Extinction and Non-simultaneous Blow-up of Solutions in Fast Diffusion Equations

快速扩散方程组解的熄灭和非同时爆破

In this paper,we deal with some fast di usion equations ut=um+au vp and vt=△v^n+bu^qv^βsubject to null Dirichlet boundary conditions.We prove that every solution vanishes in nite time for pq>(m-α)(n-β),m>αand n>β,where the exact relation of initial data is determined with the aid of some Sobolev Embedding inequalities.If pd<(m-α)(n-β),m>αand n>β,we show the barriers of the initial data which lead to the non-extinction of solutions.For the case pq=(m-α)(n-β),the solutions vanish for small initial data.The results fill in a gap for the case pq<mn in Nonlinear Anal.Real World Appl.4(2013)1931-1937.The coecients a and b play a vital role in the existence of non-extinction weak solution provided that a and b are large enough.At last,we use the scaling methods to determine some exponent regions where one of the components would blow up alone for some suitable initial data.