Critical exponent for semilinear damped wave equations with weighted nonlinear terms and data from Sobolev spaces of negative order

Technical Summary: Critical Exponent for Semilinear Damped Wave Equations with Weighted Nonlinear Terms and Data from Sobolev Spaces of Negative Order

Problem Statement
This paper investigates the Cauchy problem for semilinear damped wave equations featuring weighted nonlinear terms with Coulomb-type singularities and initial data belonging to Sobolev spaces of negative order. The specific equation under consideration is:
$$\begin{cases} u_{tt}(t, x) - \Delta u(t, x) + u_t(t, x) = |x|^{-\alpha}|u(t, x)|^p, & x \in \mathbb{R}^n, t > 0, \\ u(0, x) = \varepsilon u_0(x), \quad u_t(0, x) = \varepsilon u_1(x), & x \in \mathbb{R}^n, \end{cases}$$
where $p > 1$, $\alpha \geq 0$, and the initial data $(u_0, u_1)$ belong to the product space $\dot{H}^{-\beta}(\mathbb{R}^n) \times \dot{H}^{-\beta}(\mathbb{R}^n)$ with $\beta \geq 0$. The parameter $\varepsilon > 0$ represents the size of the initial data. The authors aim to determine the critical exponent $p_c(\alpha, \beta, n)$ that separates the global existence of small data solutions from the finite-time blow-up of weak solutions, and to derive lifespan estimates in the subcritical case.

Methodology
The authors employ a combination of linear decay estimates, fixed-point arguments, and test function methods.

1. Linear Decay Estimates: The analysis begins with the linear damped wave equation. The authors utilize the Fourier transform to derive sharp $L^m-L^q$ decay estimates for the linear kernels $K(t, x)$ and their time derivatives. These estimates bridge the diffusive structure of the damped wave equation with the heat equation, showing that solutions decay similarly to heat equation solutions as $t \to \infty$.
2. Global Existence (Small Data): For the global existence result, the authors define a suitable function space $X(T)$ equipped with time-weighted norms involving the decay rates derived from the linear theory. They construct a solution map $\Phi$ using Duhamel's principle and apply the Banach fixed-point theorem.
   • To handle the singularity $|x|^{-\alpha}$ in the nonlinearity, the authors utilize the Caffarelli-Kohn-Nirenberg inequality (Proposition 2.2) to control the decay rate of the weighted term.
   • The Hardy-Littlewood-Sobolev inequality is used to manage the negative order Sobolev spaces.
   • The proof relies on establishing contraction mapping properties for the operator $\Phi$ under specific conditions on $p$, $\alpha$, and $\beta$.
3. Blow-up Result: For the non-existence of global solutions, the authors use the test function method. They construct a specific test function $\Phi_R(t, x)$ involving a cut-off function in time and a spatial weight related to the singularity. By integrating the weak formulation against this test function and applying Hölder's inequality, they derive a contradiction for large $R$ when $p$ is below the critical threshold.
4. Lifespan Estimates: In the subcritical case ($1 < p < p_c$), the authors derive both lower and upper bounds for the lifespan $T_{\alpha, \beta}(\varepsilon)$. The lower bound is obtained by analyzing the contraction condition in the fixed-point argument, while the upper bound is refined using the test function method with logarithmic corrections.

Key Contributions and Results

• Critical Exponent Determination: The primary result is the identification of the critical exponent for the weighted damped wave equation with negative order data:
  $$p_c(\alpha, \beta, n) := 1 + \frac{4 - 2\alpha}{n + 2\beta}$$
  This formula holds for dimensions $1 \leq n \leq 4$ and parameters $0 \leq \alpha, \beta < n/2$. The authors demonstrate that the weight $|x|^{-\alpha}$ and the negative order $\beta$ both shift the critical exponent to the left compared to the standard Fujita exponent ($1 + 2/n$) or the case with $\alpha=0$.

• Global Existence Theorem (Theorem 1.1): For $p \geq p_c(\alpha, \beta, n)$, the authors prove the existence of a unique global solution for sufficiently small initial data $\varepsilon$. The solution belongs to $C([0, \infty), H^1)$ and satisfies specific time-decay estimates:
  $$\|u(t, \cdot)\|_{L^2} \lesssim \varepsilon (1+t)^{-\frac{\beta}{2}}, \quad \|\nabla u(t, \cdot)\|_{L^2} \lesssim \varepsilon (1+t)^{-\frac{1+\beta}{2}}.$$
  Additional constraints on $p$ are provided for dimensions $n=3, 4$ to ensure the validity of the Sobolev embeddings and inequalities used.

• Blow-up Theorem (Theorem 1.2): For $1 < p < p_c(\alpha, \beta, n)$, the authors prove that no global weak solution exists, even for small data, provided the initial data satisfies a specific lower bound condition involving logarithmic decay.

• Lifespan Estimates: In the subcritical regime, the lifespan $T_{\alpha, \beta}(\varepsilon)$ is estimated as:
  $$\varepsilon^{-\frac{4(p-1)}{4 - 2\alpha - (n+2\beta)(p-1)}} \lesssim T_{\alpha, \beta}(\varepsilon) \lesssim \varepsilon^{-\frac{4(p-1)}{4 - 2\alpha - (n+2\beta)(p-1)}} (\log(\varepsilon^{-1}))^{\frac{4(p-1)}{4 - 2\alpha - (n+2\beta)(p-1)}}.$$
  The authors note that the presence of the weight $|x|^{-\alpha}$ contributes to increasing the lifespan of blow-up solutions compared to the unweighted case.

• Corollaries for $L^m$-based Data: The authors extend their results to initial data in $\dot{H}^{-\bar{\beta}}_m \times \dot{H}^{-\bar{\beta}}_m$ for $m \in (1, 2]$, deriving a corresponding critical exponent $p_c(\alpha, \bar{\beta} + \theta(m), n)$.

Significance and Claims
The paper claims to extend previous results on semilinear damped wave equations by simultaneously incorporating two generalizations:

1. Weighted Nonlinearity: Addressing the Coulomb-type singularity $|x|^{-\alpha}$, which had been studied in isolation (e.g., in [26] for $\beta=0$) but not combined with negative order data in this specific framework.
2. Negative Order Data: Extending the initial data space to $\dot{H}^{-\beta}$, building upon work by [5] and [8] which considered $\alpha=0$.

The authors state that their results generalize the findings of [5, 8, 26]. Specifically, they show that the critical exponent is shifted by both the singularity strength $\alpha$ and the regularity deficit $\beta$. The paper also highlights an improvement in the upper bound estimate for the lifespan compared to previous works (specifically [15, Theorem 1.8]) in the case $\alpha = \beta = 0$, by including a logarithmic correction factor. The work is presented as a rigorous mathematical analysis establishing the threshold between global existence and blow-up for this specific class of equations.