[Paper Review] Nonexistence of global solutions of nonlinear wave equations with weak time-dependent damping related to Glassey conjecture

N. Lai, Hiroyuki Takamura|arXiv (Cornell University)|Nov 21, 2017

Advanced Mathematical Physics Problems9 references3 citations

TL;DR

This paper establishes the nonexistence of global solutions for nonlinear wave equations with weak time-dependent damping in both the scattering ($\beta > 1$) and scale-invariant ($\beta = 1$) regimes, proving the same upper bound for the lifespan as in the undamped case—confirming a key part of the Glassey conjecture. By introducing tailored multipliers to absorb the damping term, the authors derive blow-up results and lifespan estimates matching the non-damped case, even under weak damping.

ABSTRACT

This work is devoted to the nonexistence of global-in-time energy solutions of nonlinear wave equation of derivative type with weak time-dependent damping in the scattering and scale invariant range. By introducing some multipliers to absorb the damping term, we succeed in establishing the same upper bound of the lifespan for the scattering damping as the non-damped case, which is a part of so-called Glassey conjecture on nonlinear wave equations. We also study an upper bound of the lifespan for the scale invariant damping with the same method.

Motivation & Objective

• To investigate the nonexistence of global solutions for nonlinear wave equations with weak time-dependent damping.
• To extend the Glassey conjecture—concerning blow-up for $ p \leq p_c(n) $—to the case with damping.
• To establish upper bounds for the lifespan of energy solutions in both scattering ($\beta > 1$) and scale-invariant ($\beta = 1$) damping regimes.
• To overcome the challenge of damping terms by introducing specialized multipliers that absorb the damping effect.

Proposed method

• Introduce a multiplier $ m(t) $ for the scattering case ($\beta > 1$) to absorb the $ \frac{\mu}{(1+t)^\beta} u_t $ damping term.
• Use a different multiplier $ m_1(t) = (1+t)^\mu $ for the scale-invariant case ($\beta = 1$), which is unbounded but effective in handling the damping.
• Apply integration by parts and test function techniques to derive energy-type inequalities involving the solution and its time derivative.
• Use Hölder's inequality and pointwise estimates on the nonlinear term $ |u_t|^p $ to derive differential inequalities for a functional $ H(t) $ or $ H_1(t) $.
• Establish a differential inequality of the form $ H'(t) \geq C (1+t)^{-\alpha} H^p(t) $, which leads to finite-time blow-up.
• Use comparison arguments and initial data positivity to derive explicit upper bounds for the lifespan $ T $.

Experimental results

Research questions

• RQ1 Does the Glassey conjecture on blow-up for $ p \leq p_c(n) $ hold under weak time-dependent damping?
• RQ2 Can the same upper bound for the lifespan as in the undamped case be achieved when damping is present?
• RQ3 How does the choice of multiplier affect the analysis of damped wave equations with $ \beta = 1 $ or $ \beta > 1 $?
• RQ4 Can the damping term be effectively absorbed using multiplier techniques to recover blow-up results similar to the non-damped case?

Key findings

• For $ \beta > 1 $ (scattering case), the lifespan $ T $ satisfies $ T \leq C \varepsilon^{-(p-1)/(1 - (n-1)(p-1)/2)} $ for $ 1 < p < p_c(n) $, matching the non-damped case.
• For $ p = p_c(n) $ and $ n \geq 2 $, the lifespan satisfies $ T \leq \exp(C \varepsilon^{-(p-1)}) $, again matching the non-damped result.
• In the scale-invariant case ($ \beta = 1 $), the lifespan upper bound depends on the damping parameter $ \mu $, with $ H_1'(t) \geq C (1+t)^{-(n+2\mu-1)(p-1)/2} H_1^p(t) $, leading to a modified lifespan estimate.
• The use of the multiplier $ m_1(t) = (1+t)^\mu $ allows the derivation of blow-up results even when the multiplier is unbounded.
• The results confirm that weak time-dependent damping does not prevent blow-up for $ p \leq p_c(n) $, supporting the Glassey conjecture in the damped setting.
• The upper bounds for the lifespan are identical to those in the undamped case, demonstrating that weak damping does not extend global existence.