Dispersive estimates for wave-type equations with time-dependent damping

Technical Summary: Dispersive Estimates for Wave-Type Equations with Time-Dependent Damping

Problem Statement
This paper investigates the Cauchy problem for a class of semilinear evolution equations featuring scale-invariant time-dependent damping. The primary equation under study is:
$$\begin{cases} u_{tt} + L_{w^2}u + \frac{\mu}{1+t}u_t = \Delta^\theta f(u), & t > 0, \, x \in \mathbb{R}^n, \\ u(0, x) = 0, \\ u_t(0, x) = u_1(x), \end{cases}$$
where $\mu > 0$, $\theta \in \{0, 1\}$, and $f(u)$ represents a power-type nonlinearity (specifically $|u|^\alpha$ or $\Delta|u|^\alpha$). The operator $L_{w^2}$ is defined via the Fourier multiplier $w(\xi)^2$. The authors focus on the regime $0 < \mu < 1$, which corresponds to weakly damped hyperbolic behavior where dissipative effects do not dominate dispersive effects. The central objective is to establish the global (in time) existence of small data solutions and to determine the critical exponents ($\alpha_{\text{crit}}$) that distinguish between global existence and finite-time blow-up.

Methodology
The analysis relies fundamentally on deriving sharp $L^p-L^q$ dispersive decay estimates for the corresponding linear problem. The methodology proceeds as follows:

1. Fourier Analysis and ODE Reduction: By applying the spatial Fourier transform, the linear evolution equation is reduced to a second-order ordinary differential equation in time with a parameter-dependent frequency. Through a change of variables $\tau = (1+t)w(\xi)$, this ODE is transformed into a Bessel-type equation.
2. Representation via Hankel Functions: The solution to the transformed ODE is expressed using Hankel functions ($H^\pm_\rho$), where $\rho = 1 - \mu/2$. This representation allows for the precise characterization of the solution's oscillatory and decaying behavior across different frequency regimes.
3. Phase Space Decomposition: The extended phase space is divided into distinct zones based on the magnitude of the frequency $\xi$ and the time variable:
   • High frequencies ($Z_{\text{high}}$): Where $|\xi| \geq 1$.
   • Low frequencies ($Z_1, Z_2, Z_3$): Decomposed based on the relationship between $(1+s)w(\xi)$ and $(1+t)w(\xi)$.
     In each zone, the authors utilize the Mikhlin-Hörmander multiplier theorem to estimate the Fourier multipliers. Special attention is paid to non-homogeneous symbols (e.g., in the Boussinesq case) where low and high frequencies exhibit different behaviors.
4. Fixed Point Argument: For the semilinear problem, the solution is constructed as a fixed point of an operator $N$ defined via Duhamel's principle. The global existence is proven using the contraction mapping principle in a suitably weighted Banach space $X(T)$, provided the initial data is sufficiently small and the nonlinearity exponent $\alpha$ lies in the supercritical range.

Key Contributions and Results

1. Plate-Type Equations ($\sigma$-evolution equations)
For operators where $w(\xi) = |\xi|^\sigma$ with $\sigma \geq 2$ (including the plate equation where $\sigma=2$), the authors establish global existence for small initial data in $L^1 \cap L^2$.

• Improved Parameters: Theorem 2.1 extends previous results by relaxing the dimension constraint from $n < \sigma$ to $n < 2\sigma$ and simplifying the upper bound for the damping parameter $\mu$.
• Critical Exponent: The critical exponent is of Fujita type. Global existence is guaranteed for $\alpha > 1 + \frac{2}{\mu + n/\sigma - 1}$.
• Decay Estimates: The solution satisfies decay estimates of the form $\|u(t)\|_{L^q} \lesssim (1+t)^{1-\mu - \frac{n}{\sigma}(\frac{1}{1+\delta} - \frac{1}{q})}$. The authors note a slight loss of decay rate due to the lack of $L^1-L^q$ estimates in certain regimes.

2. Boussinesq-Type Equations
For operators where $w(\xi) = \sqrt{|\xi|^2 + |\xi|^4}$ (modeling the Boussinesq equation) with derivative-type nonlinearity $\Delta|u|^\alpha$, the authors derive a Strauss-type critical exponent.

• Critical Exponent: Theorem 2.2 identifies $\alpha_{\text{crit}}$ as the positive solution to the equation:
  $$\frac{n}{2}\frac{\alpha-1}{\alpha+1} + \frac{\mu}{2} = \frac{1}{\alpha}\left(1 + \frac{\mu}{2}\right).$$
  This result holds for $0 < \mu < 1$ and sufficiently small initial data in homogeneous Sobolev spaces of negative order.
• Significance of $\mu$: The paper highlights that in the weak damping regime ($0 < \mu < 1$), the competition between dispersion and weak dissipation preserves the hyperbolic nature of the equation, leading to a Strauss-type exponent (dependent on dimension and $\mu$) rather than a Fujita-type exponent.

Significance and Claims
The paper claims to fill specific gaps in the existing literature regarding global existence results for small data solutions to these damped evolution equations.

• Refinement of Plate Equation Results: The authors assert that their results for the plate-type equation improve upon previous work (specifically [15]) by broadening the admissible range of spatial dimensions and simplifying the conditions on the damping parameter $\mu$, while removing upper bounds on the nonlinearity exponent $\alpha$.
• Novelty for Boussinesq Equations: The authors state that, to their knowledge, the Cauchy problem for the Boussinesq equation with scale-invariant time-dependent damping has not been previously addressed. They provide the first derivation of the Strauss-type critical exponent for this specific model in the weakly damped regime.
• Distinction of Regimes: The work reinforces the theoretical distinction between weak damping ($0 < \mu < 1$), which yields hyperbolic-like critical exponents (Strauss), and strong damping ($\mu > 1$), which is expected to yield diffusive-like critical exponents (Fujita), although the latter is only conjectured for the Boussinesq case in this work.

The paper concludes that the interplay between dispersion and the specific structure of the time-dependent damping is the determining factor in the qualitative behavior of solutions and the determination of critical exponents.