Degenerate 3-evolution equations in Gevrey classes

Abstract: We consider the Cauchy problem for third-order evolution differential operators with variable coefficients, depending on time $t\in [0,T]$ and space $x\in\mathbb{R}$, where the leading coefficient $a_3(t)$ vanishes at $t = 0$ with finite order. We establish sufficient conditions on the behavior of the lower order coefficients $a_j(t,x)$ $j=1,2$ as $t \to 0<sup>{+}$ and $|x| \to \infty$ that ensure well-posedness in $L<sup>2(\mathbb{R})$, $H<sup>{\infty}(\mathbb{R})$ and Gevrey-type spaces.

• The paper establishes sufficient conditions for L², H∞, and Gevrey–Sobolev well-posedness when the cubic leading coefficient vanishes like t^ℓ, using an infinite-order pseudodifferential conjugation and energy estimates.
• A degeneracy gap at the Dₓ² level is more harmful than one at the Dₓ level: when ℓ>k, results require ℓ<2k+1 and σ₂>(k+1)/(2+3k−ℓ), while level-1 gaps can still permit H∞ or L² well-posedness.
• The admissible Gevrey index is determined by explicit quantities q₁ and q₂ that combine time-degeneracy gaps with spatial decay, while the sharpness of the thresholds and extension to higher dimensions remain open.

The paper studies the Cauchy problem for third-order $p$-evolution operators with real characteristics,

$$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$

on $[0,T]\times\mathbb{R}$, where the leading coefficient $a_3(t)$ vanishes at $t=0$ like $t^\ell$. The authors establish sufficient conditions on the time degeneracy and spatial decay of the lower-order coefficients that guarantee well-posedness in $L^2$, in $H^\infty$, and in Gevrey–Sobolev spaces $H^\infty_\theta(\mathbb{R})$ (2605.16576).

Setting and background

For non-degenerate leading coefficients ($|a_3(t)|\geq C>0$), the $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$0 theory for this class is complete: necessary and sufficient conditions require decay of $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$1 as $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$2 with $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$3 and of $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$4 as $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$5 with $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$6. When these decays are slower, well-posedness persists only in Gevrey-type spaces $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$7, with an upper bound on $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$8 determined by $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$9; a matching necessary condition states that $[0,T]\times\mathbb{R}$0.

Prior to this work, vanishing of the leading term had been treated only for Schrödinger-type ($[0,T]\times\mathbb{R}$1-evolution) operators, by Cicognani and Reissig, who coupled the time-degeneracy exponents $[0,T]\times\mathbb{R}$2 with the spatial decay $[0,T]\times\mathbb{R}$3 to obtain Gevrey indices. The present paper extends this program to genuine $[0,T]\times\mathbb{R}$4-evolution equations, where the interaction between the cubic principal part and two lower-order levels makes the analysis substantially more involved.

Assumptions and main results

The coefficients satisfy, for constants $[0,T]\times\mathbb{R}$5 and decay rates $[0,T]\times\mathbb{R}$6:

• $[0,T]\times\mathbb{R}$7 (finite-order vanishing at $[0,T]\times\mathbb{R}$8);
• $[0,T]\times\mathbb{R}$9 for $a_3(t)$0 (Levi-type conditions coupling time degeneracy to spatial decay);
• $a_3(t)$1 bounded with all derivatives.

The central notion is the degeneracy gap: the case $a_3(t)$2 (resp. $a_3(t)$3) means the level-$a_3(t)$4 (resp. level-$a_3(t)$5) coefficient degenerates faster than the leading term as $a_3(t)$6.

Degeneracy gap at level 2 ($a_3(t)$7). Defining

$a_3(t)$8

with analogous formulas when $a_3(t)$9, and $t=0$0, the first main theorem gives well-posedness in $t=0$1 for all $t=0$2, provided $t=0$3, $t=0$4, and $t=0$5. Both the time gap and the slow spatial decay jointly determine the admissible Gevrey index. Notably, when $t=0$6 no $t=0$7 well-posedness is obtainable for any $t=0$8: the level-2 degeneracy gap is intrinsically stronger than a level-1 gap.

Degeneracy gap only at level 1 ($t=0$9). Here the threshold $t^\ell$0 — known to be necessary from the non-degenerate theory — replaces the more restrictive condition above. If $t^\ell$1, well-posedness holds for $t^\ell$2; if $t^\ell$3 and $t^\ell$4, for $t^\ell$5; and if $t^\ell$6 with $t^\ell$7, one even recovers $t^\ell$8 well-posedness (for $t^\ell$9) or $L^2$0 well-posedness (for $L^2$1) despite the positive gap $L^2$2. This contrast confirms the asymmetry between the two levels.

