We consider the semilinear parabolic system with absorption terms in a bounded domain Ω of
$\mathbb{R}^{N}$
$\[\left\{\begin{array}{l@{\quad}l}u_{t}-\Delta u+\vert v\vert ^{p}\vert u\vert ^{k-1}u=0,&\hbox{in }\varOmega\times(0,\infty),\\v_{t}-\Delta v+\vert u\vert ^{q}\vert v\vert ^{\ell -1}v=0,&\hbox{in }\varOmega\times(0,\infty),\\u(0)=u_{0},\quad v(0)=v_{0},&\hbox{in }\varOmega,\end{array}\right.\]$
where p,q>0 and k,ℓ≥0, with Dirichlet or Neuman conditions on
$\curpartial\varOmega\times(0,\infty)$
. We study the existence and uniqueness of the Cauchy problem when the initial data are L1 functions or bounded measures. We find invariant regions when u0, v0 are nonnegative, and give sufficient conditions for positivity or extinction in finite time.
