Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 06, pp. 1–28. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu GEVREY REGULARITY OF THE SOLUTIONS OF INHOMOGENEOUS NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS PASCAL REMY Abstract. In this article, we are interested in the Gevrey properties of the formal power series solutions in time of some inhomogeneous nonlinear par- tial differential equations with analytic coefficients at the origin of Cn+1. We systematically examine the cases where the inhomogeneity is s-Gevrey for any s ≥ 0, in order to carefully distinguish the influence of the data (and their de- gree of regularity) from that of the equation (and its structure). We thus prove that we have a noteworthy dichotomy with respect to a nonnegative rational number sc fully determined by the Newton polygon of a convenient associated linear partial differential equation: for any s ≥ sc, the formal solutions and the inhomogeneity are simultaneously s-Gevrey; for any s < sc, the formal solu- tions are generically sc-Gevrey. In the latter case, we give an explicit example in which the solution is s′-Gevrey for no s′ < sc. As a practical illustration, we apply our results to the generalized Burgers-Korteweg-de Vries equation. 1. Problem setting The nonlinear evolution equations are often used to represent the motion of the isolated waves, localized in a small part of space in many fields such as optical fibers, neural physics, solid state physics, hydrodynamics, diffusion process, plasma physics and nonlinear optics (nonlinear heat equation, nonlinear Klein-Gordon equation, nonlinear Euler-Lagrange equation, Burgers equation, Korteweg-de Vries equation, Boussinesq equation, etc.). When studying such equations, one of the major challenges is the determination of exact solutions, if any exists, and the precise analysis of their properties (dy- namic, asymptotic behavior, etc.) in order to have a better understanding of the mechanism of the underlying physical phenomena and dynamic processes. To do that, one possible way is given by the summation theory which allows to construct analytic solutions from formal ones. The present work is devoted to the study of these formal solutions (existence, unicity, Gevrey properties), which is the first step towards this approach. More precisely, we consider an inhomogeneous nonlinear partial differential equation with 2020 Mathematics Subject Classification. 35C10, 35G20, 35Q53. Key words and phrases. Gevrey order; inhomogeneous partial differential equation; nonlinear partial differential equation; generalized Burgers-KdV equation; Newton polygon; formal power series; divergent power series. ©2023. This work is licensed under a CC BY 4.0 license. Submitted November 6, 2021. Published January 19, 2023. 1 2 P. REMY EJDE-2023/06 a 1-dimensional time variable t ∈ C and a n-dimensional spatial variable x = (x1, . . . , xn) ∈ Cn of the form ∂κt u− ∑ i∈K ∑ q∈Qi ( ∑ p∈Pi,q tvi,q,pai,q,p(t, x)up ) ∂it∂ q xu = f̃(t, x) ∂jt u(t, x)|t=0 = ϕj(x), j = 0, . . . , κ− 1 (1.1) where • κ ≥ 1 is a positive integer; • K is a nonempty subset of {0, . . . , κ− 1}; • Qi is a nonempty finite subset of Nn for all i ∈ K (N denotes the set of the nonnegative integers); • Pi,q is a nonempty finite subset of N for all i ∈ K and q ∈ Qi; • vi,q,p ≥ 0 is a nonnegative integer for all i ∈ K, q ∈ Qi and p ∈ Pi,q; • the coefficients ai,q,p(t, x) are analytic on a polydisc Dρ0,...,ρn := Dρ0 × Dρ1 × · · · ×Dρn centered at the origin of Cn+1 (Dρ denotes the disc with center 0 ∈ C and radius ρ > 0) and satisfy ai,q,p(0, x) 6≡ 0 for all i ∈ K, q ∈ Qi and p ∈ Pi,q; • ∂qx denotes the derivative ∂q1x1 . . . ∂qnxn while q := (q1, . . . , qn) ∈ Nn; • the inhomogeneity f̃(t, x) is a formal power series in t with analytic coef- ficients in Dρ1,...,ρn (we denote by f̃(t, x) ∈ O(Dρ1,...,ρn)[[t]]) which may be smooth, or not; (We denote f̃ with a tilde to emphasize the possible divergence of the series) • the initial conditions ϕj(x) are analytic on Dρ1,...,ρn for j = 0, . . . , κ− 1. Looking for a formal solution ũ(t, x) ∈ O(Dρ1,...,ρn)[[t]], and writing any element g̃(t, x) ∈ O(Dρ1,...,ρn)[[t]] in the form g̃(t, x) = ∑ j≥0 gj,∗(x) tj j! with gj,∗(x) ∈ O(Dρ1,...,ρn) for all j, we easily get that the coefficients uj,∗(x) of ũ(t, x) are uniquely determined by the recurrence relations uj+κ,∗(x) = fj,∗(x) + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `0+`1+···+`p+ `p+1=j−vi,q,p j!ai,q,p;`0,∗(x)u`1,∗(x) . . . u`p,∗(x)∂qxu`p+1+i,∗(x) `0!`1! . . . `p!`p+1! (1.2) together with the initial conditions uj,∗(x) = ϕj(x) for j = 0, . . . , κ− 1. As usual, we use the classical conventions that the fourth sum is zero as soon as j−vi,q,p < 0, and that the product u`1,∗(x)...u`p,∗(x) `1!...`p! is 1 as soon as p = 0. The purpose of the paper is to answer to the question: What relationship exists between the Gevrey order of the formal solution ũ(t, x) and the Gevrey order of the inhomogeneity f̃(t, x)? Indeed, according to the algebraic structure of the s-Gevrey spacesO(Dρ1,...,ρn)[[t]]s (see Section 2 for the exact definition of theses spaces), it is classical one has ũ(t, x) ∈ O(Dρ1,...,ρn)[[t]]s ⇒ f̃(t, x) ∈ O(Dρ1,...,ρn)[[t]]s. EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 3 But, what can we say about the converse? A first answer was provided by the author in [37] (see also [30, 35] for some particular examples) in the linear case (case Pi,q = {0} for all i and q) and in the semilinear case with a polynomial nonlinearity in u (case P0,0 ∩ (N\{0}) 6= ∅ and Pi,q = {0} for all (i, q) 6= (0, 0)); that is for the equations of the form L(u)− P (u) = f̃(t, x), (1.3) where L is a linear partial differential operator of the form L = ∂κt − ∑ i∈K ∑ q∈Qi ai,q(t, x)∂it∂ q x, and where P (X) is either zero or a polynomial with valuation ≥ 2 in X and with analytic coefficients at the origin of Cn+1. In particular, he proved that the Gevrey orders of the formal solution ũ(t, x) and the inhomogeneity f̃(t, x) are closely related through a special value, called the critical value of (1.3) and denoted in the sequel by sc, which is fully determined by the Newton polygon at t = 0 of the linear operator L. Proposition 1.1 ([37]). Let sc denote the nonnegative rational number equal to the inverse of the smallest positive slope of the Newton polygon at t = 0 of the linear operator L if any exists, and equal to 0 otherwise. Then (1) ũ(t, x) and f̃(t, x) are simultaneously s-Gevrey for any s ≥ sc; (2) ũ(t, x) is generically sc-Gevrey while f̃(t, x) is s-Gevrey with s < sc. Remark 1.2. When the inhomogeneity f̃(t, x) is s-Gevrey with s < sc, the hy- potheses made on (1.3) do not allow in general to specify the exact Gevrey order of the formal solution ũ(t, x) as in the opposite case s ≥ sc (Point 1). However, the second point of Proposition 1.1 asserts that this order is always less