Gevrey Asymptotics for Logarithmic Type Solutions to Singularly Perturbed Problems with Nonlocal Nonlinearities

In this paper, we examine a family of singularly perturbed nonlinear partial differential equations modeled as for vanishing initial data $u(0,z,ϵ)\equiv 0$. The constants $d_D,{\delta }_D\ge 1$ are natural numbers and $Q\left(X\right)$,$R_D\left(X\right)$,$P_j\left(X\right)$ for $j\in J_1$, $Q_j\left(X\right)$ for $j\in J_2$, where $J_1,J_2$ are two finite subsets of the positive integers ${\mathbb{N}}^{*}$, stand for polynomials with complex coefficients. The linear differential operator $P(t,z,ϵ,t{\partial }_t,{\partial }_z)$ depends analytically in a perturbation parameter $ϵ$ on a disc $D_{{ϵ}_0}$ with radius ${ϵ}_0>0$ centered at 0 and relies polynomially in the complex time t and holomorphically with respect to the space variable z on a horizontal strip framed as $H_{\beta }=\{z\in \mathbb{C}/|\mathrm{Im}\left(z\right)|<\beta \}$ in $\mathbb{C}$, for some given width $2\beta >0$. The forcing term $f(t,z,ϵ)$ is a map of logarithmic type represented as a sum were $f_j(t,z,ϵ)$, $j=1,2$, are polynomials in t, with holomorphic coefficients in z on $H_{\beta }$ and in $ϵ$ on $D_{{ϵ}_0}$. The map $H(v_0,z,ϵ,{\left\{v_j\right\}}_{j\in J_1},{\left\{w_j\right\}}_{j\in J_2})$ is a specific polynomial of degree at most 2 in its arguments $v_0$, ${\left\{v_j\right\}}_{j\in J_1}$ and ${\left\{w_j\right\}}_{j\in J_2}$, which relies holomorphically in z on $H_{\beta }$ and in $ϵ$ on $D_{{ϵ}_0}$. The precise shape of H is framed in (14).

The nonlinear term H of (1) involves not only powers of $P_j\left({\partial }_z\right)u(t,z,ϵ)$, $j\in J_1$, but also powers of derivatives of ${\gamma }_{ϵ}^{*}u(t,z,ϵ)$ where ${\gamma }_{ϵ}^{*}$ is a nonlocal operator acting on $u(t,z,ϵ)$ which represents the so-called monodromy operator around 0 relatively to $ϵ$. In the literature, the concept of formal monodromy around a point a in $\mathbb{C}$ appears in the construction of formal fundamental solutions to linear systems of differential equations with so-called irregular singularity at the given point a, known as the Levelt-Turrittin theorem, see [1]. It asserts that a differential system of the form for analytic coefficients matrix $A\left(x\right)\in M_n\left(\mathbb{C}\right)\left\{x\right\}$ near 0 with $n\ge 1$, for an integer $r\ge 2$, with an irregular singularity at 0, possesses a formal fundamental solution with the shape for some well chosen integer $e\ge 1$, where $\widehat{P}\left(y\right)\in {\mathrm{GL}}_n\left(\mathbb{C}\left[\left[y\right]\right][1/y]\right)$ is a formal meromorphic invertible matrix, $\phi \left(x^{1/e}\right)$ is a diagonal matrix whose coefficient are polynomials in $x^{-1/e}$ with complex coefficients and $C\in M_n\left(\mathbb{C}\right)$ is related to the so-called formal monodromy matrix $M\in {\mathrm{GL}}_n\left(\mathbb{C}\right)$ by the formula $M=exp\left(2\pi iC\right)$. It is worth remarking that this formal monodromy matrix extends in the formal settings the so-called monodromy matrix that appear in the representation of fundamental matrix solutions to systems (2) with regular singularity of the form where H is an invertible matrix with meromorphic coefficients near 0, for a matrix E giving rise to the monodromy matrix $N\in {\mathrm{GL}}_n\left(\mathbb{C}\right)$ by means of $N=exp\left(2\pi iE\right)$. The matrix N is obtained as analytic continuation of the fundamental matrix solution $Y\left(x\right)$ along a simple loop $\gamma$ going counterclockwise around the origin 0 with base point x by means of the identity where ${\gamma }^{*}Y$ denotes the analytic continuation along $\gamma$, see [2]. In the same manner as the analytic continuation operator ${\gamma }^{*}$ acting on analytic functions, a formal monodromy operator ${\gamma }^{*}$ acting on various spaces and rings (such as the so-called Picard-Vessiot rings) through the formulas ${\gamma }^{*}\left(z^{\lambda }\right)=e^{2i\pi \lambda }{z}^{\lambda }$ for complex numbers $\lambda \in \mathbb{C}$ and ${\gamma }^{*}\left(l\right)=l+2i\pi$ where l is the symbol for the Log function, has been introduced and studied from an abstract and algebraic point of view in the textbook [1].

