On semilinear damped wave equations with initial data in homogeneous Sobolev spaces

1. Problem Statement

The paper investigates the Cauchy problem for the semilinear damped wave equation in $\mathbb{R}^n$:
$$\begin{cases} \partial_t^2 u + \partial_t u - \Delta u = |u|^p, & x \in \mathbb{R}^n, \, t > 0, \\ (u(0, x), \partial_t u(0, x)) = (\epsilon u_0, \epsilon u_1), & x \in \mathbb{R}^n, \end{cases}$$
where $p > 1$ and $0 < \epsilon \ll 1$ is a small parameter.

The primary focus is on determining the critical exponent $p_c$ that separates the regime of global existence of solutions from finite-time blow-up, specifically when the initial data $(u_0, u_1)$ belongs to the intersection of homogeneous and inhomogeneous Sobolev spaces:
$$(u_0, u_1) \in (\dot{H}^{-\gamma} \cap H^s) \times (\dot{H}^{-\gamma} \cap L^2),$$
with $\gamma > 0$ and $s \in (0, 1]$.

Context:
Previous work by Chen and Reissig [1] established the critical exponent for the case $0 < \gamma < n/2$, finding $p_{crit} = 1 + \frac{4}{n+2\gamma}$. This paper addresses the previously unhandled case where $\gamma \geq n/2$, a regime where the initial data has lower regularity (more singular) in the frequency domain.

2. Methodology

The author employs a combination of functional analysis, harmonic analysis, and test function methods.

A. Global Existence (Theorem 1)

• Strategy: The proof relies on a fixed-point argument in a suitable Banach space of functions with specific decay rates.
• Key Tool: The author utilizes a lemma from Chen-Reissig [1] (Lemma 1 in the paper) as a foundational step.
• Monotonicity Argument: A crucial technical step involves Lemma 2, which proves the monotonicity of the space $\dot{H}^{-\gamma} \cap H^s$ with respect to $\gamma$. Specifically, if $0 \leq \tilde{\gamma} < \gamma$, then $\dot{H}^{-\gamma} \subset \dot{H}^{-\tilde{\gamma}}$.
• Application: By choosing an auxiliary parameter $\tilde{\gamma}$ such that $\tilde{\gamma} < n/2$ (satisfying the conditions of the known Chen-Reissig result) and ensuring the initial data space embeds into the space required by Lemma 1, the author extends the global existence result to the case $\gamma \geq n/2$.
• Decay Estimates: The solution is shown to satisfy specific time-decay estimates in $L^2$ and $H^s$ norms, dependent on the chosen $\tilde{\gamma}$.

B. Blow-up Analysis (Theorems 2 & 3)

• Strategy: The author uses the test function method (also known as the Kaplan method or Fujita-type test function approach) to derive lifespan estimates and prove non-existence of global weak solutions.
• Construction of Test Functions:
  • A non-trivial bump function $\phi \in C_0^\infty(\mathbb{R}^n)$ is constructed (Lemma 3) satisfying specific positivity and scaling properties ($\hat{\phi} \geq 0$ and $\phi(rx) \leq \phi(Rx)$).
  • Space-time test functions are defined as $\phi_R(x) = \tilde{\phi}(R^{-1}x)$ and $\eta_R(t) = \tilde{\eta}(R^{-2}t)$.
• Integral Inequalities: The author defines an integral functional $I(R)$ involving the solution and the test functions. By applying Hölder's inequality and Young's inequality to the weak formulation of the equation, differential inequalities are derived.
• Initial Data Conditions:
  • Theorem 2: Assumes the Fourier transform of the initial data behaves like $|\xi|^{-n/2+\gamma}(\log|\xi|)^{-1}$ near the origin. This specific singularity is tailored to the $\gamma \geq n/2$ regime.
  • Theorem 3: Assumes the Fourier transform of the initial data is strictly positive on a ball $B(r)$. This recovers the classical Fujita-type blow-up behavior.

3. Key Contributions and Results

A. Extension of Global Existence (Theorem 1)

For dimensions $1 \leq n \leq 6$, the paper establishes global existence for small initial data in $(\dot{H}^{-\gamma} \cap H^s) \times (\dot{H}^{-\gamma} \cap L^2)$ under the condition:
$$\gamma \geq \min\left(\frac{n}{2}, \frac{\sqrt{n^2+16n}-n}{4}\right).$$
The critical exponent $p$ must satisfy:
$$p > \max\left(1+\frac{2}{n}, \frac{\sqrt{n^2+16n}+n}{2n}\right).$$
This result effectively bridges the gap for $\gamma \geq n/2$, showing that global solutions exist even for highly singular initial data, provided the nonlinearity $p$ is sufficiently large.

B. Blow-up Results and Lifespan Estimates

The paper provides sharp upper bounds for the lifespan $T$ of solutions when they blow up:

1. Theorem 2 (Generalized Blow-up):
   If $1 < p < 1 + \frac{4}{n+2\gamma}$ and the initial data has the specific logarithmic singularity described in (7), the solution blows up in finite time.

   • Lifespan Estimate: $T \lesssim \epsilon^{-\frac{1}{p' - 1 - \frac{n}{4} - \frac{\gamma}{2}}}$.
   • This exponent matches the inverse of the difference between the conjugate exponent $p'$ and the critical conjugate $p'_{crit}$.

2. Theorem 3 (Fujita-type Blow-up):
   If $1 < p < 1 + \frac{2}{n}$ (the Fujita exponent $p_F$) and the initial data is positive in the Fourier domain, the solution blows up.

   • Lifespan Estimate: $T \lesssim \epsilon^{-\frac{p}{p' - 1 - \frac{n}{2}}}$.
   • This recovers the classical Fujita exponent behavior even in the presence of the $\dot{H}^{-\gamma}$ regularity.

C. The Critical Exponent Landscape

The paper clarifies the phase diagram in the $(\gamma, p)$ plane (illustrated in Figure 1 of the paper):

• For $n=1, 2$, the critical exponent coincides with the Fujita exponent $p_F = 1 + 2/n$ regardless of $\gamma$.
• For $n \geq 3$, there is a transition. The critical exponent is determined by the minimum of the Fujita exponent and the Chen-Reissig exponent $p_{crit} = 1 + \frac{4}{n+2\gamma}$.
• The results confirm that for $\gamma \geq n/2$, the blow-up behavior is dominated by the Fujita exponent $p_F$ in the low $p$ regime, while global existence is possible for higher $p$.

4. Significance

• Completing the Theory: This work fills a critical gap in the theory of damped wave equations by resolving the case of initial data with $\gamma \geq n/2$, a regime where standard embedding theorems used in previous literature fail.
• Sharp Lifespan Estimates: The derivation of precise lifespan estimates for both the Fujita regime and the generalized critical regime provides a complete picture of the solution's behavior near the blow-up time.
• Methodological Rigor: The construction of specific test functions with logarithmic corrections (Lemma 6) demonstrates a sophisticated handling of singular initial data in the frequency domain, offering a template for future studies on equations with low-regularity data.
• Unification: The paper unifies the understanding of critical exponents, showing how the interplay between the damping term, the dimension $n$, and the regularity parameter $\gamma$ dictates the global solvability of the system.

In summary, Matsunaga successfully extends the known critical exponent theory for semilinear damped wave equations to the highly singular initial data regime ($\gamma \geq n/2$), providing rigorous proofs for both global existence and finite-time blow-up with precise lifespan estimates.