Influence of the damping constant on critical-modulus nonlinearities

Determine whether the multiplicative damping constant μ influences the critical behavior for nonlinearities of the forms |u|^{pStr(n)}ν(|u|) and |u_t|^{pGla(n)}ν(|u_t|), where ν is a modulus of continuity satisfying τ^{-αν(τ)}→+∞ as τ→0^+ for every α∈(0,1], in the case k≥2 of the damping coefficient b(t)=μ/((e^{[k]}+t)∏_{j=1}^k ln^{[j]}(e^{[k]}+t)).

Background

For damping coefficients with k≥2, the paper proves blow-up for the pure-power nonlinearities |u|^p in the range 1<p≤pStr(n) and |u_t|^p in the range 1<p≤pGla(n), but the role of the multiplicative constant μ is not reflected in these power scales, apart from its appearance in lifespan upper bounds.

Remark 2.4 proposes that μ may instead affect the critical behavior for nonlinearities obtained by multiplying the Strauss- or Glassey-critical powers by a modulus of continuity ν that is weaker than every Hölder modulus, as expressed by the condition τ^{-αν(τ)}→+∞ for all α∈(0,1]. The paper does not establish this conjectured influence.