Global existence of small data solutions to 3-D semilinear Euler-Poisson-Darboux equations

Problem Statement
The paper addresses the global existence of small data weak solutions to the $n$-dimensional semilinear Euler-Poisson-Darboux equation:
$$\square u + \frac{\mu}{t} \partial_t u = |u|^p, \quad t \ge 1, \, x \in \mathbb{R}^n$$
where $\mu > 0$ and $p > 1$. The central open question concerns the critical exponent $p_{\text{crit}}(n, \mu) = \max\{p_s(n+\mu), p_f(n)\}$, where $p_s$ is the Strauss exponent and $p_f(n) = 1 + 2/n$ is the Fujita exponent. While blow-up results are established for $1 < p \le p_{\text{crit}}(n, \mu)$ and global existence is solved for dimensions $n=1, 2$, the case for $n=3$ remains partially open. Specifically, the paper focuses on the regime where $\mu \ge 14/5$, for which the critical exponent is determined by the Fujita exponent $p_f(3) = 5/3$. The authors aim to prove global existence for small initial data when $p > \max\{5/3, 1 + 2/\mu\}$.

Methodology
The authors employ a combination of the vector field method and delicate time-decay analysis tailored to the 3-D linear Euler-Poisson-Darboux operator $\square + \frac{\mu}{t} \partial_t$.

1. Explicit Representation via Special Functions: The linear homogeneous equation is solved explicitly using Fourier transforms. The solution operators are expressed in terms of Bessel functions ($J_\nu, Y_\nu$) and Hankel functions ($H^\pm_\nu$). The authors carefully distinguish between cases where the order parameter $\rho = -(\mu+1)/2$ is an integer (requiring second-kind Bessel functions) and non-integer cases, utilizing asymptotic properties and recurrence relations of these special functions to derive precise estimates.
2. Vector Field Norms: To handle the lack of commutativity between the operator $\square + \frac{\mu}{t} \partial_t$ and standard vector fields (a limitation noted in previous works like [16]), the authors introduce specific vector field norms $\|v\|_{Z, s, (p,q)}$. These norms involve derivatives generated by the set $Z = \{\partial, L_0, L_j, \Omega_{kl}\}$, where $L_0$ is the scaling vector field, $L_j$ are Lorentz boosts, and $\Omega_{kl}$ are rotations.
3. Sobolev Embeddings on $S^2$: A key technical innovation is the derivation of a Sobolev embedding on the sphere $S^2$ adapted to these norms. Unlike the 2-D case where $\|u\|_{(2,\infty)} \lesssim \|u\|_{Z,1,2}$, the 3-D case requires the estimate $\|u\|_{(2,\infty)} \lesssim \|u\|_{Z,1,2} + \|\partial u\|_{Z,1,2}$, which is crucial for controlling the nonlinearity.
4. Time-Decay Estimates: The authors establish sharp time-decay estimates for both homogeneous and inhomogeneous linear equations in different frequency zones (low, medium, and high frequencies). These estimates depend on the damping coefficient $\mu$ and the dimension $n=3$.
5. Fixed Point Argument: The nonlinear problem is reformulated as a fixed point problem using Duhamel's principle. A contraction mapping principle is applied in a weighted Banach space $X(T)$, defined by norms involving time weights $t^{\mu/2}$ (for $14/5 \le \mu < 3$) or $t^{3/2 - 3/(1+\epsilon_1)}$ (for $\mu \ge 3$).

Key Contributions and Results
The main result, Theorem 1.1, establishes the global existence of a unique small data solution $u$ for the 3-D problem (1.2) under the following conditions:

• Damping: $\mu \ge 14/5$.
• Nonlinearity: $p > \max\{5/3, 1 + 2/\mu\}$.
• Regularity: The solution belongs to $C([1, \infty); H^2(\mathbb{R}^3)) \cap C^1([1, \infty); H^1(\mathbb{R}^3)) \cap C^2([1, \infty); L^2(\mathbb{R}^3))$.

The paper provides a detailed table of known results and unsolved cases, highlighting that this work resolves the open question for $\mu \ge 3$ (where $p > 5/3$ is sufficient) and significantly advances the case for $14/5 \le \mu < 3$.

Significance and Limitations
The paper claims to solve the open question for $n=3$ in the regime $\mu \ge 14/5$, provided $p$ satisfies the derived lower bound. The authors explicitly note a technical gap for the range $14/5 \le \mu < 3$: the condition $p > 1 + 2/\mu$ is slightly more restrictive than the conjectured critical exponent $p_f(3) = 5/3$. The difference, $p_f(3) - (1 + 2/\mu) = 2(3-\mu)/(3\mu)$, arises from the integrability condition of the time integral in the Duhamel term (specifically, the convergence of $\int_1^t \tau^{\mu/2 + (-\mu/2)p} d\tau$).

The authors state that the remaining gap for $14/5 < \mu < 3$ and the case $0 < \mu < 14/5$ (where the Strauss exponent dominates) will be addressed in forthcoming work using delicate space-time weighted Strichartz estimates. The current work relies on the vector field method combined with Bessel function analysis, which is sufficient for the $\mu \ge 14/5$ regime but encounters technical limitations in the intermediate damping range.