In our context, the action of the formal monodromy ${\gamma }_{ϵ}^{*}$ on $u(t,z,ϵ)$ can be reformulated as a shift mapping on angles $\theta \mapsto \theta +2\pi$ in polar coordinates by means of the change of functions for $ϵ=r{e}^{\sqrt{-1}\theta }$, with radius $r>0$ and angle $\theta \in \mathbb{R}$, through the formula In this way, the main equation (1) can be recast as some nonlinear mixed type partial difference-differential equation for the map $v(t,z,r,\theta )$. In the framework of nonlinear difference equations in the complex domain with the shape for ${\mathbb{C}}^n-$valued analytic maps F in a neighborhood of $(\infty ,y_0)$ for some $y_0\in {\mathbb{C}}^n$, we notice that important results concerning asymptotic features of their solutions have been obtained by several authors, see [3,4,5]. In comparison with these results, we do not reach asymptotic expansions as $\theta$ goes to infinity in the equation fulfilled by v but we rather plan to get exact asymptotics as the real singular perturbation parameter $r>0$ approaches the origin.

We highlight our premise that the main equation (1) counts in powers of the basic differential operator $t{\partial }_t$ which is labelled of Fuchsian type. We refer to [6] for many sharp results about Fuchsian ordinary and partial differential equations. However, under the sufficient conditions required on (1) listed in Subsection 2.3 it pans out that (1) will be reduced throughout the work to a coupling of two partial differential equations, stated in (47) and (48), that comprise only powers of the basic differential operator $u_1^{k_1+1}{\partial }_{u_1}$, for a well chosen integer $k_1\ge 1$, of irregular type in a complex variable $u_1$. The definition of irregular type differential operators is given in the classical textbook [7] in the ordinary differential equations settings displayed in (2) and in the work [8] in the framework of partial differential equations.

In the present contribution, we aim to cook up a set of holomorphic solutions to (1) and to describe their asymptotic expansions as $ϵ$ tends to 0 (stated in Theorem 1 of Subsection 8.2). These solutions are shaped as logarithmic type maps that involve Fourier/Laplace transforms. Namely, under the list of requirements which mould (1) and detailed in Subsection 2.3, one can outline

Furthermore, owing to their Laplace integral structure, the components ${\left\{u_{j,p}\right\}}_{p\in I_1}$ own asymptotic expansions of Gevrey type in the parameter $ϵ$. Indeed, for given $j=1,2$, all the partial functions $ϵ\mapsto u_{j,p}(t,z,ϵ)$, $p\in I_1$, share a common asymptotic formal power series expansion on ${\mathcal{E}}_p$, with bounded holomorphic coefficients ${\mathbb{G}}_{n,j}$ on $\mathcal{T}\times H_{\beta }$. These asymptotic expansions turn out to be of Gevrey order $1/k_1$ on every sectors ${\mathcal{E}}_p$, meaning that constants $K_{p,j},M_{p,j}>0$ can be singled out for which the error bounds hold for all integers $N\ge 0$, all $ϵ\in {\mathcal{E}}_p$, uniformly in $t\in \mathcal{T}$ and $z\in H_{\beta }$. At last, we verify that the formal logarithmic type expression itself obeys the main equation (1).

Throughout the proof of our main result, we show that the components $u_{j,p}(t,z,ϵ)$, $j=1,2$ of the built up solutions $u_p$, $p\in I_1$, to (1) turn out to be embedded in a larger family of maps $u_{j,p}(t,z,ϵ)$, $j=1,2$, for all integers $0\le p\le \varsigma -1$ for some integer $\varsigma \ge 2$. These maps are bounded holomorphic on products $\mathcal{T}\times H_{\beta }\times {\mathcal{E}}_p$ where $\underline{\mathcal{E}}={\left\{{\mathcal{E}}_p\right\}}_{0\le p\le \varsigma -1}$ stands for a set of bounded sectors, entailing ${\mathcal{E}}_p$ for $p\in I_1$, which represents a good covering in ${\mathbb{C}}^{*}$ (see Definition 7). Each map $u_{j,p}(t,z,ϵ)$, $j=1,2$, is modeled as a rescaled version of a bounded holomorphic map $(u_1,z)\mapsto U_{j,d_p}(u_1,z,ϵ)$ through on domains $U_{1,d_p}\times H_{\beta }$ for any fixed $ϵ\in D_{{ϵ}_0}\setminus \left\{0\right\}$, where $U_{1,d_p}$ are bounded sectors bisected by the direction $d_p$, depicted in Definition 8 of the work. The set of maps ${\left\{U_{2,d_p}\right\}}_{0\le p\le \varsigma -1}$ is shown to solve a specific nonlinear partial differential equation with coefficients that are polynomial in $u_1$, holomorphic with respect to $ϵ$ on $D_{{ϵ}_0}$ and relatively to z on $H_{\beta }$ displayed in (36). The set of maps ${\left\{U_{1,d_p}\right\}}_{0\le p\le \varsigma -1}$ conforms a particular nonlinear partial differential equation stated in (37) whose coefficients and forcing term bring in not only polynomials in $u_1$ and holomorphic dependence relatively to $ϵ$ on $D_{{ϵ}_0}$ and to z on $H_{\beta }$ but also polynomial reliance on the maps ${\left\{U_{2,d_p}\right\}}_{0\le p\le \varsigma -1}$ and their derivatives with respect to $u_1$ and z. In this sense, the maps ${\left\{U_{j,d_p}\right\}}_{0\le p\le \varsigma -1}$, $j=1,2$, solve a coupling of nonlinear partial differential equations. The asymptotic property for the components $u_{j,p}(t,z,ϵ)$, $j=1,2$, of $u_p(t,z,ϵ)$ stems from sharp exponential bound estimates for the differences of neighboring maps $u_{j,p+1}-u_{j,p}$ reached in Proposition 10, for which a classical statement for the existence of asymptotic expansions of Gevrey type can be applied, see Subsection 8.1.

