On the semilinear damped wave equation with Riesz potential-type power nonlinearity and initial data in pseudo-measure spaces

1. Problem Statement

The paper investigates the Cauchy problem for the semilinear damped wave equation coupled with a Riesz potential-type power nonlinearity. The equation is given by:

$$\begin{cases} u_{tt} - \Delta u + u_t = I_\gamma(|u|^p), & x \in \mathbb{R}^n, \, t > 0, \\ u(0, x) = \varepsilon u_0(x), \quad u_t(0, x) = \varepsilon u_1(x), & x \in \mathbb{R}^n, \end{cases}$$

where:

• $n \ge 1$ is the spatial dimension.
• $\gamma \in [0, n)$ is the order of the Riesz potential $I_\gamma$.
• $p > 1$ is the power of the nonlinearity.
• $\varepsilon > 0$ is a small parameter scaling the initial data.
• The Riesz potential is defined via convolution: $I_\gamma(f) = C_{n,\gamma} |x|^{-(n-\gamma)} * f$, or equivalently as the inverse fractional Laplacian $(-\Delta)^{-\gamma/2}$.

Key Novelty: Unlike previous studies that typically assume initial data in $L^1$, $L^2$, or Sobolev spaces, this paper considers initial data $(u_0, u_1)$ belonging to pseudo-measure spaces $Y_q$.
The space $Y_q(\mathbb{R}^n)$ is defined as:
$$Y_q(\mathbb{R}^n) := \left\{ f \in \mathcal{S}'(\mathbb{R}^n) : \hat{f} \in L^1_{loc}(\mathbb{R}^n) \text{ and } \|f\|_{Y_q} := \sup_{\xi \in \mathbb{R}^n} |\xi|^q |\hat{f}(\xi)| < \infty \right\}.$$
This framework focuses on the behavior of the Fourier transform near the low-frequency region ($\xi = 0$), where the parameter $q \in (0, n/2)$ quantifies the singularity or decay rate of the Fourier transform.

2. Methodology

The authors employ a combination of Harmonic Analysis and Nonlinear Evolution Equation techniques:

1. Linear Decay Estimates:

   • The authors first analyze the corresponding linear damped wave equation ($I_\gamma(|u|^p) = 0$).
   • They derive $L^2$-framework decay estimates for the linear solution operator $K(t, x)$ acting on initial data in $Y_q$.
   • Crucially, they establish that the decay rate depends explicitly on the low-frequency index $q$. Specifically, for the solution $u_{lin}$, the decay behaves like $(1+t)^{-\frac{n}{4} - \frac{q}{2}}$ in $L^2$, which is faster than the standard $(1+t)^{-n/4}$ decay when $q > 0$.

2. Fixed Point Argument (Global Existence):

   • For the supercritical and critical cases ($p \ge p_{crit}$), the authors construct a solution space $X_s(T)$ equipped with a weighted norm involving decay rates.
   • They utilize Duhamel's principle to rewrite the problem as a fixed-point equation $u = N[u]$.
   • The proof relies on fractional Gagliardo-Nirenberg inequalities and the Hardy-Littlewood-Sobolev inequality to handle the Riesz potential term $I_\gamma(|u|^p)$.
   • The contraction mapping principle (Banach Fixed Point Theorem) is used to prove the existence of unique global small-data solutions.

3. Test Function Method (Blow-up):

   • For the subcritical case ($1 < p < p_{crit}$), the authors prove finite-time blow-up.
   • They employ the test function method (also known as the Kaplan's method or Fujita's method adapted for wave equations).
   • By constructing specific test functions $\chi_R(t)$ and $\phi_R(x)$ involving the Riesz potential structure and the initial data condition $u_0 + u_1 \gtrsim \langle x \rangle^{-n+q}$, they derive a contradiction if a global solution is assumed to exist.