or equal to sc (This is obvious due to the filtration of the Gevrey spaces (see Section 3.1) and the first point of Proposition 1.1.) and that this inequality is the best possible. Indeed, one can easily find cases for which ũ(t, x) is exactly sc-Gevrey (see [37, Prop. 4.11] for instance). In this article, we propose to extend the results of Proposition 1.1 to the very general (1.1). Similarly as the equations of the form (1.3), the critical value sc of this equation is fully determined by the Newton polygon at t = 0 of a convenient linear partial differential operator, called the associated linear operator. Section 3 is devoted to the study of this operator: after a heuristic approach using two simple examples (Section 3.1), we describe the general geometric structure of its Newton polygon at t = 0, and we derive from this the value of sc as well as some fundamental associated inequalities (Section 3.2). In Section 4, we state our main result (Theorem 4.1) and some direct consequences. The proof of Theorem 4.1 is developed in Sections 5 and 6. Before remembering some results about the Gevrey formal series (Section 2), let us mention here that a similar problem has already been studied by Tahara in [46] in the case of real variables. However, the calculations we develop in this paper are based on a very different approach. Let us also mention that other slightly different works have also been done for several years by many authors towards the convergence [21, 23, 43] and the Gevrey order [14, 15, 22, 39, 40, 41, 42, 44, 45] of the formal power series solutions of 4 P. REMY EJDE-2023/06 some singular nonlinear partial differential equations, and towards the summability [17, 19, 26, 34, 36, 38] of the formal power series solution of some nonlinear partial differential equations. Furthermore, in [12, 13], Lastra and Malek considered some parametric nonlinear partial differential equations; in [18], Malek investigated the Gevrey properties of some nonlinear integro-differential equations. Of course, given the technical and computational difficulties inherent in the nonlinearity, the known results, especially in the framework of the summability or the multisummability, are currently far fewer than in the linear case. 2. Gevrey formal series All along the article, we consider t as the variable and x as a parameter. Thereby, to define the notion of Gevrey classes of formal power series in O(Dρ1,...,ρn)[[t]], one extends the classical notion of Gevrey classes of formal power series in C[[t]] to families parametrized by x in requiring similar conditions, the estimates be- ing however uniform with respect to x. Doing that, any formal power series of O(Dρ1,...,ρn)[[t]] can be seen as a formal power series in t with coefficients in a convenient Banach space defined as the space of functions that are holomorphic on a polydisc Dρ,...,ρ (0 < ρ ≤ min(ρ1, . . . , ρn)) and continuous up to its boundary, equipped with the usual supremum norm. For a general study of the formal power series with coefficients in a Banach space, we refer for instance to [2]. In the sequel, we endow Cn with the maximum norm: for x = (x1, . . . , xn) ∈ Cn, ‖x‖ = max `∈{1,...,n} |x`|. Definition 2.1. Let s ≥ 0. A formal power series ũ(t, x) = ∑ j≥0 uj,∗(x) tj j! ∈ O(Dρ1,...,ρn)[[t]] is said to be Gevrey of order s (in short, s-Gevrey) if there exist three positive constants 0 < ρ < min(ρ1, . . . , ρn), C > 0 and K > 0 such that the inequalities sup ‖x‖≤ρ |uj,∗(x)| ≤ CKjΓ(1 + (s+ 1)j) hold for all j ≥ 0. In other words, Definition 2.1 means that ũ(t, x) is s-Gevrey in t, uniformly in x on a neighborhood of x = (0, . . . , 0) ∈ Cn. We denote by O(Dρ1,...,ρn)[[t]]s the set of all the formal series in O(Dρ1,...,ρn)[[t]] which are s-Gevrey. Observe that the set C{t, x} of germs of analytic functions at the origin of Cn+1 coincides with the union ∪ρ1>0,...,ρn>0O(Dρ1,...,ρn)[[t]]0. In particular, any element of O(Dρ1,...,ρn)[[t]]0 is convergent and C{t, x}∩O(Dρ1,...,ρn)[[t]] = O(Dρ1,...,ρn)[[t]]0. Observe also that the sets O(Dρ1,...,ρn)[[t]]s are filtered as follows O(Dρ1,...,ρn)[[t]]0 ⊂ O(Dρ1,...,ρn)[[t]]s ⊂ O(Dρ1,...,ρn)[[t]]s′ ⊂ O(Dρ1,...,ρn)[[t]] for all s and s′ satisfying 0 < s < s′ < +∞. The following proposition specifies the algebraic structure of O(Dρ1,...,ρn)[[t]]s. Proposition 2.2. Let s ≥ 0. Then the set (O(Dρ1,...,ρn)[[t]]s, ∂t, ∂x1 , . . . , ∂xn) is a C-differential algebra. EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 5 Proof. Since (O(Dρ1,...,ρn)[[t]], ∂t, ∂x1 , . . . , ∂xn) is a C-differential algebra, it is suffi- cient to prove that O(Dρ1,...,ρn)[[t]]s is stable under multiplication and derivations. The proof of the stability under the multiplication and the derivation ∂t is similar to the one already detailed in [31, Prop. 1] (see also [2, p. 64]) in the case n = 1. To prove the stability under the derivation ∂x` with ` ∈ {1, . . . , n}, we proceed as follows. Let ũ(t, x) ∈ O(Dρ1,...,ρn)[[t]]s as in Definition 2.1 and w̃(t, x) = ∂x` ũ(t, x). For a given 0 < ρ′ < ρ, the Cauchy Integral Formula gives us, for all j ≥ 0 and all ‖x‖ ≤ ρ′: wj,∗(x) = ∂x`uj,∗(x) = 1 (2iπ)n ∫ γ(x) uj,∗(x ′) (x′` − x`)2 ∏n k=1 k 6=` (x′k − xk) dx′, where γ(x) := {x′ = (x′1, . . . , x ′ n) ∈ Cn; |x′k − xk| = ρ − ρ′ for all k ∈ {1, . . . , n}}. Hence, the inequalities sup ‖x‖≤ρ′ |wj,∗(x)| ≤ C ′KjΓ(1 + (s+ 1)j) with C ′ = C ρ− ρ′ for all j ≥ 0. Indeed, the definition of the path γ(x) implies ‖x′‖ ≤ ρ. The proof is complete. � Observe that the stability under the derivation ∂x` would not be guaranteed without the condition “there exist 0 < ρ < min(ρ1, . . . , ρn) . . . ” in Definition 2.1. 3. Associated linear operator As we said in Section 1, the critical value of (1.1) is fully determined by the Newton polygon at t = 0 of a convenient linear partial differential operator. In Section 3.1 below, we investigate two simple examples to heuristically introduce this operator. 3.1. Two preliminaries examples. Example 1: a fundamental equation. In this first example, we consider the equation ∂tu = tvup∂qxu, v, p ≥ 0, q ≥ 2 u(0, x) = 1 1− x ∈ O(D1) (3.1) in two variables (t, x) ∈ C2. When p = 0, (3.1) is linear and the Newton polygon at t = 0 of its associated linear operator ∂t− tv∂qx (see Figure 1) has a unique positive slope, which is equal to k = 1+v q−1 [30, 37]. - −1 - 1 • - q -v • Figure 1. Newton polygon at t = 0 of ∂t − tv∂qx 6 P. REMY EJDE-2023/06 Hence, applying Proposition 1.1, its formal solution ũ(t, x) is generically sc- Gevrey with sc = 1 k = q−1 1+v . In fact, using a method similar to the one developed in [30, Prop. 4.11], one can even show that ũ(t, x) is exactly sc-Gevrey. When p ≥ 1, (3.1) is no longer linear, but the particular value sc continues to play an important role as shown by the following result. Proposition 3.1. The formal solution ũ(t, x) of (3.1) is exactly sc-Gevrey for any p ≥ 0. Proof. According to the general identities (1.2), the coefficients uj,∗(x) of ũ(t, x) are recursively determined from the initial conditions u0,∗(x) = 1 1−x by the relations uj+1,∗(x) = j! ∑ `1+···+`p+ `p+1=j−v u`1,∗(x) . . . u`p,∗(x)∂qxu`p+1,∗(x) `1! . . . `p!