Degenerate 3-evolution equations in Gevrey classes

Technical Summary: Degenerate 3-Evolution Equations in Gevrey Classes

Problem Statement
This paper investigates the well-posedness of the Cauchy problem for third-order evolution differential operators with variable coefficients, specifically those that degenerate at $t=0$. The operator $P$ is defined as:
$$P = D_t + a_3(t)D_x^3 + a_2(t, x)D_x^2 + a_1(t, x)D_x + a_0(t, x)$$
where the leading coefficient $a_3(t)$ vanishes at $t=0$ with a finite order $\ell$ (i.e., $|a_3(t)| \sim t^\ell$). The lower-order coefficients $a_j(t, x)$ for $j=1, 2$ are complex-valued and exhibit specific decay rates as $|x| \to \infty$ and degeneracy behaviors as $t \to 0^+$.

The central challenge addressed is that the vanishing of the leading term $a_3(t)$, combined with the behavior of the lower-order terms, can destroy well-posedness in standard Sobolev spaces ($H^\infty(\mathbb{R})$) or even $L^2(\mathbb{R})$. The authors seek to establish sufficient conditions on the coefficients to ensure well-posedness in Gevrey-type spaces $H^\infty_\theta(\mathbb{R})$, which interpolate between Sobolev spaces and Gevrey classes.

Methodology
The proof strategy relies on a conjugation technique using a change of variables involving pseudodifferential operators of infinite order. The core transformation is:
$$v(t, x) = e^{k(t)\langle D \rangle_h^{1/\theta}} e^{\Lambda(t, x, D)} u(t, x)$$
where:

1. Weighting: The term $e^{k(t)\langle D \rangle_h^{1/\theta}}$ (with $k(t) = \rho_0(T+1-t)$) introduces a time-dependent Gevrey weight to absorb errors and control the growth of the solution.
2. Conjugation: The operator $e^{\Lambda(t, x, D)}$ is constructed with a symbol $\Lambda(t, x, \xi)$ designed to counteract the degeneracy of $a_3(t)$ and the slow decay of $a_1, a_2$.

The symbol $\Lambda$ is constructed by splitting the phase space into "evolution zones" and "pseudodifferential zones" based on the interplay between time $t$ and frequency $\xi$. The construction involves:

• Defining cut-off functions $\psi, \chi$ to localize the symbol behavior.
• Constructing $\Lambda$ as a sum of terms $\Lambda_e$ (handling the degeneracy gap) and $\Lambda_\psi$ (handling the pseudodifferential region).
• Determining specific exponents $q_1$ and $q_2$ (derived from the degeneracy orders $\ell, k, k'$ and decay rates $\sigma_1, \sigma_2$) to minimize the total order of the symbol $\Lambda$.

The transformed problem is analyzed in $L^2(\mathbb{R})$ using energy estimates. The authors employ:

• Sharp-Gårding and Fefferman-Phong inequalities: To establish lower bounds for the real parts of the symbols of the transformed operator, ensuring the positivity required for energy estimates.
• Pseudodifferential Calculus: Utilizing classes of symbols $S^\infty_{\mu;\theta}$ and $SG^\infty_{\mu;\theta}$ to handle operators of infinite order and their compositions.

Key Contributions and Results
The paper establishes necessary and sufficient conditions for well-posedness in $H^\infty_\theta(\mathbb{R})$ based on the interplay between the time-degeneracy of the leading term and the spatial decay of the lower-order terms.

1. Degeneracy Gaps and Gevrey Indices:
   The authors define critical indices $q_1$ and $q_2$ which determine the upper bound for the Gevrey index $\theta$ (specifically, well-posedness holds for $1 < \theta < 1/q$, where $q = \max\{q_1, q_2\}$).

   • $q_2$ depends on the degeneracy gap $\ell - k$ (where $k$ is the order of vanishing of the imaginary part of $a_2$) and the decay rate $\sigma_2$.
   • $q_1$ depends on the degeneracy gap $\ell - k'$ (where $k'$ is the order of vanishing of the imaginary part of $a_1$) and the decay rate $\sigma_1$.

2. Main Theorems:

   • Theorem 2.2 (General Degenerate Case): If the leading term degenerates faster than the lower-order terms ($\ell > k$ and $\ell > k'$), well-posedness in $H^\infty_\theta$ is achieved provided $\theta$ is sufficiently small. Specifically, if $\ell < 2k+1$ and $\sigma_2$ satisfies a specific lower bound, the problem is well-posed for $\theta \in (1, 1/q)$. If $\ell \ge 2k+1$, well-posedness in $L^2$ or $H^\infty$ is generally lost regardless of the decay of lower-order terms.
   • Theorem 2.3 (Partial Degeneracy): If the degeneracy gap exists only at the first order ($\ell \le k$ but $\ell > k'$), the conditions for well-posedness are less restrictive. If the decay of $a_2$ is fast enough ($\sigma_2 \ge 1$), well-posedness can be recovered in $H^\infty(\mathbb{R})$ or even $L^2(\mathbb{R})$ under certain conditions on $q_1$.
   • Theorem 2.6 (Non-Degenerate Leading Term): In the case where the leading term does not degenerate faster than the lower-order terms ($\ell \le \min\{k, k'\}$), the results generalize previous work on non-degenerate equations, showing that well-posedness depends solely on the spatial decay rates $\sigma_1$ and $\sigma_2$.

3. Sharpness of Conditions:
   The paper demonstrates that if the degeneracy gap is too large (e.g., $\ell \ge 2k+1$), no amount of spatial decay in the lower-order coefficients can restore well-posedness in Gevrey spaces with $\theta > 1$. This highlights the "effective" nature of the degeneracy in time.

Significance and Claims
The authors claim that this work provides the first answer in the literature regarding the well-posedness of $p$-evolution equations (specifically $p=3$) where the leading coefficient vanishes at $t=0$ and lower-order terms are complex-valued.

• Extension of Literature: The results extend known theories for non-degenerate operators (where $a_3(t) \ge C > 0$) and Schrödinger-type degenerate equations (where $p=2$) to the third-order case.
• Interplay of Parameters: The paper explicitly quantifies how the "degeneracy gap" in time ($\ell - k$) and the "decay rate" in space ($\sigma$) interact to determine the optimal Gevrey index $\theta$. It shows that a larger degeneracy gap requires a smaller Gevrey index (closer to 1, i.e., less regularity) for well-posedness.
• Limitations: The authors note that the condition $\ell < 2k+1$ in Theorem 2.2 is a technical constraint of their current proof method and whether it is sharp remains an open question for future study. Similarly, the sharpness of the Gevrey spaces obtained is noted as a topic for future investigation.

The paper concludes by stating that the derived conditions are sufficient to ensure the existence and uniqueness of solutions in the specified Gevrey-Sobolev spaces, thereby characterizing the precise regularity required to handle the degeneracy of the operator.