4. Lifespan Estimates:

   • By combining the upper bound (from the blow-up proof) and the lower bound (from the local existence proof), they derive sharp estimates for the lifespan $T_\varepsilon$ of solutions in the subcritical regime.

3. Key Contributions and Results

A. Determination of the Critical Exponent

The paper establishes a new critical exponent $p_{crit}(n, q, \gamma)$ that separates global existence from finite-time blow-up. The exponent is given by:
$$p_{crit}(n, q, \gamma) := 1 + \frac{2 + \gamma}{n - q}.$$

• Constraints: $1 \le n \le 4$ and $0 \le \gamma < q < n/2$.
• Significance: This generalizes the classical Fujita exponent ($p_F = 1 + 2/n$). When $\gamma = 0$ and $q = 0$, the result recovers the standard Fujita exponent. The presence of $q$ (pseudo-measure regularity) and $\gamma$ (Riesz potential strength) shifts the critical threshold.

B. Global Existence (Supercritical and Critical Cases)

• Theorem 1.1 & 1.2: If $p \ge p_{crit}(n, q, \gamma)$, there exists a unique global (in time) solution for sufficiently small initial data $\varepsilon$.
• Decay Rates: The solution satisfies specific decay estimates in $L^2$ and homogeneous Sobolev norms $\dot{H}^s$:
  $$\|u(t)\|_{L^2} \lesssim \varepsilon (1+t)^{-\frac{n}{4} - \frac{q}{2}}, \quad \|u(t)\|_{\dot{H}^1} \lesssim \varepsilon (1+t)^{-\frac{n}{4} - \frac{q}{2} - \frac{1}{2}}.$$
  The decay is accelerated by the parameter $q$, reflecting the influence of the low-frequency behavior of the initial data.

C. Finite-Time Blow-up (Subcritical Case)

• Theorem 1.3: If $1 < p < p_{crit}(n, q, \gamma)$, no global weak solution exists, even for arbitrarily small initial data, provided the initial data satisfies a sign condition ($u_0 + u_1 \gtrsim \langle x \rangle^{-n+q}$).
• This confirms that $p_{crit}$ is indeed the sharp threshold.

D. Sharp Lifespan Estimates

• Proposition 3.1 & 3.2: In the subcritical regime ($1 < p < p_{crit}$), the lifespan $T_\varepsilon$ of the solution satisfies the sharp estimate:
  $$T_\varepsilon \sim \varepsilon^{-\frac{2(p-1)}{2 + \gamma - p(n-q)}}.$$
  This formula precisely quantifies how the lifespan scales with the size of the initial data $\varepsilon$, the nonlinearity power $p$, the Riesz potential order $\gamma$, and the pseudo-measure index $q$.

4. Significance and Impact

1. Extension of Functional Framework: The paper successfully extends the theory of damped wave equations to pseudo-measure spaces $Y_q$. This is significant because these spaces are naturally suited for problems where long-time dynamics are governed by low-frequency phenomena, a concept well-established in fluid dynamics (Navier-Stokes) but less explored in wave equations.
2. Unification of Effects: It provides a unified framework to understand the interplay between damping (which induces diffusion-like behavior), Riesz potentials (non-local interactions), and initial data regularity (specifically low-frequency decay).
3. Refinement of Critical Thresholds: The derived critical exponent $p_{crit} = 1 + \frac{2+\gamma}{n-q}$ demonstrates that:
   • Increasing the Riesz potential order $\gamma$ (stronger non-locality) lowers the critical exponent, making blow-up more likely.
   • Increasing the pseudo-measure index $q$ (stronger control/decay at low frequencies) raises the critical exponent, making global existence more likely.
4. Sharpness: The results are sharp not only in terms of the exponent but also in the lifespan estimates, providing a complete picture of the solution's behavior near the critical threshold.

In conclusion, this work fills a gap in the literature regarding damped wave equations with non-local nonlinearities and initial data in spaces characterized by Fourier decay, offering new insights into the stability and blow-up mechanisms of such systems.