On the blow-up of solutions to scale-invariant wave equations with damping and mass: Beyond the positive discriminant restriction

1. Problem Statement

The paper investigates the finite-time blow-up behavior of solutions to the following scale-invariant semilinear wave equation with time-dependent damping and mass terms, featuring a derivative-type nonlinearity:

$$\begin{cases} \partial_t^2 u - \Delta u + \frac{\mu}{1+t}\partial_t u + \frac{\nu^2}{(1+t)^2}u = |\partial_t u|^p, & (x, t) \in \mathbb{R}^n \times [0, \infty), \\ (u(x, 0), \partial_t u(x, 0)) = \varepsilon(f(x), g(x)), & x \in \mathbb{R}^n, \end{cases}$$

Key Parameters:

• $\mu \ge 0$: Damping coefficient.
• $\nu^2 \ge 0$: Mass coefficient.
• $p > 1$: Power of the time-derivative nonlinearity.
• $\varepsilon > 0$: Smallness parameter of the initial data (assumed positive and compactly supported).

The Core Challenge:
The dynamics of this equation are governed by the linear operator $L_{\mu, \nu}$. A crucial quantity in previous literature is the discriminant:
$$\delta = (\mu - 1)^2 - 4\nu^2$$
Historically, blow-up results for this equation were restricted to the regime $\delta \ge 0$. The case $\delta < 0$ (where the mass term dominates the damping in a specific spectral sense) was considered an open problem or required different techniques, often leading to the assumption that the sign of $\delta$ was a structural prerequisite for the blow-up mechanism.

2. Methodology

The author employs a test function method combined with a perturbative approach to extend existing results to the $\delta < 0$ regime.

A. Construction of Test Functions

Traditional methods for $\delta \ge 0$ rely on finding a test function $U$ that satisfies the adjoint equation $L^*_{\mu, \nu} U = 0$ exactly, often involving modified Bessel functions. This construction fails when $\delta < 0$ because the indices of the Bessel functions become complex.

The Novel Approach:
Hamza relaxes the requirement. Instead of seeking an exact solution to the adjoint equation, he constructs a family of test functions $\psi_d^\eta(x, t)$ that satisfy a differential inequality (specifically, the adjoint operator applied to the function is non-positive).

• Ansatz: Separation of variables $\psi_d^\eta(x, t) = \rho_d^\eta(t) \phi_\eta(x)$.
• Spatial Component: $\phi_\eta(x)$ is a standard exponential-type function satisfying $\Delta \phi_\eta = \eta^2 \phi_\eta$.
• Temporal Component: $\rho_d^\eta(t) = (1+t)^{d/2} e^{-\eta t}$, where $d$ is a parameter chosen such that $d > \mu$.
• Key Insight: By choosing the parameter $\eta$ sufficiently large, the author ensures that $L^*_{\mu, \nu} \psi_d^\eta \le 0$ regardless of the sign of $\delta$. This bypasses the need for Bessel functions and the restriction $\delta \ge 0$.

B. Functional Analysis

The proof utilizes two primary functionals defined via the test functions:

1. $F_d^\eta(t) = \int_{\mathbb{R}^n} u(x, t) \psi_d^\eta(x, t) dx$
2. $G_d^\eta(t) = \int_{\mathbb{R}^n} \partial_t u(x, t) \psi_d^\eta(x, t) dx$

The author derives differential-integral inequalities for these functionals. Crucially, they establish lower bounds for $F_d^\eta(t)$ and $G_d^\eta(t)$ that are strictly positive for small $\varepsilon$ and large $t$.

C. Iterative Blow-up Argument

Using the lower bounds and Hölder's inequality, the author derives a differential inequality for an auxiliary functional $L_d^\eta(t)$ related to the nonlinear term:
$$\frac{d}{dt} L_d^\eta(t) \ge C (L_d^\eta(t))^p (1+t)^{-\frac{(n+d-1)(p-1)}{2}}$$
By analyzing the exponent of the time decay, the author proves that if the power $p$ is below a critical threshold, the functional $L_d^\eta(t)$ must blow up in finite time, implying the solution $u$ cannot exist globally.

3. Key Contributions

1. Removal of the $\delta \ge 0$ Restriction:
   The most significant contribution is demonstrating that the condition $\delta \ge 0$ is not an intrinsic physical requirement for blow-up but rather a technical artifact of previous methods (specifically the reliance on exact solutions to the adjoint equation involving Bessel functions). The blow-up region remains valid even when $\delta < 0$.

2. Unification of the Critical Exponent:
   The paper establishes that the critical threshold for blow-up is determined solely by the shifted dimension $n + \mu$, aligning with the Glassey conjecture. The critical exponent is:
   $$p_{Gla}(n+\mu) = 1 + \frac{2}{n+\mu-1}$$
   The blow-up occurs for $1 < p < p_{Gla}(n+\mu)$, regardless of the value of $\nu$ (and thus regardless of the sign of $\delta$).

3. Refined Lifespan Estimates:
   The author provides an upper bound for the lifespan $T_\varepsilon$ of the solution:
   $$T_\varepsilon \le C \varepsilon^{-\frac{2(p-1)}{2 - (n+d-1)(p-1)}}$$
   where $d$ is a parameter in the range $(\mu, \frac{p+1}{p-1} - n)$. This estimate improves upon previous results for the $\delta \ge 0$ case by optimizing the parameter $d$.

4. Main Results

Theorem 2.2:
Let $\mu > 0$, $\nu^2 > 0$, and $p > 1$. Assume initial data are non-negative, compactly supported, and not identically zero. If $1 < p < p_{Gla}(n+\mu)$, then for sufficiently small $\varepsilon$, the energy solution $u$ blows up in finite time $T_\varepsilon$. The lifespan satisfies the upper bound:
$$T_\varepsilon \le C \varepsilon^{-\frac{2(p-1)}{2 - (n+d-1)(p-1)}}$$
for any $d \in (\mu, \frac{p+1}{p-1} - n)$.

Note on Critical Case:
The result is currently established for the strictly subcritical case ($p < p_{Gla}(n+\mu)$). The critical case $p = p_{Gla}(n+\mu)$ requires a more refined analysis beyond the perturbative method used here.

5. Significance and Implications

• Theoretical Breakthrough: This work resolves a long-standing ambiguity regarding the role of the discriminant $\delta$ in scale-invariant wave equations. It confirms that the mass term $\frac{\nu^2}{(1+t)^2}u$, even when dominant ($\delta < 0$), does not prevent the blow-up phenomenon driven by the derivative nonlinearity if the damping is strong enough to shift the effective dimension.
• Methodological Advancement: The construction of test functions that satisfy an inequality rather than an equality provides a robust tool for analyzing equations where exact spectral solutions are unavailable or complex. This approach is applicable to coupled systems and Tricomi-type models.
• Physical Interpretation: The results suggest that the "effective dimension" of the wave equation is shifted by the damping coefficient $\mu$ (to $n+\mu$), and this shift dictates the blow-up dynamics, rendering the mass parameter $\nu$ secondary in determining the existence of the blow-up region, though it influences the specific constants.

In summary, Hamza's paper extends the Glassey-type blow-up theory to the full parameter space of scale-invariant damping and mass, proving that the sign of the discriminant is irrelevant to the fundamental blow-up mechanism.