Precise asymptotics below the critical exponent

Characterize the precise asymptotic behavior of the radial function v(r) for the parabolic k-Hessian equation with separable right-hand side f(|x|)g(t) when k>n/2 and 0<\alpha<(2k-n)/(kn).

Background

The paper constructs separable-variable radial entire solutions u(x,t)=w(t)v(|x|) to a class of parabolic k-Hessian equations with positive continuous right-hand side terms f(|x|) and g(t). Under periodicity assumptions and the supercritical condition \alpha>(2k-n)/(kn), Theorem 2 establishes refined asymptotic behavior for v(r), including its leading power-law exponent.

The authors explicitly state that, in the parameter range k>n/2 and 0<\alpha<(2k-n)/(kn), only rough bounds are available and the precise asymptotic behavior of v(r) is unresolved. Determining this behavior would extend the asymptotic analysis beyond the parameter regime treated by the paper.