Summary: This paper deals with a coupled system of semilinear heat equations with Dirichlet boundary condition, \[ u_t=v^{\alpha_1} u^{\alpha_2} (\Delta u+u),\;v_t=u^{\beta_1}v^{\beta_2}(\Delta v+v), \;u=v|_{\partial\Omega}=0, \]
\[ u(x,0) =u_0(x),\quad v(x,0)= v_0 (x) \;(x\in\Omega,t>0). \] Local existence of a solution for the system is shown by using the regularization method and upper-lower solution technique. The global existence of a solution is discussed.