In this work, as mentioned above, we restrict ourselves to quadratic nonlinearities. Besides, they are chosen in a way to respect the natural triangular structure of the systems of partial differential equations satisfied by the components $u_{j,p}(t,z,ϵ)$, $j=1,2$ stated in (193), (194), which stems from the linear part of (1). It means that its resolution is reduced to the study of a coupling of two equations which comprise one single equation satisfied by $u_{2,p}(t,z,ϵ)$ and a second equation for $u_{1,p}(t,z,ϵ)$ with coefficients and forcing term that involve $u_{2,p}(t,z,ϵ)$. The treatement of a more general case with non triangular structure is postponed to a futur paper.

The approach developped in this work can be extended to the construction of both formal and genuine holomorphic solutions to comparable problems as (1) with higher order logarithmic terms for $n\ge 2$, for suitable nonlinear terms and forcing terms chosen properly in a similar way as the ones in the present work. We focus on the complete description for the case $n=1$ for the sake of simplicity in order to give the readers a clear idea of the main purpose of the study and avoiding cumbersome notations and computations.

Logarithmic type solutions have been extensively studied in the framework of nonlinear partial differential equations with so-called Fuchsian type and described in the Chapter 8 of the textbook by R. Gérard and H. Tahara [6]. Namely, these authors consider nonlinear partial differential equations with the shape where $I_m=\{(j,\alpha )\in \mathbb{N}\times {\mathbb{N}}^n/j+\left|\alpha \right|\le m,j<m\}$ for some integers $m,n\ge 1$, for analytic maps $F(t,x,Z)$ near the origin in $\mathbb{C}\times {\mathbb{C}}^n\times {\mathbb{C}}^{\mathrm{card}\left(I_m\right)}$. Under conditions of non resonance of the characteristic exponents at $x=0$ combined with some Poincaré condition on the characteristic polynomial associated to (3), they have described the holomorphic solutions to (3) with at most polynomial growth in t on bounded sectors centered at 0, for x near the origin in ${\mathbb{C}}^n$ as the maps written in the form of a convergent logarithmic type expression for $J_m=\{(i,j,k)\in \mathbb{N}\times {\mathbb{N}}^{\mu }\times \mathbb{N}/i+2m\left|j\right|\ge k+2m,\left|j\right|\ge 1\}$ where

In the case of so-called equations of irregular type or non Fuchsian type, in which our present work falls, less results are known and represents a favourable breeding ground for upcoming research. Nonetheless, in that trend, we mention the remarkable recent general result [9] obtained by H. Tahara. This work extends a paper by H. Yamazawa which treats linear partial differential equations, see [10]. Therein, the author examines nonlinear partial differential equations with $L_m=\{(j,\alpha )\in \mathbb{N}\times {\mathbb{N}}^K/j+\left|\alpha \right|\le m\}$, for some integers $m,K\ge 1$, which possess a formal series (which is divergent in the generic situation) solution where each term $u_n$, $n\ge 1$, is analytic with respect to t on some appropriate bounded sector S centered at 0 in $\mathbb{C}$ and holomorphic near 0 relatively to x on some disc $D_R$ in ${\mathbb{C}}^K$. In general, these expressions $u_n$ might involve combinations of functions of the form $t^{\lambda \left(x\right)}$ for holomorphic maps $\lambda$, powers of t and $log\left(t\right)$ and analytic functions with respect to x on $D_R$. The author introduced a so-called Newton polygon associated to the equation (4) along the formal solution $\widehat{u}(t,x)$. In the case this Newton polygon possesses $p\ge 1$ slopes and under some additional technical requirements, the author builds up a new formal solution to (4) which is subjected to the next two features

Thereupon, it turns out that $w(t,x)$ admits $\widehat{u}(t,x)$ as an asymptotic expansion as t tends to 0 on S in the sense that for any $A>0$, there exists $N_0\ge 1$ such that for all $t\in S$, some constant $C>0$, any $N\ge N_0$.

At last, in the linear setting, some general results reaching beyond the structure of logarithmic type solutions have been achieved. Namely, for Cauchy problems involving linear differential operators $a(x,D)$ of order $m\ge 1$ with holomorphic coefficients in $x={\left(x_j\right)}_{0\le j\le n}$ in ${\mathbb{C}}^{n+1}$, existence and uniqueness results for so-called ramified solutions around certain characteristic hypersurfaces K in ${\mathbb{C}}^{n+1}$, provided that v is ramified around K, have been obtained by several authors, see [11,12,13].