Pointwise or norm-specific blow-up rate

Establish a corresponding pointwise or norm-specific blow-up rate for solutions of the semilinear wave equation with nonlinear damping, with the formal amplitude exponent \(1/\vartheta(p,q)\) suggested by the dominant-balance heuristic and a possible change of branch at \(q_{c}=2p/(p+1)\).

Background

The paper determines the large-amplitude lifespan exponent but does not establish the temporal rate at which the solution becomes unbounded near its finite blow-up time. A heuristic based on balancing the inertial/source terms or damping/source terms predicts an amplitude exponent 1/ϑ(p,q)1/\vartheta(p,q), with different formulas below and above the threshold qcq_c.

The open problem is to turn this formal prediction into a rigorous pointwise or norm-specific blow-up estimate for the original bounded-domain equation.

References

The second is the blow-up rate: the heuristic of Remarks \ref{rem:main} $(ii)$ suggests the formal amplitude exponent $1/\vartheta(p,q)$, changing branch at $q_{c}$, but establishing a corresponding pointwise or norm-specific blow-up rate for eq:main remains open.

eq:main:

{utt−Δu=u∣u∣p−1−ut∣ut∣q−1,(x,t)∈Ω×(0,T∗),u(x,0)=u0(x)=ϱf(x),ut(x,0)=u1(x)=ϱg(x),x∈Ω,u(x,t)=0,(x,t)∈∂Ω×(0,T∗),\begin{cases} u_{tt}-\Delta u=u|u|^{p-1}-u_{t}|u_{t}|^{q-1}, & (x,t)\in\Omega\times(0,T^{*}),\\ u(x,0)=u_{0}(x)=\varrho f(x),\quad u_{t}(x,0)=u_{1}(x)=\varrho g(x), & x\in\Omega,\\ u(x,t)=0, & (x,t)\in\partial\Omega\times(0,T^{*}), \end{cases}

— Sharp Lifespan Estimates for a Semilinear Wave Equation with Nonlinear Damping  (2609.10819 - Kaabi, 9 Sep 2026) in Section 5, Concluding remarks and open problems; see also Remark 1( ii)