On a semilinear parabolic system of reaction–diffusion with absorption

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.