Extinction of solutions to a class of fast diffusion systems with nonlinear sources

الأصل · EN

In this paper, the finite time extinction of solutions to the fast diffusion system uₜ=div(|∇ u|ᵖ⁻²∇ u)+vᵐ, vₜ=div(|∇ v|q⁻²∇ v)+uⁿ is investigated, where 1<p,q<2, m,n>0 and Ω⊂ Rⁿ(N≥1) is a bounded smooth domain. After establishing the local existence of weak solutions, the authors show that if mn>(p-1)(q-1), then any solution vanishes in finite time provided that the initial data are ``comparable"; if mn=(p-1)(q-1) and Ω is suitably small, then the existence of extinction solutions for small initial data is proved by using the De Giorgi iteration process and comparison method. On the other hand, for 1<p=q<2 and mn<(p-1)², the existence of at least one non-extinction solution for any positive smooth initial data is proved.