`p+1! . From this, we first derive that • the terms uj(1+v)+`,∗(x) are zero for all j ≥ 0 and all ` ∈ {1, . . . , v}; • the terms uj(1+v),∗(x) read for all j ≥ 0 in the form uj(1+v),∗(x) = aj (1− x)j(q+p)+1 , where the coefficients aj is positive and satisfies the inequalities Aj ≤ aj ≤ Cj,p+1Aj with Aj = (j(q + p))!∏j `=1 ∏p m=1(`q + (`− 1)p+m) j∏ `=1 v∏ m=1 ((`− 1)(1 + v) +m); and C0,p+1 = 1 and, for all j ≥ 1, Cj,p+1 = the number of the nonzero terms in the sum∑ `1+···+`p+ `p+1=(j−1)(1+v) u`1,∗(x) . . . u`p,∗(x)∂qxu`p+1,∗(x) `1! . . . `p!`p+1! . As previously, we use in the definition of Aj the classical convention that the prod- ucts are 1 as soon as j (or p, or v) is zero. The constants Aj ’s can be bound for all j ≥ 0 by applying technical Lemmas 3.2 and 3.3. To bound the constants Cj,p+1’s, it suffices to observe that the property u`,∗(x) 6≡ 0⇔ ` is a multiple of 1 + v implies that the Cj,p+1’s can be recursively determined from C0,p+1 = 1 by the relation Cj,p+1 = ∑ `′1+···+` ′ p+` ′ p+1=j−1 C`′1,p+1 . . . C`′p,p+1C`′p+1,p+1. (3.2) Thereby, the Cj,p+1’s are the generalized Catalan numbers of order p + 1 and we have Cj,p+1 = 1 jp+ 1 ( j(p+ 1) j ) ≤ 2j(p+1) for all j ≥ 0 (see [10, 11, 27] for instance). These numbers were named in honor of the mathematician Eugène Charles Catalan (1814-1894). They appear in many EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 7 probabilist, graphs and combinatorial problems. For example, they can be seen as the number of (p + 1)-ary trees with j source-nodes, or as the number of ways of associating j applications of a given (p + 1)-ary operation, or as the number of ways of subdividing a convex polygon into j disjoint (p + 2)-gons by means of non-intersecting diagonals. They also appear in theoretical computers through the generalized Dyck words. See for instance [10] and the references inside. Therefore, (j(q + p))!(jv)! (q + p)jp(j!)p ≤ aj ≤ (2p+1(1 + v)v)j (j(q + p))!(j!)v (j!)p . (3.3) We are now able to prove Proposition 3.1. Choosing 0 < r < 1, we first derive from the second inequaliy of (3.3) the relations |uj(1+v),∗(x)| ≤ 1 1− r (2p+1(1 + v)v (1− r)q+p )j (j(q + p))!(j!)v (j!)p for all j ≥ 0 and |x| ≤ r. On the other hand, applying the Stirling’s Formula, we obtain the equivalence (j(q + p))!(j!)v (j!)pΓ(1 + j(sc + 1)(1 + v)) ∼ j→+∞ √ 2π v−p √ q + p q + v j v−p 2 ( (q + p)q+p (q + v)q+v )j when j tends to infinity (we have (sc+1)(1+v) = q+v). Consequently, there exist two convenient positive constants C,K > 0 such that |uj(1+v),∗(x)| ≤ CKjΓ(1 + j(sc + 1)(1 + v)) for all j ≥ 0 and |x| ≤ r, and we can conclude that ũ(t, x) is sc-Gevrey (recall indeed that the coefficients u`,∗(x) are zero as soon as ` is not a multiple of 1 + v). It is left to prove that ũ(t, x) is s-Gevrey for no s < sc. To do that, let us suppose the opposite. Then, Definition 2.1 and the first inequality of (3.3) imply the relations (j(q + p))!(jv)! (q + p)jp(j!)p ≤ uj(1+v),∗(0) ≤ CKjΓ(1 + j(s+ 1)(1 + v)) and, consequently, the inequalities 1 ≤ C(K(q + p)p)j (j!)pΓ(1 + j(s+ 1)(1 + v)) (j(q + p))!(jv)! for all j ≥ 0 and some convenient positive constants C and K independent of j. The result follows since such inequalities are impossible. Indeed, applying the Stirling’s formula, we obtain C(K(q + p)p)j (j!)pΓ(1 + j(s+ 1)(1 + v)) (j(q + p))!(jv)! ∼ j→+∞ C ′j p−1 2 (K ′ jσ )j (3.4) with • C ′ = C √ (s+1)(1+v)(2π)p−1 (q+p)v ; • K ′ = Keσ((s+1)(1+v))(s+1)(1+v) (q+p)qvv ; • σ = q + v − (s+ 1)(1 + v), and the right hand-side of (3.4) goes to 0 when j tends to infinity (we have indeed σ > q+ v− (sc + 1)(1 + v) = 0 by assumption on s). This completes the proof. � 8 P. REMY EJDE-2023/06 Lemma 3.2. Let j ≥ 0. Then (j!)p ≤ j∏ `=1 p∏ m=1 (`q + (`− 1)p+m) ≤ (q + p)jp(j!)p. Proof. The statement is clear when j = 0 or p = 0, and it stems from the inequalities j∏ `=1 `p ≤ j∏ `=1 p∏ m=1 (`q + (`− 1)p+m) ≤ j∏ `=1 (`q + `p)p when j, p ≥ 1. � Lemma 3.3. Let j ≥ 0. Then (jv)! ≤ j∏ `=1 v∏ m=1 ((`− 1)(1 + v) +m) ≤ (1 + v)jv(j!)v. (3.5) Proof. Since Lemma 3.3 is obvious for v = 0, we assume v ≥ 1 in calculations below. The first inequality of (3.5) is proved by induction on j. It is clear for j = 0 and j = 1. Let us now suppose that it holds for a certain j ≥ 1. Then j+1∏ `=1 v∏ m=1 ((`− 1)(1 + v) +m) ≥ (jv)! v∏ m=1 (j(1 + v) +m) = ((j + 1)v + j)! (jv + j)! (jv)! and the conclusion follows from the inequality( (j + 1)v + j j ) ≥ ( jv + j j ) . As for the second inequality of (3.5), it is clear when j = 0, and it stems from the inequality j∏ `=1 v∏ m=1 ((`− 1)(1 + v) +m) ≤ j∏ `=1 (`(1 + v))v when j ≥ 1. � Proposition 3.1 tells us that the power up does not affect the Gevrey order of the formal solution ũ(t, x), that is it is the same for any p ≥ 0. In particular, it is fully determined by the Newton polygon at t = 0 of the linear operator ∂t − tv∂qx. Changing besides the initial condition u(0, x) = 1 1−x by another simple initial condition (e.g. ex, or any other condition provided that the estimates on the coefficients uj,∗(x) of ũ(t, x) are sufficiently simple to calculate), one can check that this property remains valid, that is the Gevrey order of ũ(t, x) is the same for any p ≥ 0 (here, generically sc-Gevrey according to Proposition 1.1). It is then reasonable to think that this is also true for any arbitrary analytic initial condition at the origin x = 0. Let us now look at another fundamental equation. EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 9 Example 2: the gBKdV equation. In this second example, we focus on the general- ized Burgers-Korteweg-de Vries equation (in short, the gBKdV equation) ∂tu = tv1up1∂q1x u+ tv2up2∂q2x u, v1, v2, p1, p2, q2 ≥ 0, q1 ≥ 2 u(0, x) = ϕ(x) (3.6) in two variables (t, x) ∈ C2 and with analytic initial condition. When (v1, p1, q1, v2, p2, q2) = (0, 0, 2, 0, 1, 1), we have the Burgers equation, and when (v1, p1, q1, v2, p2, q2) = (0, 0, 3, 0, 1, 1), we have the Korteweg-de Vries equa- tion. (3.6) is fundamental in many physical, mechanical and chemical problems. For instance, it allows to model nonlinear waves in dispersive-dissipative media with instabilities, waves arising in thin films flowing down an inclined surface, moderate- amplitude shallow-water surface waves, changes of the concentration of substances in chemical reactions, etc. Various investigations in real variables were already done by many authors to- wards the Gevrey properties of the formal solution ũ(t, x) of (3.6) under more or less generic assumptions (see [5, 6, 7, 8, 9] for instance, and the references inside). In particular, it was proved for the Korteweg-de Vries equation that ũ(t, x) is 2- Gevrey. In our present study, we can prove in a similar way of (3.1) the following result. Proposition 3.4. Let us assume ϕ(x) = 1 1−x and let us set sc = max ( q1 − 1 1 + v1 , q2 − 1 1 + v2 ) . Then, the formal solution ũ(t, x) of (3.6) is exactly sc-Gevrey for any p1, p2 ≥ 0. A much more general statement including variable coefficients, general analytic initial condition and inhomogeneous part will be given later in Example 4.3. As in the previous example, Proposition 3.4 tells us that the Gevrey order of ũ(t, x) does not depend on either the power up1 or the power up2 , and that it is fully determined by the smallest positive slope of the Newton polygon at t = 0 of a convenient linear operator; namely, the operator ∂t − tv1∂q1x − tv2∂q2x [37]. Remark 3.5. In the case where q1 = q2 = q, the value sc given in Proposition 3.4 is more precisely equal to sc = q − 1 1 + min(v1, v2) , and we can actually choose the operator ∂t − tmin(v1,v2)∂qx as the associated linear operator. 