IJPAM: Volume 20, No. 1 (2005)

ON A NONLINEAR COUPLED SYSTEM

M.R. Clark$^1$, H.R. Clark$^2$, O.A. Lima$^3$
$^1$Universidade Federal do Piauí
DM, PI, BRASIL
e-mail: mclark@ufpi.br
$^2$Instituto de Matemática (IM)
Universidade Federal Fluminense (UFF)
Rua Mário Santos Braga S/N
Valonguinho, 24.020-140, Niterói, Rio de Janeiro, BRASIL
e-mail: hclark@vm.uff.br
$^3$Universidade Estadual da Paraíba
PB, BRASIL
e-mail: osmundo@opentina.com.br


Abstract.We prove the existence and uniqueness of the weak solutions of the Cauchy problem for the system

\begin{eqnarray*} &\displaystyle u''-\Delta u+f(u, v)u=h_1, &\\ &\displaystyle v''-\Delta v+g(u, v)v=h_2, \end{eqnarray*}

assuming only that the functions $\,f\,$ and $\,g\,$ are continuous in the first variable and Lipschitz in the second one. In one dimensional case, this non-linear system describes the motion of charged meson in an electromagnetic field. In previous investigations, this system has been studied supposing the functions $\,f\,$ and $\,g\,$ more regular then in our case. Therefore, we improve the earlier results, as we can see in Introduction, by weakening the regularity assumptions on the functions $\,f\,$ and $\,g.\,$

Received: March 5, 2005

AMS Subject Classification: 35F25, 35L70

Key Words and Phrases: Cauchy problem, existence, uniqueness, weak local solutions

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2005
Volume: 20
Issue: 1