Hidetoshi Tahara

On the analytic continuation of solutions to nonlinear partial differential equations

Abstract We consider the analytic continuation of solutions to the nonlinear partial differential equation ∂ ∂t m u=F t,x, ∂ ∂t j ∂ ∂x α u j+|α|⩽m j⩽m−1 in the complex domain. Suppose a solution u(t,x) is known to be holomorphic in the domain {(t,x)∈ C × C n ; |x| 0 and |argt|

On the Holomorphic Solution of Non–Linear Totally Characteristic Equations

The paper deals with a non linear singular partial di erential equation E t t F t x u u x in the holomorphic category When E is of Fuchsian type the existence of the unique holomorphic solution was established by G erard Tahara In this paper under the assumption that E is of totally characteristic type the authors give a su cient condition for E to have a unique holomorphic solution The result is extended to higher order case

Coupling of two partial differential equations and its application

The paper considers the coupling of the following two nonlinear partial differential equations

Holomorphic and singular solutions of non linear singular partial differential equations

\frac{{\partial u}} {{\partial t}} = F\left( {t,x,u,\frac{{\partial u}} {{\partial x}}} \right) and \frac{{\partial w}} {{\partial t}} = G\left( {t,x,w,\frac{{\partial w}} {{\partial x}}} \right),

Operators with regular singularities: Several variables case

and establishes the equivalence of them. The result is applied to the problem of analytic continuation of the solution.

Maillet’s type theorems for non linear singular integro—differential equations

This paper deals with singular nonlinear partial differential equations of the form t∂u/∂t = F (t, x, u, ∂u/∂x), with independent variables (t, x) ∈ R × C, and where F (t, x, u, v) is a function continuous in t and holomorphic in the other variables. Using the Banach fixed point theorem, we show that a unique solution u(t, x) exists under the condition that F (0, x, 0, 0) = 0, Fu(0, x, 0, 0) = 0 and Fv(0, x, 0, 0) = x γ(x) with Re γ(0) < 0. 2010 Mathematics Subject Classification: Primary 35A01; Secondary 35A10, 35A20, 35F20.

Maillet’s type theorems for non linear singular partial differential equations without linear part

In chapter 5, we have studied holomorphic and singular solutions of non linear singular partial differential equations of the first order called Briot-Bouquet type. In the present chapter, we are extending the results of chapter 5 to equations of higher order.

On the existence of holomorphic solutions of the Cauchy problem for non linear partial differential equations

In this chapter, we introduce linear and non linear singular operators D acting on formal power series and we study the operators having the property of “regular singularity”. This means that we give conditions on D to have the following property: “if u is a formal power series such that Du converges then u is a convergent power series”. This study gives us then very interesting applications to differential equations and gives new proofs for classical results.