sigma=0$ and in dimension $N=1$ (see ). However, we were unable to prove the non-existence of any bounded self-similar solution satisfying beh.zero and beh.inf for $K$ very large with our techniques.

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 Remark immediately following Theorem 3.3 (Theorem \ref{th.SSSzero})