3.2. Associated linear operator. According to the above calculations, it is nat- ural to introduce the following definition. Definition 3.6. We call linear operator associated with (1.1) the operator ∂κt − ∑ i∈K ∑ q∈Qi tvi,q,p∗∂it∂ q x, (3.7) where vi,q,p∗ = minp∈Pi,q vi,q,p for all i ∈ K and q ∈ Qi. Let us now describe the general structure of the Newton polygon at t = 0 of the operator (3.7). We choose here the definition of Miyake [20] (see also Yonemura [49] or Ouchi [25]) which is an analogue to the one given by Ramis [29] for the linear 10 P. REMY EJDE-2023/06 ordinary differential equations. Recall that, Tahara and Yamazawa use in [47] a slightly different one. Then, denoting by C(a, b) the domain C(a, b) = {(x, y) ∈ R2 : x ≤ a and y ≥ b} for any (a, b) ∈ R2, the Newton polygon at t = 0 of the operator (3.7) is defined as the convex hull of C(κ,−κ) ⋃⋃ i∈K ⋃ q∈Qi C(λ(q) + i, vi,q,p∗ − i), where λ(q) = q1 + · · ·+ qn denotes the length of q = (q1, . . . , qn) ∈ Nn. The geometric structure of this domain is specified as follows. Proposition 3.7. Let S = {(i, q) such that i ∈ K, q ∈ Qi and λ(q) > κ− i}. (1) Suppose S = ∅. Then, the Newton polygon at t = 0 of the operator (3.7) is reduced to the domain C(κ,−κ). In particular, it has no side with a positive slope (see Figure 2(a)). (2) Suppose S 6= ∅. Then, the Newton polygon at t = 0 of the operator (3.7) has at least one side with a positive slope. Moreover, its smallest positive slope k is given by k = min (i,q)∈S ( κ− i+ vi,q,p∗ λ(q)− κ+ i ) = κ− i∗ + vi∗,q∗,p∗ λ(q∗)− κ+ i∗ , where the pair (i∗, q∗) ∈ S stands for any convenient pair, chosen and fixed once and for all, so that the edge with slope k be the segment with end points (κ,−κ) and (λ(q∗) + i∗, vi∗,q∗,p∗ − i∗) (see Figure 2(b)). - −κ - κ • • •• • • • • • - −κ - κ - λ(q∗) + i∗-vi∗,q∗,p∗ − i∗ • slo pe k • • • • • • • • • • • • • • • • • (a) Case S = ∅ (b) Case S 6= ∅ Figure 2. Newton polygon at t = 0 of the operator (3.7) Proof. The first point is obvious from the fact that S = ∅ implies C(λ(q)+i, vi,q,p∗− i) ⊂ C(κ,−κ) for all i ∈ K and q ∈ Qi. As for the second point, it suffices to remark, on one hand, that C(λ(q) + i, vi,q,p∗ − i) ⊂ C(κ,−κ) for all pairs (i, q) 6∈ S, and, on the other hand, that the segment with the two end points (κ,−κ) and (λ(q) + i, vi,q,p∗ − i) has a positive slope equal to (κ− i+ vi,q,p∗)/(λ(q)− κ+ i) for all pairs (i, q) ∈ S. � The critical value of (1.1) is defined as in [37] for the case of the equations of the form (1.3). EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 11 Definition 3.8. Let S = {(i, q) : such that i ∈ K, q ∈ Qi and λ(q) > κ−i}. Then, the critical value sc of (1.1) is the nonnegative rational number sc := 0 if S = ∅ 1 k = λ(q∗)− κ+ i∗ κ− i∗ + vi∗,q∗,p∗ if S 6= ∅ In other words, the critical value of (1.1) is equal to the inverse of the smallest positive slope of the Newton polygon at t = 0 of its associated linear operator (3.7) if any exists, and 0 otherwise. According to the definition of sc, we derive in particular from Proposition 3.7 the following inequalities which will play a fundamental role in the proof of our main theorem (see Section 5). Lemma 3.9. Let s ≥ sc. Then (s+ 1)(κ− i+ vi,q,p) ≥ λ(q) + vi,q,p for all i ∈ K, all q ∈ Qi and all p ∈ Pi,q. Proof. Let us first assume S = ∅. Then, since s ≥ sc = 0, we have (s+ 1)(κ− i+ vi,q,p) ≥ κ− i+ vi,q,p and the result follows from the inequality λ(q) ≤ κ− i. Let us now assume S 6= ∅. From the definition of sc and the definition of the vi,q,p∗ ’s, we first have s ≥ sc ≥ λ(q)− κ+ i κ− i+ vi,q,p∗ ≥ λ(q)− κ+ i κ− i+ vi,q,p > 0 for all (i, q) ∈ S, and next s ≥ λ(q)− κ+ i κ− i+ vi,q,p (3.8) for all i and q, since λ(q) − κ + i ≤ 0 while (i, q) 6∈ S. Lemma 3.9 follows by first adding +1 to both sides of (3.8) and then by multiplying by the positive term κ− i+ vi,q,p. � 4. Main result We are now able to state the main result of the article. Theorem 4.1. Let sc be the critical value of (1.1) (see Definition 3.8). Then (1) ũ(t, x) and f̃(t, x) are simultaneously s-Gevrey for any s ≥ sc; (2) ũ(t, x) is generically sc-Gevrey while f̃(t, x) is s-Gevrey with s < sc. Observe that Theorem 4.1 coincides with Proposition 1.1 in the case of (1.3). It generalizes therefore the results stated in [37], but also those already obtained by the author in the linear case [31, 32, 33]. Observe also that Theorem 4.1 yields a result similar to the Maillet-Ramis Theo- rem for the ordinary linear differential equations [28, 29] (see also [16, Thm. 4.2.7]). Corollary 4.2. Assume that the inhomogeneity f̃(t, x) is convergent. Then, the formal solution ũ(t, x) of (1.1) is either convergent or 1/k-Gevrey, where k stands for the smallest positive slope of the Newton polygon at t = 0 of the associated linear operator (3.7). 12 P. REMY EJDE-2023/06 Before starting the proof of Theorem 4.1, let us illustrate it with a simple exam- ple. Example 4.3 (Back to the gBKdV equation). Let us consider the general inho- mogeneous gBKdV equation ∂tu− tv1a1(t, x)up1∂q1x u− tv2a2(t, x)up2∂q2x u = f̃(t, x) v1, v2, p1, p2, q2 ≥ 0, q1 ≥ 2 (4.1) in two variables (t, x) ∈ C2 and with analytic initial condition u(0, x) = ϕ(x). Let us also consider the following two particular cases: • the inhomogeneous Burgers equation: ∂tu− tv1a1(t, x)∂2xu− tv2a2(t, x)u∂xu = f̃(t, x); (4.2) • the inhomogeneous Korteweg-de Vries equation: ∂tu− tv1a1(t, x)∂3xu− tv2a2(t, x)u∂xu = f̃(t, x). (4.3) Applying successively Definitions 3.6 and 3.8, the linear operator associated with (4.1) is defined as ∂t − tv1∂q1x − tv2∂q2x if q1 6= q2 ∂t − tmin(v1,v2)∂qx if q1 = q2 = q, and consequently the critical value of (4.1) is given by sc = max ( q1−1 1+v1 , q2−11+v2 ) if q1 6= q2 q−1 1+min(v1,v2) if q1 = q2 = q Theorem 4.1 yields the characterization of the Gevrey order of the formal solution ũ(t, x) of (4.1) in terms of the one of the inhomogeneity