Nonexistence for effectively damped waves with time-dependent mass

Technical Summary: Nonexistence for Effectively Damped Waves with Time-Dependent Mass

Problem Statement
This paper investigates the Cauchy problem for semilinear wave equations featuring time-dependent damping and a time-dependent mass term:
$$\begin{cases} u_{tt} - \Delta u + b(t)u_t + m^2(t)u = |u|^p, & t \ge 0, \, x \in \mathbb{R}^n, \\ u(0, x) = \varepsilon f(x), \quad u_t(0, x) = \varepsilon g(x), \end{cases}$$
where $n \ge 1$, $p > 1$, and $\varepsilon > 0$. The damping $b(t)$ is assumed to be "effective," and the mass $m(t)$ is dominated by the damping as $t \to \infty$.

Previous work by D'Abbicco, Girardi, and Reissig established global small-data existence for supercritical powers $p$ and identified a critical scale $p_{\beta, \eta}(n) = 1 + \frac{2\eta}{n + 2\eta\beta}$ for initial data in $L^\eta \cap H^1 \times L^\eta \cap L^2$ (with $1 \le \eta < 2$). Here, $\beta$ is the lower mass index defined by $\liminf_{t\to\infty} B(t)m^2(t)$, where $B(t) = \int_0^t b(\tau)^{-1} d\tau$. While they proved nonexistence for the corresponding nonlinear diffusion equation in the subcritical range, the analogous nonexistence result for the wave equation remained an open problem. This paper addresses that gap specifically for the case $\eta = 1$, aiming to prove the nonexistence of global weak solutions for $1 < p < p_{\beta, 1}(n)$ and treat the critical case $p = p_{\beta, 1}(n)$.

Methodology
The authors employ a modified test-function method, relying on the construction of a specific positive adjoint multiplier. The core of the methodology involves:

1. Coefficient Assumptions: Beyond the standard effective damping and dominated mass hypotheses (Hypotheses 1–2 from prior literature), the paper introduces two new conditions:

   • Intrinsic Accumulated-Mass Balance (Hypothesis 3): This controls the deviation of the accumulated mass integral $Q(t, r) = \int_r^t \frac{m^2(\tau)}{b(\tau)} d\tau$ from the logarithmic growth dictated by the index $\beta$. It defines an "intrinsic excess" $\Omega_{\beta, T_0}(L)$ which must be finite.
   • Global Liouville Nonoscillation (Hypothesis 4): The condition $m^2(t) \le \frac{b^2(t)}{4} + \frac{b'(t)}{2}$ is imposed to ensure the non-oscillation of the adjoint equation.

2. Construction of the Adjoint Mode: A key technical step is the construction of a global positive "slow" solution $\rho(t)$ to the adjoint equation $\rho'' - (b\rho)' + m^2\rho = 0$.

   • For large times, $\rho$ is constructed via a terminal Volterra equation, yielding an asymptotic behavior proportional to $b(t)^{-1} e^{Q(t, T^*)}$.
   • For the compact initial interval, the Liouville nonoscillation condition (Hypothesis 4) is used to prevent the solution from acquiring zeros, ensuring $\rho(t) > 0$ globally.

3. Test-Function Argument: Using $\rho(t)$ as a weight and a spatial-temporal cutoff function $\psi_R$ based on the diffusion scale $B(t)$, the authors derive a fundamental inequality. This inequality relates the initial data functional $J_0$ to integrals of the solution $|u|^p$.

4. Osgood Argument: By applying an Osgood-type lemma to the derived inequality, the authors establish conditions under which the integral of the test function must diverge, leading to a contradiction if a global solution were to exist.

Key Results
The paper establishes the following main theorems:

• Nonexistence in the Strictly Subcritical Range: If the initial data satisfies a sign condition ($J_0 > 0$) and the intrinsic excess $\Omega_{\beta, T_0}$ is sublinear, there are no global weak solutions for $1 < p < p_{\beta, 1}(n) = 1 + \frac{2}{n + 2\beta}$.
• Nonexistence in the Critical Case: At the critical exponent $p = p_{\beta, 1}(n)$, global weak solutions do not exist provided the integral $\int^\infty \exp(-(p-1)\Omega_{\beta, T_0}(L)) dL$ diverges (an Osgood divergence condition).
• Lifespan Estimates: For solutions that exist only locally, the paper provides conditional upper bounds on the lifespan $T_\varepsilon$:
  • For $1 < p < p_{\beta, 1}(n)$, $1 + B(T_\varepsilon) \le C(1 + \varepsilon^{-\frac{1}{\theta}})$ where $\theta = \frac{1}{p-1} - \frac{n}{2} - \beta$.
  • For $p = p_{\beta, 1}(n)$, $1 + B(T_\varepsilon) \le C \exp(C \varepsilon^{-(p-1)})$.
• Admissible Coefficients: Section 7 provides explicit examples of coefficient families (polynomial and rational decay/growth) that satisfy both the existence conditions of D'Abbicco et al. and the nonexistence conditions of this paper, demonstrating the sharpness of the derived scale.

Significance and Claims
The paper claims to resolve an open problem regarding the sharpness of the existence scale for effectively damped wave equations with time-dependent mass. Specifically:

• It confirms that the scale $p_{\beta, 1}(n)$, previously established for global existence, is indeed the threshold for nonexistence in the subcritical regime, mirroring the behavior of the associated diffusion equation.
• The proof technique is distinct in that it does not postulate the existence of a positive adjoint multiplier but rather constructs it rigorously using the Liouville nonoscillation condition and a terminal Volterra approach.
• The introduction of the "intrinsic excess" $\Omega_{\beta, T_0}$ allows for a more refined analysis of the mass accumulation, showing that the lower index $\beta$ alone is insufficient to control the nonexistence threshold without additional balance conditions.
• The results are presented as a definitive answer for the globally nonoscillatory subclass of coefficients, bridging the gap between the diffusion comparison model and the wave equation counterpart.

The authors maintain a modest tone, noting that their results apply to the specific subclass of coefficients satisfying the Liouville nonoscillation condition and intrinsic balance, and that the proof relies on the modified test-function approach developed in prior literature.