The initial value problem for a semi-linear partial differential equation of the 1st order in the sense of Frechet derivative

Abstract

The initial value problem for a semi-linear partial differential equation in the unknown function u(x, t) of the two independent variables t and x was studied. The variable t and x move in some real Banach spaces. The values of the unknown function u are in a third real Banach space. Partial differentiation with respect to t or x is considered in the sense of Frechet. Petrowsky gave the introduction to the theory of hyperbolic systems of first order linear partial differential equations with two independent real variables. In this note, the extension to the quasi-linear case was planed. At first, some preparations from the theory of Frechet derivatives were given. Then another preparation from the theory of ordinary differential equations in the sense of Frechet derivatives was made. Finally the main result was stated and proved. When dim (E) is greater than 1, where E is the domain of x, the differential equation to be solved is regarded as an 'overdetermined system' of equations. Therefore complete integrability conditions were imposed. Assuming the complete integrability conditions, the existence theorem of the initial value problem was given.

バナッハ空間の変数x,tを2独立変数とする、バナッハ空間値関数uの半線形偏微分方程式の初期値問題を論じた。x,tの偏微分はフレシエ微分の意味である。ペトロフスキーは実2変数双曲型1階線形偏微分方定式の理論を与えた。これの擬線形への拡張を論じた。はじめに、フレシエ微分の理論について準備した。ついで、フレシエ微分の意味での常微分方程式について準備した。最後に、主要結果を述べ、証明した。xの定義域空間Eが、dim(E)>1の場合、解くべき微分方程式は、'過剰決定系'であるため、完全可積分条件と呼ぶ付加条件を与える必要がある。この完全可積分条件を仮定して、初期値問題に対する解の存在定理を述べた。