Results for blow-up and sharp lifespan estimates to a weakly coupled system of structurally damped wave equations with critical nonlinearities

1. Problem Statement

The paper investigates a weakly coupled system of semilinear structurally damped wave equations with critical nonlinearities involving moduli of continuity. The system is defined on $\mathbb{R}^n \times [0, \infty)$ as follows:

$$\begin{cases} u_{tt} - \Delta u + (-\Delta)^\sigma u_t = |v|^{p^*}\mu_1(|v|), \\ v_{tt} - \Delta v + (-\Delta)^\sigma v_t = |u|^{q^*}\mu_2(|u|), \end{cases}$$

with initial data $(u_0, u_1, v_0, v_1)$.

Key Parameters and Assumptions:

• Structural Damping: The parameter $\sigma \in [0, 1/2]$. This range corresponds to "effective damping" (parabolic-like behavior), distinguishing it from the "noneffective damping" case ($\sigma \in (1/2, 1]$), which remains an open problem for critical exponents.
• Critical Curve: The exponents $p^*, q^* > 1$ lie on the critical curve defined by:
  $$\frac{1 + \max\{p^*, q^*\}}{p^*q^* - 1} = \frac{n - 2\sigma}{2}$$
  (Assuming $p^* \leq q^*$, this simplifies to $\frac{1+q^*}{p^*q^*-1} = \frac{n-2\sigma}{2}$).
• Nonlinearities: The terms $|v|^{p^*}\mu_1(|v|)$ and $|u|^{q^*}\mu_2(|u|)$ include moduli of continuity $\mu_1, \mu_2$. These functions are introduced to refine the regularity of the nonlinearity beyond simple power laws, allowing for a more precise determination of the critical threshold between global existence and finite-time blow-up.
• Initial Data: Small data in the space $D = ((H^{[n/2]+1} \cap L^1) \times (H^{[n/2]} \cap L^1))^2$.

2. Methodology

The authors employ a combination of test function methods (for blow-up) and fixed-point arguments in weighted evolution spaces (for global existence and lifespan estimates).

A. Test Function Construction (for Blow-up)

A major novelty of this work is the construction of non-compact support test functions to handle the fractional Laplacian operator $(-\Delta)^\sigma$ when $\sigma > 0$.

• Challenge: Previous methods (e.g., Dao-Reissig) used test functions with compact support, which fail for non-local operators like $(-\Delta)^\sigma$ because the operator acts globally.
• Solution: The authors construct specific test functions $\phi_R(x)$ and $\eta_R(t)$ based on a smooth cut-off function $\phi(x)$ that decays algebraically rather than vanishing compactly.
  • Spatial part: $\phi_R(x) = \phi(R^{-1/2}x)$ where $\phi(x) \sim (1+|x|^4)^{-1}$ for large $|x|$.
  • Temporal part: Scaled smooth functions $\eta_R(t)$.
• Key Estimates: They derive precise scaling properties for the fractional Laplacian applied to these functions, specifically showing how $(-\Delta)^\sigma$ interacts with the scaling parameter $R$. This allows them to control the non-local terms in the integral inequalities.

B. Integral Inequalities and ODE Systems

• Blow-up Proof: By multiplying the system by the test functions and integrating, they derive a system of ordinary differential inequalities (ODIs) for auxiliary functionals $G_p(R)$ and $G_q(R)$ (related to the $L^1$ norms of the solutions).
• Contradiction: Under the assumption of a global solution, they show that the integral condition on the moduli of continuity leads to a contradiction (the integral diverges), proving that no global solution exists.

C. Lifespan Estimates (Upper and Lower Bounds)

• Upper Bound: Derived using the same test function method as the blow-up proof, but applied to local solutions with small initial data scaled by $\epsilon$. This yields an implicit upper bound for the lifespan $T_\epsilon$ involving the inverse of a function $\psi$.
• Lower Bound: Established using a fixed-point argument in a Banach space $X(T)$ equipped with weighted norms.
  • The weights involve decay rates determined by the linearized problem (Proposition 2.1) and the moduli of continuity.
  • A crucial step involves optimizing the weight function $\ell(t)$ to match the specific decay of the nonlinear terms, ensuring the contraction mapping principle holds up to the estimated lifespan.