f̃(t, x). (1) Assume that f̃(t, x) is s-Gevrey with s ≥ sc. Then ũ(t, x) is also s-Gevrey. (2) Assume that f̃(t, x) is s-Gevrey with s < sc. Then ũ(t, x) is generically sc-Gevrey. In the two special cases of (4.2) and (4.3), these two points apply respectively with sc = 1 1+v1 and sc = 2 1+v1 . In particular, for the classical analytic initial condition Korteweg-de Vries equation ∂tu+ ∂3xu+ 6u∂xu = 0 u(0, x) = ϕ(x) we find the well-known fact that the formal solution ũ(t, x) is 2-Gevrey. Let us now turn to the proof of Theorem 4.1. This one is detailed in the following two sections. The first point is the most technical and the most complicated. Its proof is based on the Nagumo norms, a technique of majorant series and a fixed point procedure (see Section 5). As for the second point, it stems both from the first one and from Proposition 6.1 that gives an explicit example for which ũ(t, x) is s′-Gevrey for no s′ < sc while f̃(t, x) is s-Gevrey with s < sc (see Section 6). EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 13 5. Proof of the first item of Theorem 4.1 According to Proposition 2.2, it is clear that ũ(t, x) ∈ O(Dρ1,...,ρn)[[t]]s ⇒ f̃(t, x) ∈ O(Dρ1,...,ρn)[[t]]s. Reciprocally, let us fix s ≥ sc and let us suppose that the inhomogeneity f̃(t, x) is s-Gevrey. By assumption, its coefficients fj,∗(x) ∈ O(Dρ1,...,ρn) satisfy the following condition (see Definition 2.1): there exist three positive constants 0 < ρ < min(ρ1, . . . , ρn), C > 0 and K > 0 such that the inequalities |fj,∗(x)| ≤ CKjΓ(1 + (s+ 1)j) (5.1) for all j ≥ 0 and all ‖x‖ ≤ ρ. We must prove that the coefficients uj,∗(x) ∈ O(Dρ1,...,ρn) of ũ(t, x) satisfy sim- ilar inequalities. The approach we present below is analogous to the ones already developed in [3, 31, 32, 33] in the framework of linear partial and integro-differential equations, and in [30, 35, 37] in the case of semilinear equations of the form (1.3). It is based on the Nagumo norms [4, 24, 48] and on a technique of majorant series. However, as we shall see, our calculations appear to be much more technical and complicated. Furthermore, the nonlinear polynomial terms associated with each derivation ∂it∂ q x generate several new technical combinatorial situations. Before starting the calculations, let us first recall for the convenience of the reader the definition of the Nagumo norms and some of their properties which are needed in the sequel. 5.1. Nagumo norms. Definition 5.1. Let f ∈ O(Dρ1,...,ρn), p ≥ 0 and 0 < r < min(ρ1, . . . , ρn). Then the Nagumo norm, with indices (p, r), of f is ‖f‖p,r := sup ‖x‖≤r |f(x)dr(x)p|, where dr(x) denotes the Euclidian distance dr(x) := r − ‖x‖. Proposition 5.2. Let f, g ∈ O(Dρ1,...,ρn), p, p′ ≥ 0 and 0 < r < min(ρ1, . . . , ρn). Then (1) ‖ · ‖p,r is a norm on O(Dρ1,...,ρn). (2) |f(x)| ≤ ‖f‖p,rdr(x)−p for all ‖x‖ < r . (3) ‖f‖0,r = sup ‖x‖≤r |f(x)| is the usual sup-norm on the polydisc Dr,...,r. (4) ‖fg‖p+p′,r ≤ ‖f‖p,r‖g‖p′,r. (5) ‖∂x`f‖p+1,r ≤ e(p+ 1)‖f‖p,r for all ` ∈ {1, . . . , n}. Proof. Properties 1–4 are straightforward and are left to the reader. To prove Property 5, we proceed as follows. Let ` ∈ {1, . . . , n} and x ∈ Cn such that ‖x‖ < r and 0 < R < dr(x). Using the Cauchy Integral Formula, we have ∂x`f(x) = 1 (2iπ)n ∫ γ(x) f(x′) (x′` − x`)2 ∏n k=1 k 6=` (x′k − xk) dx′, where γ(x) := {x′ = (x′1, . . . , x ′ n) ∈ Cn; |x′k − xk| = R for all k ∈ {1, . . . , n}}. Since x′ ∈ γ(x)⇒ ‖x′‖ < r, 14 P. REMY EJDE-2023/06 we can apply Property 2 of Proposition 5.2; hence |∂x`f(x)| ≤ 1 R max x′∈γ(x) |f(x′)| ≤ 1 R ‖f‖p,r max x′∈γ(x) dr(x ′)−p = 1 R ‖f‖p,r(dr(x)−R)−p. Observe that the last equality stems from the relations dr(x ′) = r − ‖x′‖ = r − ‖x+ x′ − x‖ ≥ dr(x)− ‖x′ − x‖ = dr(x)−R > 0. When p = 0, the choice R = dr(x) e implies the inequality |∂x`f(x)| ≤ e‖f‖0,rdr(x)−1; hence, |∂x`f(x)|dr(x) ≤ e‖f‖0,r. (5.2) When p > 0, the choice R = dr(x) p+1 and( 1− 1 p+ 1 )−p = ( 1 + 1 p )p < e, bring us to the inequalities |∂x`f(x)| ≤ ‖f‖p,rdr(x)−p−1(p+ 1) ( 1− 1 p+ 1 )−p ≤ e(p+ 1)‖f‖p,rdr(x)−p−1 and then to the inequality |∂x`f(x)|dr(x)p+1 ≤ e(p+ 1)‖f‖p,r. (5.3) Property 5 follows since inequalities (5.2) and (5.3) remain valid when ‖x‖ = r. This completes the proof. � Remark 5.3. Inequalities 4–5 of Proposition 5.2 are the most important properties. Also observe that the same index r occurs on both of their sides, allowing thus to get estimates for the product fg in terms of f and g, and for the derivatives ∂x`f for any ` ∈ {1, . . . , n} in terms of f without having to shrink the polydisc Dr,...,r. Let us now turn to the proof of the first item of Theorem 4.1. 5.2. Some inequalities. Let σs denote the positive rational number defined by σs = (s+ 1)(κ+ v), where v ≥ 0 stands for the maximum of the vi,q,p’s. The purpose of this section is to construct from the recurrence relations (1.2) some inequalities relating the terms ‖u`,∗‖`σs,ρ Γ(1 + (s+ 1)`) for ` = 0, . . . , j + κ, in order to apply a technique of majorant series in Section 5.3 below. EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 15 Preliminary inequalities. From the recurrence relations (1.2), the relations uj+κ,∗(x) Γ(1 + (s+ 1)(j + κ)) = fj,∗(x) Γ(1 + (s+ 1)(j + κ)) + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `0+`1+···+`p+ `p+1=j−vi,q,p j!ai,q,p;`0,∗(x)u`1,∗(x) . . . u`p,∗(x)∂qxu`p+1+i,∗(x) `0!`1! . . . `p!`p+1!Γ(1 + (s+ 1)(j + κ)) , starting with u`,∗(x) = ϕ`(x) for ` = 0, . . . , κ− 1, hold for all j ≥ 0. As usual, we use the classical conventions that the fourth sum is zero when j − vi,q,p < 0, and that the product u`1,∗(x)...u`p,∗(x) `1!...`p! is 1 when p = 0. Applying then the Nagumo norm of indices ((j+κ)σs, ρ), we derive successively from Property 1 and from Properties 4− 5 of Proposition 5.2 the relations ‖uj+κ,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) ≤ ‖fj,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `0+`1+···+`p+ `p+1=j−vi,q,p j! ∥∥ai,q,p;`0,∗u`1,∗ . . . u`p,∗∂qxu`p+1+i,∗ ∥∥ (j+κ)σs,ρ `0!`1! . . . `p!`p+1!Γ(1 + (s+ 1)(j + κ)) and, next, the inequalities ‖uj+κ,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) ≤ ‖fj,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `0+`1+···+`p+`p+1=j−vi,q,p Ai,q,p,j,`0,`1,...`p,`p+1,s × ‖u`1,∗‖`1σs,ρ . . . ‖u`p,∗‖`pσs,ρ‖u`p+1+i,∗‖(`p+1+i)σs,ρ Γ(1 + (s+ 1)`1) . . .Γ(1 + (s+ 1)`p)Γ(1 + (s+ 1)(`p+1 + i)) (5.4) for all j ≥ 0, where Ai,q,p,j,`0,`1,...`p,`p+1,s is nonnegative and defined by Ai,q,p,j,`0,`1,...`p,`p+1,s = Γ(1 + (s+ 1)`1) . . .Γ(1 + (s+ 1)`p) `0!`1! . . . `p!`p+1! × j!eλ(q) ∏λ(q)−1 `=0 ((`p+1 + i)σs + λ(q)− `)Γ(1 + (s+ 1)(`p+1 + i)) Γ(1 + (s+ 1)(j + κ)) × ‖ai,q,p;`0,∗‖(`0+κ−i+vi,q,p)σs−λ(q),ρ. for all j ≥ vi,q,p. Remark 5.4. Observe that all the norms written in the inequality (5.4), and especially the norms ‖ai,q,p;`0,∗‖(`0+κ−i+vi,q,p)σs−λ(q),ρ are well-defined. Indeed, due 16 P. REMY EJDE-2023/06 to Lemma 3.9, we have the inequality (`0 + κ− i+ vi,q,p)σs − λ(q) ≥ (κ− i+ vi,q,p)(s+ 1)− λ(q) ≥ vi,q,p ≥ 0. Bounds of Ai,q,p,j,`0,`1,...