Introduction and Main Results

In [3], H. Chen and H. Tahara studied the following equation:

$$\left\{\begin{array}{l} t\partial_tu=f(t,x,u,\partial_xu),\\ u(0,x)\equiv 0,\end{array}\right.$$ (1.1)

where $(t,x)\in{\bf C}^2$, and $f(t,x,u,\xi)$ is a holomorphic function in a neighbourhood of the origin of ${\bf C}_t \times{\bf C}_x\times {\bf C}_u\times{\bf C}_{\xi}$ satisfying

$$f(0,x,0,0)\equiv 0,~~~~~{\mbox{near}}\ \ x=0.$$ (1.2)

By the condition (1.2), $f(t,x,u,\xi)$ is written as follows:

$$f(t,x,u,\xi)=\alpha(x)t+\beta(x)u+\gamma(x)\xi+ \sum_{p+q+r\geq 2} f_{pqr}(x)t^pu^q\xi^r,$$

where $\alpha(x)$, $\beta(x)$ and $\gamma(x)$ are holomorphic functions, and $p$, $q$, $r\in{\bf Z}_{\geq 0}=\{0,1,2,\ldots\}$.

If $\gamma(x)\equiv 0$, equation is called of nonlinear Fuchsian type (or of Briot-Bouquet type). In this case, many mathematicians studied the various theories. For example, convergence of formal solutions ([4, Chapters 3,5] ), the Maillet type theorem ([4, Chapter 6] , [7]), asymptotic expansions ([6]), singular solutions ([4, Chapters 4,5] ).

If $\gamma(x)\not\equiv0$ and $\gamma(0)\ne 0$, we can see that the equation is solvable in $\partial u/\partial x$. Therefore, we have a unique holomorphic solution with arbitrary holomorphic initial data $u(t,0)=\varphi(t)$ satisfying $\varphi(0)=0$ by Cauchy-Kowalevski's theorem, where $u(0,x)\equiv 0$ is automatically satisfied.

In the other case, that is, $\gamma(x)\not\equiv0$ and $\gamma(0)=0$, the equation is called of totally characteristic type. If $\gamma(x)=xc(x),\ c(0)\ne0$, Chen-Tahara obtained the conditions for the formal solution to converge ([3]). This result was generalized to several space variables by Chen-Luo ([1]) in the case where $b_k(x)=\mu_kx_k+ {\mbox{higher order}}$ (see (1.4) below), but $t$ variable is still restricted to be one dimensiomal.

If $\gamma(x)=x^jc(x),\ c(0)\ne0,\ j\geq2$, Chen-Luo-Tahara proved the Maillet type theorem, that is, they gave the formal Gevrey class in which the formal solution belongs ([2]).

In [3], Chen-Tahara obtained the following result:


Theorem (Chen-Tahara) Assume (1.2) and that $\gamma(x)=xc(x)$ with $c(0)\ne0$. Then, if

$$\vert l-\beta(0)-c(0)m\vert\geq \sigma(m+1)\ \ \ {\mbox{for ... ...in{\bf N}\times{\bf Z}_{\geq 0}$}},\ ({\bf N}=\{1,2,\ldots\})$$ (1.3)

holds for some $\sigma>0$, then the equation (1.1) has a unique holomorphic solution.

In this paper, we consider a generalization of this Chen-Tahara's theorem to the case of several time-space variables.

Let $(t,x)=(t_1,\ldots,t_d,x_1,\ldots,x_n)\in{\bf C}^d \times{\bf C}^n$ be $(d+n)$-dimensional complex variables ( $d\geq 1,\ n\geq 1$). The following equation seems to be a natural extension of (1.1) to several time-space variables:

$$\left\{\begin{array}{l} \displaystyle{\sum_{i,j=1}^da_{ij}(x... ...ial_{x_k}u\})},\\ u(t,x)=O(\vert t\vert^K),\end{array}\right.$$ (1.4)

where $\vert t\vert=t_1+\cdots+t_d$, $K$ is a fixed positive integer satisfying $K\geq 2$ and $a_{ij}(x)$, $b_k(x)$, $c(x)$ and $d_l(x)$ are holomorphic in a neighbourhood of the origin, and $f_{K+1} (t,x,u,\tau,\xi)$ $(\tau=(\tau_j)\in{\bf C}^d,\ \xi=(\xi_k)\in{\bf C}^n)$ is also holomorphic in a neighbourhood of the origin with the following Taylor expansion:

$$f_{K+1}(t,x,u,\tau,\xi)=\sum_{\vert p\vert+Kq+(K-1)\vert r\vert+K\vert s\vert \geq K+1} f_{pqrs}(x)t^pu^q\tau^r\xi^s,$$

where $q\geq 0$, $p=(p_1,\ldots,p_d) \in({\bf Z}_{\geq 0})^d$, $r=(r_1,\ldots,r_d)\in({\bf Z}_{\geq 0})^d$, $s=(s_1,\ldots,s_n)\in({\bf Z}_{\geq 0})^n$,

$$\vert p\vert=p_1+\cdots+p_d,\ \ \ \vert r\vert=r_1+\cdots+r_d,\ \ \ \vert s\vert=s_1+\cdots+s_n,$$

and

$$t^p=\prod_{j=1}^d{t_j}^{p_j},\ \ \ \tau^r=\prod_{j=1}^d{\tau_j}^{r_j},\ \ \ \xi^s=\prod_{k=1}^n{\xi_k}^{s_k}.$$

Here we remark that the assumption $K\geq 2$ implies $\partial_{t_j}u(0,0)=0\ (j=1,2,\ldots,d)$ which assures that $(0,0,u(0,0),\{\partial_{t_j}u(0,0)\}, \{\partial_{x_k}u(0,0)\})$ belongs to the domain of definition of $f_{K+1} (t,x,u,\tau,\xi)$.

