SOME CLASS OF INFINITE DIMENSIONAL HYPERBOLIC SYSTEMS OF PDES

Abstract

The initial value problem for a system of PDEs of the form $(*)\frac{\partial u_i(t,x)}{\partial t}+\frac{\partial u_i\left(t,x\right)}{\partial x}{\lambda }_i\left(t,x,u\left(t,x\right)\right)=f_i\left(t,x,u\left(t,x\right)\right),1\le i\le N.$ is solved. In $(*)t$ is a real variable, $x$ is a variable moving in a real Banach space $X$. $u_1,\dots ,u_N$ are unknown functions whose values are in a real Banach space $G$. $u$ denotes the ‘vector’ $(u_1,\dots ,u_N)$. Differentiation in $x$ is taken in the sense of the Fréchet derivative, $f_1,\dots ,f_N$ are $G$-valued given functions and ${\lambda }_1,\dots ,{\lambda }_N$ are $X$-valued given functions.