`p,`p+1,s. We shall prove the following. Proposition 5.5. Let i ∈ K, q ∈ Qi, p ∈ Pi,q, j ≥ vi,q,p, and `0, `1, . . . , `p, `p+1 ∈ N such that `0 + `1 + · · ·+ `p + `p+1 = j − vi,q,p. Then Ai,q,p,j,`0,`1,...`p,`p+1,s ≤ (e(κ+ v))λ(q) ‖ai,q,p;`0,∗‖(`0+κ−i+vi,q,p)σs−λ(q),ρ Γ(1 + (s+ 1)`0) . Proof. From the definition of Ai,q,p,j,`0,`1,...`p,`p+1,s and Lemma 5.6, we first derive the inequality Ai,q,p,j,`0,`1,...`p,`p+1,s ≤ Γ(1 + (s+ 1)(j − vi,q,p))Γ(1 + (s+ 1)(`p+1 + i)) Γ(1 + (s+ 1)`p+1) × j!eλ(q) ∏λ(q)−1 `=0 ((`p+1 + i)σs + λ(q)− `) (j − vi,q,p)!Γ(1 + (s+ 1)(j + κ)) ‖ai,q,p;`0,∗‖(`0+κ−i+vi,q,p)σs−λ(q),ρ Γ(1 + (s+ 1)`0) . Next, applying successively Lemmas 5.8 and 5.9, we obtain respectively Ai,q,p,j,`0,`1,...`p,`p+1,s ≤ Γ(1 + (s+ 1)(j − vi,q,p))Γ(1 + (s+ 1)(`p+1 + i)) Γ(1 + (s+ 1)`p+1) × eλ(q) ∏λ(q)−1 `=0 ((`p+1 + i)σs + λ(q)− `) Γ(1 + (s+ 1)(j + κ)− vi,q,p) ‖ai,q,p;`0,∗‖(`0+κ−i+vi,q,p)σs−λ(q),ρ Γ(1 + (s+ 1)`0) , and Ai,q,p,j,`0,`1,...`p,`p+1,s ≤ Γ(1 + (s+ 1)(j − vi,q,p))Γ(1 + (s+ 1)(`p+1 + i)) Γ(1 + (s+ 1)(j − vi,q,p + i))Γ(1 + (s+ 1)`p+1) × (e(κ+ v))λ(q) ‖ai,q,p;`0,∗‖(`0+κ−i+vi,q,p)σs−λ(q),ρ Γ(1 + (s+ 1)`0) , Proposition 5.5 follows then from Lemma 5.10 and that `p+1 ≤ j − vi,q,p. � Lemma 5.6. The inequality Γ(1 + (s+ 1)k1) . . .Γ(1 + (s+ 1)km) Γ(1 + (s+ 1)(k1 + · · ·+ km)) ≤ k1! . . . km! (k1 + · · ·+ km)! holds for all m ≥ 1 and k1, . . . , km ∈ N. Proof. It suffices to prove that the function fk1,...,km defined on [1,+∞[ by fk1,...,km(x) = Γ(1 + k1x) . . .Γ(1 + kmx) Γ(1 + (k1 + · · ·+ km)x) is decreasing. Computing its derivative with respect to x, we have f ′k1,...,km(x) = fk1,...,km(x) ( m∑ `=1 k`ψ(1 + k`x)− ( m∑ `=1 k` ) ψ ( 1 + ( m∑ `=1 k` ) x )) EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 17 for all x ≥ 1, where ψ := Γ′/Γ is the Digamma function (or Psi function). Applying then the classical relation (see [1, p. 259] for instance) ψ(1 + q) = −γ + +∞∑ h=1 q h(h+ q) , where q ≥ 0 and γ is the Euler’s constant, we obtain that for all x ≥ 1, f ′k1,...,km(x) = xfk1,...,km(x) +∞∑ h=1 ( m∑ `=1 k2` h(h+ k`x) − (∑m `=1 k` )2 h ( h+ (∑m `=1 k` ) x )). Lemma 5.7 below complete the proof by showing that f ′k1,...,km(x) ≤ 0 for all x ≥ 1. � Lemma 5.7. Let x, h ≥ 1 be. Then m∑ `=1 k2` h+ k`x ≤ (∑m `=1 k` )2 h+ (∑m `=1 k` ) x for all m ≥ 1 and all k1, . . . , km ∈ N. Proof. We proceed by induction on m. Lemma 5.7 is obvious for m = 1. For m = 2, we clearly have k21 h+ k1x + k22 h+ k2x − (k1 + k2)2 h+ (k1 + k2)x = −k1k2h(k1x+ k2x+ 2h) (h+ k1x)(h+ k2x)(h+ (k1 + k2)x) ≤ 0. Let us now suppose that Lemma 5.7 is true for all k ∈ {1, . . . ,m} for a certain m ≥ 1. Then, the successive relations m+1∑ `=1 k2` h+ k`x ≤ (∑m `=1 k` )2 h+ (∑m `=1 k` ) x + k2m+1 h+ km+1x ≤ (∑m `=1 k` + km+1 )2 h+ (∑m `=1 k` + km+1 ) x = (∑m+1 `=1 k` )2 h+ (∑m+1 `=1 k` ) x hold for any k1, . . . , km+1 ∈ N, which completes the proof. � Lemma 5.8. Let i ∈ K, q ∈ Qi, p ∈ Pi,q and j ≥ vi,q,p be. Then j! (j − vi,q,p)!Γ(1 + (s+ 1)(j + κ)) ≤ 1 Γ(1 + (s+ 1)(j + κ)− vi,q,p) . Proof. It is clear that the inequality holds when vi,q,p = 0. Let us now assume vi,q,p ≥ 1 and let us write the quotient j!/(j − vi,q,p)! in the form j! (j − vi,q,p)! = vi,q,p−1∏ `=0 (j − `). (5.5) On the other hand, applying vi,q,p times the recurrence relation Γ(1 + z) = zΓ(z) to Γ(1 + (s+ 1)(j + κ)), we obtain Γ(1+(s+1)(j+κ)) = Γ(1+(s+1)(j+κ)−vi,q,p) vi,q,p−1∏ `=0 ((s+1)(j+κ)− `). (5.6) 18 P. REMY EJDE-2023/06 Combining then the identities (5.5) and (5.6), we obtain j! (j − vi,q,p)!Γ(1 + (s+ 1)(j + κ)) = ∏vi,q,p−1 `=0 j−` (s+1)(j+κ)−` Γ(1 + (s+ 1)(j + κ)− vi,q,p) , and Lemma 5.8 follows from the inequalities 0 < j − ` < (s+ 1)(j + κ)− `. � Lemma 5.9. Let i ∈ K, q ∈ Qi, p ∈ Pi,q, j ≥ vi,q,p, and k ∈ {0, . . . , j − vi,q,p}. Then ∏λ(q)−1 `=0 ((k + i)σs + λ(q)− `) Γ(1 + (s+ 1)(j + κ)− vi,q,p) ≤ (κ+ v)λ(q) Γ(1 + (s+ 1)(j − vi,q,p + i)) . Proof. Let us first assume j = vi,q,p and i = 0. Then, we have k = 0, and we must prove the inequality ∏λ(q)−1 `=0 (λ(q)− `) Γ(1 + (s+ 1)(vi,q,p + κ)− vi,q,p) ≤ (κ+ v)λ(q). (5.7) Let us note that λ(q)−1∏ `=0 (λ(q)− `) = (λ(q))! = Γ(1 + λ(q)) for all λ(q), including the case λ(q) = 0 since the product is 1 by convention. On the other hand, Lemma 3.9 implies in the case λ(q) > 0 the inequalities 1 + (s+ 1)(vi,q,p + κ)− vi,q,p ≥ 1 + λ(q) ≥ 2; hence, Γ(1 + (s+ 1)(vi,q,p + κ)− vi,q,p) ≥ Γ(1 + λ(q)) since the Gamma function is increasing on [2,+∞[. In the special case λ(q) = 0, we observe that the increase of the Gamma function applied to the inequalities 1 + (s+ 1)(vi,q,p + κ)− vi,q,p ≥ 1 + κ ≥ 2 implies Γ(1 + (s+ 1)(vi,q,p + κ)− vi,q,p) ≥ Γ(2) = Γ(1) = Γ(1 + λ(q)). Consequently, the left hand-side of (5.7) is ≤ 1 and Lemma 5.9 follows then from the inequality κ+ v ≥ 1. Let us now assume (j, i) 6= (vi,q,p, 0). According to the definition of σs, we first have the identity λ(q)−1∏ `=0 ((k + i)σs + λ(q)− `) = (κ+ v)λ(q) λ(q)−1∏ `=0 ( (s+ 1)(k + i) + λ(q)− ` κ+ v ) . (5.8) On the other hand, applying λ(q) times the recurrence relation Γ(1 + z) = zΓ(z) to Γ(1 + (s+ 1)(j + κ)− vi,q,p), we have Γ(1 + (s+ 1)(j + κ)− vi,q,p) = Γ(1 + (s+ 1)(j + κ)− vi,q,p − λ(q)) λ(q)−1∏ `=0 ((s+ 1)(j + κ)− vi,q,p − `). (5.9) EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 19 Observe that this identity makes sense since the inequality j ≥ vi,q,p and Lemma 3.9 imply (s+ 1)(j + κ)− vi,q,p − λ(q) ≥ (s+ 1)(vi,q,p + κ)− vi,q,p − λ(q) ≥ (s+ 1)(vi,q,p + κ− i)− vi,q,p − λ(q) ≥ 0. Observe also that (s+ 1)(k + i) + λ(q)− ` κ+ v ≤ (s+ 1)(j + κ)− vi,q,p − ` for all ` ∈ {0, . . . , λ(q) − 1} when λ(q) > 0. Indeed, according to the inequalities k ≤ j − vi,q,p and κ+ v ≥ 1, and Lemma 3.9, we have (s+ 1)(j + κ)− vi,q,p − `− (s+ 1)(k + i)− λ(q)− ` κ+ v = (s+ 1)(j + κ− i− k)− vi,q,p − `− λ(q)− ` κ+ v ≥ (s+ 1)(κ− i+ vi,q,p)− vi,q,p − `− λ(q)− ` κ+ v ≥ (λ(q)− `) ( 1− 1 κ+ v ) ≥ 0. Consequently, combining the identities (5.8) and (5.9), we finally obtain∏λ(q)−1 `=0 ((k + i)σs + λ(q)− `) Γ(1 + (s+ 1)(j + κ)− vi,q,p) ≤ (κ+ v)λ(q) Γ(1 + (s+ 1)(j + κ)− vi,q,p − λ(q)) for all λ(q), including the case λ(q) = 0 since the product is 1 by convention. Lemma 5.9 follows then from 1 + (s+ 1)(j + κ)− vi,q,p − λ(q) ≥ 1 + (s+ 1)(j + κ)− (s+ 1)(κ− i+ vi,q,p) = 1 + (s+ 1)(j − vi,q,p + i) ≥ 2 and from Gamma function on [2,+∞[ being increasing. Observe that the first inequality stems again from Lemma 3.9. Observe also that, without the condition (j, i) 6= (vi,q,p, 0), the second inequality is no longer valid. This completes the proof. � Lemma 5.10. Let a ≥ 0. The function f defined on [0,+∞[ by f(x) = Γ(1 + a+ x) Γ(1 + x) is increasing. Proof. Denoting by f ′ the derivative of f with respect to x, we have f ′(x) = f(x)(ψ(1 + a+ x)− ψ(1 + x)) for all x ≥ 0, where ψ denotes as in the proof of Lemma 5.6 the Digamma function. The latter being increasing on [0,+∞[ (see [1, pp. 258-260] for instance), we deduce that f ′(x) ≥ 0 for all x ≥ 0, which completes the proof. � 20 P. REMY EJDE-2023/06 More precise inequalities. Let us apply Proposition 5.5 to the inequalities (5.4) to obtain that for all j ≥ 0, ‖uj+κ,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) ≤ gj,s + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `0+`1+···+`p+ `p+1=j−vi,q,p αi,q,p,`0,s × ‖u`1,∗‖`1σs,ρ . . . ‖u`p,∗‖`pσs,ρ‖u`p+1+i,∗‖(`p+1+i)σs,ρ Γ(1 + (s+ 1)`1) . . .