Now our first theorem is stated as follows:

Theorem 1   Let $\{\lambda_j\}_{j=1}^d$ be the eigenvalues of the matrix $(a_{ij}(0))$. We assume that $b_k(x)\not\equiv 0$ and $b_k(0)=0$ for $k=1,2,\ldots,n$, and let $\{\mu_k\}_{k=1}^n$ be the eigenvalues of Jacobi matrix of $(b_1(x),\ldots, b_n(x))$ at $x=0$. Then the formal power series solution of (1.4) exists uniquely and converges if the following conditions are satisfied:

There exists a positive constant $\sigma_0>0$, such that

$$\left\vert\sum_{j=1}^d\lambda_jl_j+\sum_{k=1}^n \mu_km_k\rig... ...\vert+\vert m\vert)\ \ \ \ \ {\mbox{(Poincar\'e condition)}} ,$$ (1.5)

and

$$\sum_{j=1}^d\lambda_jl_j+\sum_{k=1}^n\mu_km_k+c(0)\ne 0 \ \ \ \ \ {\mbox{(Non-resonance condition)}}$$ (1.6)

hold for all $(l,m)\in({\bf Z}_{\geq 0})^d\times ({\bf Z}_{\geq 0})^n$ with $\vert l\vert\geq K$ and $\vert m\vert\geq0$.

Remark 1   It is easy to show the following proposition.

The conditions (1.5) and (1.6) imply that

$$\left\vert\sum_{j=1}^d\lambda_jl_j+\sum_{k=1}^n\mu_k m_k+c(0)\right\vert\geq \sigma(\vert l\vert+\vert m\vert)$$ (1.7)

holds by some positive constant $\sigma>0$ for all $(l,m)\in({\bf Z}_{\geq 0})^d\times ({\bf Z}_{\geq 0})^n$ with $\vert l\vert\geq K$ and $\vert m\vert\geq0$. In the proof of Theorem 1, this condition will be used instead of (1.5) and (1.6).

Remark 2   The condition (1.7) seems to be stronger than the condition that

$$\left\vert\sum_{j=1}^d\lambda_jl_j+\sum_{k=1}^n\mu_k m_k+c(0)\right\vert \geq\sigma^{\prime}(\vert m\vert+1)$$ (1.8)

holds by some positive constant $\sigma^{\prime}>0$ for all $(l,m)\in({\bf Z}_{\geq 0})^d\times ({\bf Z}_{\geq 0})^n$ with $\vert l\vert\geq K$ and $\vert m\vert\geq0$, which is a direct generalization of Chen-Tahara's condition (1.3). However it is actually proved that (1.7) and (1.8) are equivalent. The proof can be seen in [1].

Next, we consider the following general equation:

$$\left\{\begin{array}{l} f(t,x,u(t,x),\{\partial_{t_j}u(t,x)\... ...artial_{x_k} u(t,x)\})=0,\\ u(0,x)\equiv 0.\end{array}\right.$$ (1.9)

Assumption 1   $f(t,x,u,\tau,\xi)$ ( $\tau=(\tau_j)\in{\bf C}^d,\ \xi=(\xi_k)\in{\bf C}^n$) is holomorphic in a neighbourhood of the origin, and is an entire function in $\tau$ variables for any fixed $t$, $x$, $u$ and $\xi$. Moreover we assume that

$$f(0,x,0,\tau,0)\equiv0$$ (1.10)

for $x\in{\bf C}^n$ near the origin and $\tau \in{\bf C}^d$, which is a generalization of the definition of singular equations defined in [5].

For the equation (1.9), we do not know whether or not the equation has a formal solution in general. Therefore, we assume the following:

Assumption 2   The equation (1.9) has a formal solution of the form

$$u(t,x)=\sum_{j=1}^d\varphi_j(x)t_j+\sum_{\vert l\vert\geq2, \vert m\vert\geq0}u_{lm}t^lx^m\in{\bf C}[[t,x]].$$ (1.11)

