Extinction for a couple of fast diffusion systems with nonlinear sources

In this article, the authors establish conditions for the extinction of solutions, in finite time, of a class of fast diffusion system ut=Δum+vp,vt=Δvn+uqut=Δum+vp,vt=Δvn+uq with 00q>0, in a bounded domain Ω⊂RNΩ⊂RN with N>2N>2. More precisely speaking, it is shown that if pq>mnpq>mn and the initial values uu and vv are “comparable”, then any solution vanishes in finite time, and if pq=mnpq=mn and λ1λ1 (the first eigenvalue of −Δ−Δ in ΩΩ with homogeneous boundary condition) is large enough, then there exists a solution vanishing in finite time for small initial data.