Extension to variable-exponent damping and source terms

Determine to what extent the rescaling, monotonicity, and quantitative damping arguments extend to variable-exponent semilinear wave equations in which the source and damping exponents \(p\) and \(q\) depend on the spatial variable \(x\), despite the loss of scaling invariance.

Background

The paper's lower-bound proof relies on a hyperbolic rescaling that transforms the equation into one of the same form while introducing a rescaled damping coefficient. For variable-exponent equations, spatial dependence of pp and qq prevents this exact scaling reduction.

The author identifies the monotonicity of the damping term and the quantitative rate associated with a large damping coefficient as local mechanisms that might survive in another form, but their applicability and the resulting lifespan theory remain unresolved.

References

The third is the extent to which the method survives a loss of scaling invariance, for instance for the variable-exponent equations of , where $p$ and $q$ depend on $x$ and the rescaling eq:scaling no longer maps the problem to one of the same form; the two mechanisms used above, the monotonicity of the damping term and the quantitative rate $A{-1/q}$, are local in nature and may have counterparts in that setting.

eq:scaling:

λ=ϱ(p−1)/2,v(y,s)=ϱ−1u(yλ,sλ),\lambda=\varrho^{(p-1)/2}, \qquad v(y,s)=\varrho^{-1}u\Bigl(\frac{y}{\lambda},\frac{s}{\lambda}\Bigr),

— 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; Remark 1(vi)

We therefore expect the two lower bounds to persist on unbounded domains, and in particular on $\Omega=\mathbb{R}{n}$ for profiles satisfying eq:data, but we have not carried out the details; whether $\vartheta(p,q)$ still describes the lifespan there is a further open question, and one which the concavity method in its present form does not answer.

eq:data:

f∈H01(Ω)∩L∞(Ω),∇f∈Ln∨2(Ω),g∈Ln∨2(Ω),f\in H_{0}^{1}(\Omega)\cap L^{\infty}(\Omega), \qquad \nabla f\in L^{n\vee2}(\Omega), \qquad g\in L^{n\vee2}(\Omega),

— 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