Γ(1 + (s+ 1)`p)Γ(1 + (s+ 1)(`p+1 + i)) , (5.10) where the terms gj,s and αi,q,p,`0,s are nonnegative and defined by gj,s = ‖fj,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) , αi,q,p,`0,s = (e(κ+ v))λ(q) ‖ai,q,p;`0,∗‖(`0+κ−i+vi,q,p)σs−λ(q),ρ Γ(1 + (s+ 1)`0) . The following result provides some bounds on these terms, and will be useful in the next section. Lemma 5.11. Let i ∈ K, q ∈ Qi and p ∈ Pi,q. Then there exist four positive constants C1, C2,K1,K2 > 0 such that 0 ≤ gj,s ≤ C1K j 1 and 0 ≤ αi,q,p,j,s ≤ C2K j 2 for all j ≥ 0. Proof. According to the analyticity of the function ai,q,p(t, x) on Dρ0,...,ρn , and the hypothesis on the coefficients fj,∗(x) of the inhomogeneity f̃(t, x) (see inequality (5.1)), we first have |fj,∗(x)| ≤ CKjΓ(1 + (s+ 1)j) and |ai,q,p;j,∗(x)| ≤ C ′K ′jj! for all j ≥ 0 and all ‖x‖ ≤ ρ, the constants C,K,C ′,K ′ > 0 being independent of j and x. Hence, applying Proposition 5.2, we have 0 ≤ gj,s ≤ CKjΓ(1 + (s+ 1)j)ρ(j+κ)σs Γ(1 + (s+ 1)(j + κ)) ≤ Cρκσs(Kρσs)j , and 0 ≤ αi,q,p,j,s ≤ (e(κ+ v))λ(q)C ′K ′jj!ρ(j+κ−i+vi,q,p)σs−λ(q) Γ(1 + (s+ 1)j) ≤ (e(κ+ v))λ(q)C ′ρ(κ−i+vi,q,p)σs−λ(q)(K ′ρσs)j . This completes the proof. � We shall now bound the Nagumo norms ‖uj,∗‖jσs,ρ for any j ≥ 0. To do that, we shall proceed similarly as in [3, 30, 31, 32, 33, 35, 37] by using a technique of majorant series. However, as we shall see, the calculations are much more compli- cated because of the nonlinear polynomial terms associated with each derivation ∂it∂ q x. EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 21 5.3. A majorant series. Let us consider the formal power series v(X) = ∑ j≥0 vjX j , the coefficients of which are recursively determined for all j ≥ 0 by the relations vj+κ = gj,s + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `0+`1+···+`d+1 =j+i−vi,q,p αi,q,p,`0,sv`1 . . . v`d+1 (5.11) starting with the initial condition v0 = 1 + ‖ϕ0‖0,ρ, and, for j = 1, . . . , κ − 1 (if κ ≥ 2): vj = ‖ϕj‖jσs,ρ Γ(1 + (s+ 1)j) + ∑ (i,q,p)∈Vj ∑ `0+`1+···+`d+1 =j−κ+i−vi,q,p αi,q,p,`0,sv`1 . . . v`d+1 , where d = max ( p ∈ ⋃ i∈K ⋃ q∈Qi Pi,q ) , and Vj = { (i, q, p) ∈ K ×Qi × Pi,q such that j − κ+ i− vi,q,p ≥ 0 } . Observe that the condition κ > i for all i ∈ K implies j − κ+ i− vi,q,p < j; hence, the initial conditions on the vj ’s with j = 1, . . . , κ − 1 make sense. Observe also that the set Vj may be empty (this is particularly the case when K = {0}, or when vi,q,p ≥ i for all i, q, p). Proposition 5.12. The inequalities 0 ≤ ‖uj,∗‖jσs,ρ Γ(1 + (s+ 1)j) ≤ vj (5.12) hold for all j ≥ 0. Proof. According to the initial conditions on the uj ’s and on the vj ’s, the inequali- ties in (5.12) hold for all j = 0, . . . , κ−1. Let us now suppose that these inequalities are true for all k ≤ j − 1 + κ for a certain j ≥ 0, and let us prove them for j + κ. First of all, applying our hypotheses to relations (5.10), we have 0 ≤ ‖uj+κ,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) ≤ gj,s + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `′0+` ′ 1+···+` ′ p+ `′p+1=j−vi,q,p αi,q,p,`′0,sv`′1 . . . v`′pv`′p+1+i (5.13) and then 0 ≤ ‖uj+κ,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) ≤ gj,s + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `0+`1+···+`p+ `p+1=j+i−vi,q,p αi,q,p,`0,sv`1 . . . v`pv`p+1 (5.14) since all the tuples (`′0, ` ′ 1, . . . , ` ′ p+1 + i) ∈ Np+2 in (5.13) satisfy the identity `′0 + `′1 + · · · + `′p+1 + i = j + i − vi,q,p, and since all the terms αi,q,p,`0,sv`1 . . . v`pv`p+1 in (5.14) are nonnegative. 22 P. REMY EJDE-2023/06 Next, let us observe that any (p+ 2)-tuple (`0, . . . , `p+1) ∈ Np+2 such that `0 + · · ·+`p+1 = j+i−vi,q,p can be seen as the (d+2)-tuple (`0, . . . , `p+1, `p+2, . . . , `d+1) ∈ Nd+2, where `p+2 = · · · = `d+1 = 0. Therefore, using the fact that v0 ≥ 1, we have 0 ≤ αi,q,p,`0,sv`1 . . . v`p+1 ≤ αi,q,p,`0,sv`1 . . . v`p+1 vd−p0 = αi,q,p,`0,sv`1 . . . v`p+1 v`p+2 . . . v`d+1 , and, consequently, 0 ≤ ∑ `0+···+`p+1 =j+i−vi,q,p αi,q,p,`0,sv`1 . . . v`p+1 ≤ ∑ `0+···+`p+1+ 0+···+0=j+i−vi,q,p αi,q,p,`0,sv`1 . . . v`d+1 ≤ ∑ `0+···+`d+1 =j+i−vi,q,p αi,q,p,`0,sv`1 . . . v`d+1 , since all the terms are nonnegative. Therefore, 0 ≤ ‖uj+κ,∗‖(j+κ)σs,ρ Γ(1 + (s+ 1)(j + κ)) ≤ gj,s + ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q ∑ `0+`1+···+`d+1 =j+i−vi,q,p αi,q,p,`0,sv`1 . . . v`d+1 = vj+κ which completes the proof. � Following proposition allows us to bound the vj ’s. Proposition 5.13. The formal series v(X) is convergent. In particular, there exist two positive constants C ′,K ′ > 0 such that vj ≤ C ′K ′j for all j ≥ 0. Proof. It is sufficient to prove the convergence of v(X). Let us start by observing that v(X) is the unique formal power series in X solution of the functional equation v(X) = Xα(X)(v(X))d+1 + h(X), (5.15) where α(X) and h(X) are the two formal power series defined by α(X) = ∑ i∈K ∑ q∈Qi ∑ p∈Pi,q Xκ−i−1+vi,q,pαi,q,p(X), h(X) = A0 +A1X + · · ·+Aκ−1X κ−1 +Xκ ∑ j≥0 gj,sX j with αi,q,p(X) = ∑ j≥0 αi,q,p,j,sX j , A0 = 1 + ‖ϕ0‖0,ρ, and, for j = 1, . . . , κ− 1 (if κ ≥ 2), Aj = ‖ϕj‖jσs,ρ Γ(1 + (s+ 1)j) . EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 23 Next we notice that, according to Lemma 5.11, α(X) and h(x) are actually two convergent power series with nonnegative coefficients. In particular, they respec- tively define two increasing functions on [0, rα[ and [0, rh[, where rα > 0 and rh > 0 stand respectively for the radius of convergence of α(X) and h(X). Besides, given the assumptions on the functions ai,q,p(t, x) (see page 2) and the fact that A0 ≥ 1, we have α(r) > 0 (resp. h(r) > 0) for all r ∈]0, rα[ (resp. ]0, rh[). When d = 0, the convergence of v(X) is obvious, since we have the identity (1−Xα(X))v(X) = h(X). When d ≥ 1, we proceed through a fixed point method as follows. Let us set V (X) = ∑ m≥0 Vm(X) and let us choose the solution of (5.15) given by the system V0(X) = h(X) Vm+1(X) = Xα(X) ∑ `1+···+`d+1=m V`1(X) . . . V`d+1 (X) for m ≥ 0. By induction on m ≥ 0, we easily check that Vm(X) = Cm,d+1X m(α(X))m(h(X))md+1, (5.16) where the Cm,d+1’s are the positive constants recursively determined from C0,d+1 := 1 by the relations Cm+1,d+1 = ∑ k1+···+kd+1=m Ck1,d+1 . . . Ckd+1,d+1. Thereby, all the Vm(X)’s are analytic functions on the disc with center 0 ∈ C and radius min(rα, rh). Moreover, identities (5.16) show us that Vm(X) is of order Xm for all m ≥ 0. Consequently, the series V (X) makes sense as a formal