By the existence of a formal solution, $\{\varphi_j(x)\}$ satisfy the following system formally:

$$f(0,x,0,\{\varphi_j(x)\},0)\equiv0\ \ \ {\mbox{(trivial relation)}},$$ (1.12)

and

$$\left.\frac{\partial}{\partial{t_i}}f(t,x,u(t,x), \{\partial_{t_j}u(t,x)\}, \{\partial_{x_k}u(t,x)\})\right\vert _{t=0}$$ (1.13)

$$=\frac{\partial f}{\partial t_i}(0,x,0, \{\varphi_j(x)\},0)+ \frac{\partial f}{\partial u}(0,x,0,\{\varphi_j(x)\},0) \varphi_i(x)$$

$$+\sum_{k=1}^n\frac{\partial f}{\partial \xi_k} (0,x,0,\{\varphi_j... ...l \varphi_i}{\partial x_k}(x)=0, ~~~~~~~~~~ {\mbox{{\rm for}}}\ i=1,2,\ldots,d.$$

The formal solution of this system is not convergent in general. Therefore, we assume

Assumption 3   The coefficients $\{\varphi_j(x)\}$ are all holomorphic in a neighbourhood of the origin of ${\bf C}^n$.

Remark 3   In the case $d=1$ ($d$ is the dimension of $t$ variables), a sufficient condition for the formal solution of (1.13) to converge has been already obtained by Miyake-Shirai ([5]). In the case $d\geq 2$, we give a sufficient condition for the formal solution of system (1.13) to be convergent, which will be given by Theorem 3 in Section 6, but for a while we consider the problem under Assumption 3 for simplicity of our arguments.

Now we put ${\bf {a}}(x)=(0,x,0,\{\varphi_{j}(x)\},0)$ for simplicity, and define

$$A_{ij}(x):=\frac{\partial^2f}{\partial t_i\partial\tau_j} ({... ...\xi_k}({\bf a}(x)) \frac{\partial \varphi_i}{\partial x_k}(x),$$ (1.14)

for $i,j=1,2,\ldots,d$. Moreover we define

$$B_k(x):=\frac{\partial f}{\partial \xi_k}({\bf a}(x)), \ \ {\mbox{for}}\ k=1,2,\ldots,n.$$ (1.15)

Remark 4   The functions $A_{ij}(x)$ and $B_k(x)$ correspond to $a_{ij}(x)$ and $b_k(x)$ in Theorem 1, respectively (see (1.17) below).

Here we assume that the equation is of totally characteristic type, that is,

Assumption 4   $B_k(x)\not\equiv 0$ and $B_k(0)=0$, for $k=1,2,\ldots,n.$

Now our second theorem which is our main result is stated as follows:

Theorem 2   Suppose Assumptions 1, 2, 3 and 4. Let $\{\lambda_j\}_{j=1}^d$ be the eigenvalues of $(A_{ij}(0))$, and let $\{\mu_k\}_{k=1}^n$ be the eigenvalues of Jacobi matrix of the vector $(B_k(x))$ at $x=0$. Then the formal solution (1.11) is convergent if the following condition is satisfied:

There exists a positive constant $\sigma_0>0$, such that,

$$\left\vert\sum_{j=1}^d\lambda_jl_j+\sum_{k=1}^n\mu_km_k\righ... ...\vert+\vert m\vert),\ \ \ \ \ ({\mbox{Poincar\'e condition}})$$ (1.16)

holds for all $(l,m)\in({\bf Z}_{\geq 0})^d\times ({\bf Z}_{\geq 0})^n$ with $\vert l\vert\geq 2,\ \vert m\vert\geq 0$.

Remark 5   Under the assumptions of Theorem 2, if the following non-resonance condition

$$\sum_{j=1}^d\lambda_jl_j+\sum_{k=1}^n\mu_km_k +\frac{\partial f}{\partial u}({\bf a}(0))\ne 0$$

holds for all $(l,m)\in({\bf Z}_{\geq 0})^d\times ({\bf Z}_{\geq 0})^n$ with $\vert l\vert\geq 2,\ \vert m\vert\geq 0$ as an additional condition, then the formal power series solution exists uniquely, after a determination of $\varphi_j(x)\ (j=1,2,\ldots,d)$.

Remark 6   By the Poincaré condition, there exists a positive integer $K$ such that the non-resonance condition

$$\sum_{j=1}^d\lambda_jl_j+\sum_{k=1}^n\mu_km_k +\frac{\partial f}{\partial u}({\bf a}(0))\ne 0$$

holds for all $(l,m)\in({\bf Z}_{\geq 0})^d\times ({\bf Z}_{\geq 0})^n$ with $\vert l\vert\geq K,\ \vert m\vert\geq 0$.

We put $v(t,x)=u(t,x)-\sum_{j=1}^d\varphi_j(x)t_j -\sum_{2\leq \vert l\vert\leq K-1} u_l(x)t^l$ as a new unknown function. By Assumptions 1, 2, 3 and 4, we can see that the coefficients $\{u_l(x)\}_{2\leq \vert l\vert\leq K-1}$ are determined as holomorphic functions which will be proved in Appendix B. Moreover, $v(t,x)$ satisfies the equation of the following form:

$$\left\{\begin{array}{l} \displaystyle{\sum_{i,j=1}^dA_{ij}(x... ...al_{x_k}v\})}, \\ v(t,x)=O(\vert t\vert^K).\end{array}\right.$$ (1.17)

This is an equation considered in Theorem 1. Therefore, it is sufficient to prove Theorem 1 in order to prove Theorem 2.