3. Key Contributions

1. Extension to Structural Damping ($\sigma \in (0, 1/2]$): While previous works (e.g., Dao-Reissig [7]) handled the case $\sigma=0$ (standard damping), this paper extends the theory to structurally damped waves with fractional damping $\sigma \in (0, 1/2]$.
2. Novel Test Functions: The authors successfully overcome the difficulty of non-local operators by constructing test functions with non-compact support, a necessary adaptation for fractional Laplacians that was previously unresolved in this specific coupled context.
3. Sharp Moduli of Continuity Conditions: They provide the sharp condition for the moduli of continuity $\mu_1, \mu_2$ that separates global existence from blow-up.
   • Global Existence: Occurs if $\int_0^c \frac{1}{s} (\mu_1(s))^{\frac{q^*}{q^*+1}} (\mu_2(s))^{\frac{1}{q^*+1}} ds < \infty$.
   • Blow-up: Occurs if the same integral diverges ($= \infty$).
4. Sharp Lifespan Estimates: They derive precise asymptotic estimates for the lifespan $T_\epsilon$ of small data solutions in the blow-up regime. The estimates depend on the inverse of a function $\psi$ defined by the moduli of continuity.

4. Main Results

Theorem 1.1 (Blow-up)

If the initial data satisfies $\int u_1 > 0, \int v_1 > 0$ and the moduli of continuity satisfy the divergence condition:
$$\int_0^c \frac{1}{s} (\mu_1(s))^{\frac{q^*}{q^*+1}} (\mu_2(s))^{\frac{1}{q^*+1}} ds = \infty,$$
then there is no global Sobolev solution. The solution blows up in finite time.

Theorem 1.2 (Lifespan Estimates)

If the divergence condition holds, the lifespan $T_\epsilon$ of the local solution satisfies the following sharp estimates as $\epsilon \to 0$:

• Case $p^* = q^*$:
  $$T_\epsilon \sim \left( \psi^{-1}\left( C \epsilon^{-(p^*-1)} \right) \right)^{1-\sigma}$$
• Case $p^* < q^*$:
  $$T_\epsilon \sim \left( \psi^{-1}\left( C \epsilon^{-\frac{q^*(p^*q^*-1)}{q^*+1}} \right) \right)^{1-\sigma}$$

Where $\psi(\tau) = \int_{R_0}^\tau \frac{1}{r} (\mu_1(Cr^{-\frac{n}{2}+\sigma}))^{\frac{q^*}{q^*+1}} (\mu_2(Cr^{-\frac{n}{2}+\sigma}))^{\frac{1}{q^*+1}} dr$.

Special Cases Recovered:

• If $\mu_1 \equiv \mu_2 \equiv 1$ and $\sigma=0$, the results recover the known exponential lifespan estimates for standard damped waves.
• If $\mu_1 \equiv \mu_2 \equiv 1$ and $\sigma \in (0, 1/2]$, the results provide the first sharp lifespan estimates for structurally damped coupled systems on the critical curve.

5. Significance and Open Problems

• Significance: This work bridges a gap in the theory of damped wave equations by handling the critical case with structural damping and general nonlinearities. It demonstrates that the "effective damping" regime ($\sigma \leq 1/2$) behaves similarly to the standard damped case regarding critical exponents, provided the non-local operator is handled correctly via new test functions.
• Open Problems:
  • The case of noneffective damping ($\sigma \in (1/2, 1]$) remains open. The critical exponent for the single equation in this regime is not fully established, making the coupled system analysis difficult.
  • The authors propose future work on $\sigma$-evolution equations with different damping orders ($\delta \neq \sigma$), which presents significant technical challenges due to the interaction of different fractional operators.

In summary, the paper provides a rigorous and complete analysis of the critical threshold for blow-up and the precise lifespan of solutions for a class of weakly coupled structurally damped wave equations, introducing innovative techniques to handle non-local operators in the context of critical nonlinearities.