power series in X and we obtain V (X) = v(X) by unicity. It is left to prove the convergence of V (X). To do that, let us choose 0 < r < min(rα, rh). By construction (see page 6), the constants Cm,d+1’s are the generalized Catalan numbers of order d+ 1. We have therefore Cm,d+1 = 1 dm+ 1 ( m(d+ 1) m ) ≤ 2m(d+1) for all m ≥ 0. On the other hand, according to the remark above on the series α(X) and h(X), we derive from the identities (5.16) the inequalities |Vm(X)| ≤ h(r) ( 2d+1α(r)(h(r))d|X| )m for all m ≥ 0 and all |X| ≤ r; hence, the fact that the series V (X) is normally convergent on any disc with center 0 ∈ C and radius 0 < r′ < min ( r, 1 2d+1α(r)(h(r))d ) . This proves the analyticity of V (X) at 0 and achieves thereby the proof of Propo- sition 5.13. � According to Propositions 5.12 and 5.13, we can now bound the Nagumo norms ‖uj,∗‖jσs,ρ. 24 P. REMY EJDE-2023/06 Corollary 5.14. Let C ′,K ′ > 0 be as in Proposition 5.13. Then ‖uj,∗‖jσs,ρ ≤ C ′K ′jΓ(1 + (s+ 1)j) for all j ≥ 0. We are now able to conclude the proof of the first item of Theorem 4.1. 5.4. Conclusion. We must prove estimates on the sup-norm of the uj,∗(x), similar to the ones on the norms ‖uj,∗‖jσs,ρ (see Corollary 5.14). To this end, we proceed by shrinking the closed polydisc ‖x‖ ≤ ρ. Let 0 < ρ′ < ρ. Then, for all j ≥ 0 and all ‖x‖ ≤ ρ′, we have |uj,∗(x)| = ∣∣uj,∗(x)dρ(x)jσs 1 dρ(x)jσs ∣∣ ≤ |uj,∗(x)dρ(x)jσs | (ρ− ρ′)jσs ≤ ‖uj,∗‖jσs,ρ (ρ− ρ′)jσs and, consequently, sup ‖x‖≤ρ′ |uj,∗(x)| ≤ C ′ ( K ′ (ρ− ρ′)σs )j Γ(1 + (s+ 1)j) by applying Corollary 5.14. This completes the proof of the first item of Theorem 4.1. 6. Proof of the second item of Theorem 4.1 In this section, we assume S 6= ∅ and we fix 0 ≤ s < sc. (Of course, this case does not occur when S = ∅). According to the filtration of the s-Gevrey spaces O(Dρ1,...,ρn)[[t]]s (see Section 3) and the first item of Theorem 4.1, it is clear that we have the following implications: f̃(t, x) ∈ O(Dρ1,...,ρn)[[t]]s ⇒ f̃(t, x) ∈ O(Dρ1,...,ρn)[[t]]sc ⇒ ũ(t, x) ∈ O(Dρ1,...,ρn)[[t]]sc . Therefore, to conclude that we can not say better about the Gevrey order of ũ(t, x), that is ũ(t, x) is generically sc-Gevrey, we need to find an example for which the formal solution ũ(t, x) of (1.1) is s′-Gevrey for no s′ < sc. Section 3.1 has already provided us with such two examples in the case κ = 1. In Proposition 6.1 below, we propose a much more general example. Proposition 6.1. Let us consider the equation ∂κt u− ∑ i∈K ∑ q∈Qi ( ∑ p∈Pi,q ai,q,pt vi,q,pup ) ∂it∂ q xu = f̃(t, x), ai,q,p > 0 ∂jt u(t, x)|t=0 = ϕj(x), j = 0, . . . , κ− 1 (6.1) where the initial condition ϕi∗(x) is the analytic function defined by ϕi∗(x) = 1 1− x1 − · · · − xn on the disc D1/n,...,1/n, and where the initial conditions ϕj(x) for j 6= i∗ are analytic functions on D1/n,...,1/n satisfying ∂`xϕj(0) ≥ 0 for all ` ∈ Nn, and ϕ0(0) > 0 when i∗ 6= 0. Suppose also that the inhomogeneity f̃(t, x) satisfies the following two conditions: • f̃(t, x) is s-Gevrey; • ∂`xfj,∗(0) ≥ 0 for all j ≥ 0 and all ` ∈ Nn. EJDE-2023/06 GEVREY REGULARITY OF INHOMOGENEOUS NONLINEAR PDES 25 Then, the formal solution ũ(t, x) of (6.1) is exactly sc-Gevrey. Proof. From the calculations above, it is sufficient to prove that ũ(t, x) is s′-Gevrey for no s′ < sc. First of all, let us rewrite the general relations (1.2) as the identities uj+κ,∗(x) = j!ai∗,q∗,p∗u p∗ 0,∗(x) (j − vi∗,q∗,p∗)! ∂q ∗ x uj−vi∗,q∗,p∗+i∗,∗(x) +Rj(x) with Rj(x) = fj,∗(x) + ∑ `1+···+`p∗+`p∗+1=j−vi∗,q∗,p∗ (`1,...,`p∗ ,`p∗+1)6=(0,...,0,j−vi∗,q∗,p∗ ) j!ai∗,q∗,p∗u`1,∗(x) . . . u`p∗ ,∗(x)∂q ∗ x u`p∗+1+i∗,∗(x) `1! . . . `p∗ !`p∗+1! + ∑ (i,q,p)∈K×Qi×Pi,q (i,q,p)6=(i∗,q∗,p∗) ∑ `1+···+`p+ `p+1=j−vi,q,p j!ai,q,pu`1,∗(x) . . . u`p,∗(x)∂qxu`p+1+i,∗(x) `1! . . . `p!`p+1! for all j ≥ 0, together with the initial conditions uj,∗(x) = ϕj(x) for j = 0, . . . , κ− 1. Using then our hypotheses on the coefficients ai,q,p, on the initial conditions ϕj(x), and on the inhomogeneity f̃(t, x), we easily check that, for all j ≥ 0, uj(vi∗,q∗,p∗+κ−i∗)+i∗,∗(x) = aji∗,q∗,p∗ϕ jp∗ 0 (x)(jλ(q∗))! (1− x1 − · · · − xn)jλ(q∗)+1 j∏ `=1 vi∗,q∗,p∗∏ m=1 ((`− 1)(vi∗,q∗,p∗ + κ− i∗) +m) + remj(x) with remj(0) ≥ 0. Hence, applying technical Lemma 6.2 below, we have the in- equality uj(vi∗,q∗,p∗+κ−i∗)+i∗,∗(0) ≥ ( ai∗,q∗,p∗ϕ p∗ 0 (0) )j (jλ(q∗))!(jvi∗,q∗,p∗)!. (6.2) Let us now suppose that ũ(t, x) is s′-Gevrey for some s′ < sc. Then, Definition 2.1 and inequality (6.2) imply 1 ≤ C ( K ai∗,q∗,p∗ϕ p∗ 0 (0) )j Γ(1 + i∗(s′ + 1) + j(s′ + 1)(vi∗,q∗,p∗ + κ− i∗)) Γ(1 + jλ(q∗))Γ(1 + jvi∗,q∗,p∗) (6.3) for all j ≥ 0 and some convenient positive constants C and K independent of j. Proposition 6.1 follows since such inequalities are impossible: applying the Stirling’s Formula, the right hand-side of (6.3) is equivalent to C ′ji ∗(s′+1)− 1 2 ( K ′ jσ )j , j → +∞ (6.4) with C ′ = C((s′ + 1)(vi∗,q∗,p∗ + κ− i∗))i ∗(s′+1) √ (s′ + 1)(vi∗,q∗,p∗ + κ− i∗) 2πλ(q∗)vi∗,q∗,p∗ , K ′ = K((s′ + 1)(vi∗,q∗,p∗ + κ− i∗))(s′+1)(vi∗,q∗,p∗+κ−i∗)eσ ai∗,q∗,p∗ϕ p∗ 0 (0)λ(q∗)λ(q∗)v vi∗,q∗,p∗ i∗,q∗,p∗ , 26 P. REMY EJDE-2023/06 σ := λ(q∗) + vi∗,q∗,p∗ − (s′ + 1)(vi∗,q∗,p∗ + κ− i∗), and (6.4) approaches 0 when j tends to infinity. 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Shirai; Alternative proof for the convergence or formal solutions of singular first order non- linear partial differential equations, Journal of the School of Education, Sugiyama Jogakuen University, 1 (2008), 91-102. [44] A. Shirai; Gevrey order of formal solutions of singular first order nonlinear partial differen- tial equations of totally characteristic type, Journal of the School of Education, Sugiyama Jogakuen University, 6 (2013), 159-172. [45] A. Shirai; Maillet type theorem for singular first order nonlinear partial differential equations of totally characteristic type, part II, Opuscula Math., 35 (2015) (5), 689-712. [46] H. Tahara; Gevrey regularity in time of solutions to nonlinear partial differential equations, J. Math. Sci. Univ. Tokyo, 18 (2011), 67-137. [47] H. Tahara, H. Yamazawa; Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations, J. Differential Equations, 255 (2013) (10), 3592- 3637. [48] W. Walter; An elementary proof of the Cauchy-Kowalevsky theorem, Amer. Math. Monthly, 92 (1985) (2), 115-126. [49] A. Yonemura; Newton polygons and formal Gevrey classes, Publ. Res. Inst. Math. Sci., 26 (190), 197-204. Pascal Remy Laboratoire de Mathématiques de Versailles, Université de Versailles Saint-Quentin, 45 avenue des Etats-Unis, 78035 Versailles cedex, France Email address: pascal.remy@uvsq.fr; pascal.remy.maths@gmail.com 1. Problem setting 2. Gevrey formal series 3. Associated linear operator 3.1. Two preliminaries examples 3.2. Associated linear operator 4. Main result 5. Proof of the first item of Theorem ?? 5.1. Nagumo norms 5.2. Some inequalities 5.3. A majorant series 5.4. Conclusion 6. Proof of the second item of Theorem ?? References