Multisummability of formal solutions to the Cauchy problem for some linear partial differential equations

Analytic continuation of solutions of some nonlinear convolution partial differential equations

“…For basic properties of k-convolution, see Balser [1,2], Ouchi [8,9] and Tahara-Yamazawa [11]. For simplicity, we use the notations: u * k 2 = u * k u, u * k 3 = u * k u * k u and so on, i=1,2 * k u i = u 1 * k u 2 , i=1,2,3 * k u i = u 1 * k u 2 * k u 3 and so on.…”

“…In the case of partial differential equations, the way of proof by Braaksma was followed by Ouchi [8,9], Tahara-Yamazawa [11] and Luo-Chen-Zhang [6] in treating various types of partial differential equations. But still there are many types of partial differential equations which have formal solutions but the summability has not been proved yet.…”

“…As is mentioned above, the arguments in [6,8,9,11] have given some answers to this problem. In this paper, we will introduce a new class of nonlinear convolution partial differential equations which has a nice application: the typical feature of this class is that the structure is very close to Maillet type theorems developed in Gérard-Tahara [4] and so we can apply a similar argument.…”

“…In this paper, we will introduce a new class of nonlinear convolution partial differential equations which has a nice application: the typical feature of this class is that the structure is very close to Maillet type theorems developed in Gérard-Tahara [4] and so we can apply a similar argument. In the case of linear equations, this class is the same as the one introduced in [11]. The application will be given in a forthcoming paper.…”

“…If b ν (t, x) ≡ 0 holds for all |ν| ≥ 2, (2.1) is a linear equation and it is just the same as the one treated in [11].…”