Entropy stability of non-constant bounded exponential self-similar profiles

Determine the entropy stability of non-constant bounded solutions of the stationary self-similar equation \(\Delta w-\frac12 y\cdot\nabla w+e^w-1=0\) for the exponential semilinear heat equation, without restricting to the additional decay assumption used in the paper’s entropy-instability theorem.

Background

The paper studies bounded stationary self-similar profiles for the exponential semilinear heat equation through a Gaussian FF-functional and its associated entropy. Non-constant profiles are shown to be entropy unstable under the decay condition ∣w(y)∣≤C(1+∣y∣)−1|w(y)|\leq C(1+|y|)^{-1}, while the constant profile w≡0w\equiv0 is shown to be quantitatively isolated among bounded solutions.

The stated unresolved issue concerns the entropy stability of non-constant bounded solutions in the exponential case in general. Thus, the result established in the paper resolves the question only for the subclass of non-constant bounded profiles satisfying the specified decay assumption; the general bounded setting remains open.

References

However, to our knowledge, the entropy stability of non-constant bounded solutions to stationary problem was still open in the exponential case.

stationary problem:

Δw−12y⋅∇w+ew−1=0,in Rn.\Delta w-\frac12 y\cdot\nabla w+e^{w}-1=0,\quad \text{in }\mathbb{R}^n.

— Entropy instability and rigidity for the exponential semilinear heat equation  (2609.34541 - Guo et al., 28 Sep 2026) in Section 1, Introduction