No degeneracy gaps ($L^2$3). Only the spatial decay matters, and the results reproduce and extend the non-degenerate theory: well-posedness holds in $L^2$4 with index bounds depending separately on $L^2$5 and $L^2$6 — notably removing the constraint $L^2$7 imposed in earlier work — and in $L^2$8 or $L^2$9 when both decays are fast enough.

Method of proof

The proof follows the energy-method strategy standard for $H^\infty$0-evolution equations, but requires a new conjugation adapted to simultaneous time degeneracy and slow decay. One sets

$H^\infty$1

where $H^\infty$2 and $H^\infty$3 is a real-valued symbol of order $H^\infty$4, constructed explicitly from Gevrey cut-offs. The symbol splits into contributions localized in evolution zones ($H^\infty$5, analogously at level 1) and pseudodifferential zones, mirroring the structure introduced for degenerate Schrödinger equations. The orders $H^\infty$6 are fixed by balancing the growth of $H^\infty$7 and $H^\infty$8, which yields exactly the formulas above; the condition $H^\infty$9 is precisely what forces $H^\infty_\theta(\mathbb{R})$0, so that the infinite-order calculus applies with index $H^\infty_\theta(\mathbb{R})$1.

Invertibility of $H^\infty_\theta(\mathbb{R})$2 is obtained via the reverse operator $H^\infty_\theta(\mathbb{R})$3 plus a Neumann series argument, using the large parameter $H^\infty_\theta(\mathbb{R})$4 to control the zero-order remainder. After conjugation, the transformed operator takes the form $H^\infty_\theta(\mathbb{R})$5, where $H^\infty_\theta(\mathbb{R})$6 has purely imaginary quadratic symbol and $H^\infty_\theta(\mathbb{R})$7 satisfies a positivity estimate. Choosing the free parameters $H^\infty_\theta(\mathbb{R})$8, $H^\infty_\theta(\mathbb{R})$9, $|a_3(t)|\geq C>0$0, $|a_3(t)|\geq C>0$1 renders the level-2 and level-1 symbols non-negative, so that the Fefferman–Phong inequality (level 2) and sharp-Gårding inequality (levels $|a_3(t)|\geq C>0$2) yield the a priori estimate

$|a_3(t)|\geq C>0$3

from which $|a_3(t)|\geq C>0$4 well-posedness of the transformed problem — hence $|a_3(t)|\geq C>0$5 well-posedness of the original one — follows by Gronwall's inequality and inversion of the change of variable. The cases $|a_3(t)|\geq C>0$6 require logarithmic modifications of the symbols $|a_3(t)|\geq C>0$7 and produce the alternative formulas $|a_3(t)|\geq C>0$8, $|a_3(t)|\geq C>0$9.

Limitations and open questions

The authors state several caveats explicitly. First, the conditions $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$00 and $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$01 may be technical rather than sharp: when $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$02 the method yields no well-posedness result at any Gevrey index, even for arbitrarily fast spatial decay, whereas a large gap $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$03 at level 1 alone never precludes some $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$04 well-posedness. Whether $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$05 is genuinely necessary remains untested. Second, all results are sufficient conditions; sharpness of the spaces of well-posedness in the degenerate regime is not established. Third, the analysis is carried out in one space dimension, and the spatial regularity assumption is analytic (though the authors note it extends to Gevrey coefficients of index $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$06, shifting the lower bound on $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$07 accordingly). These questions are deferred to future work.

Conclusion

The paper provides the first well-posedness theory for Cauchy problems associated with $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$08-evolution operators whose leading coefficient vanishes at finite order, covering all regimes of time degeneracy and spatial decay through a unified infinite-order pseudodifferential conjugation. The resulting Gevrey indices encode quantitatively how the degeneracy gaps $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$09, $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$10 interact with the decay rates $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$11, and reveal a structural asymmetry: a degeneracy gap at the level of $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$12 is strictly more damaging than one at the level of $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$13. Establishing necessity of the constraints $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$14 and the threshold on $$P = D_t + a_3(t)D_x^3 + a_2(t,x)D_x^2 + a_1(t,x)D_x + a_0(t,x),$$15, and extending the theory to higher dimensions, remain open.