Jin Yan University of Science and Technology of China, Hefei
Zhixin Cheng University of Science and Technology of China, Hefei
Ming Tao University of Science and Technology of China, Hefei
Resumen.
En este artículo usamos el teorema de punto fijo de Brouwer-Schauder para
obtener la existencia de soluciones locales de viscosidad suave al
problema de Cauchy para el sistema parabólico
\[
\left\{
\begin{aligned}
&u^1_t+f_1\left(u^1,u^2,\cdots,u^n\right)_x+g_1\left(u^1,u^2,\cdots,u^n\right) = \varepsilon u^1_{xx}
& \qquad \qquad \qquad \qquad \qquad \quad \vdots
&u^n_t+f_n\left(u^1,u^2,\cdots,u^n\right)_x+g_n\left(u^1,u^2,\cdots,u^n\right) = \varepsilon u^n_{xx},
\end{aligned}
\right.
\]
con data inicial medible acotada
$$u^1(x,0)=u^1_0(x),\quad u^2(x,0)=u^2_0(x), \cdots, \quad u^n(x,0)=u^n_0(x).$$
Luego, basados en la existencia local y el principio del máximo,
obtenemos la existencia de soluciones globales suaves para dos
sistemas especiales , uno relacionado con el sistema parabólico de
flujo cuadrático y el otro relacionado con el sistema LeRoux.
Abstract.
In this paper we use
the Brouwer-Schauder's fixed point theorem to obtain the
existence of local smooth viscosity solutions of the Cauchy
problem for the parabolic system
\[
\left\{
\begin{aligned}
&u^1_t+f_1\left(u^1,u^2,\cdots,u^n\right)_x+g_1\left(u^1,u^2,\cdots,u^n\right) = \varepsilon u^1_{xx}
& \qquad \qquad \qquad \qquad \qquad \quad \vdots
&u^n_t+f_n\left(u^1,u^2,\cdots,u^n\right)_x+g_n\left(u^1,u^2,\cdots,u^n\right) = \varepsilon u^n_{xx},
\end{aligned}
\right.
\]
with the bounded measurable initial data
$$u^1(x,0)=u^1_0(x),\quad u^2(x,0)=u^2_0(x), \cdots, \quad u^n(x,0)=u^n_0(x).$$
Then based on the local existence and the maximum principle, we
get the existence of global smooth solutions for two special
systems, one related to the hyperbolic system of quadratic flux
and the other related to the LeRoux system.
