Extinction for a couple of fast diffusion systems with nonlinear sources

Abstract In this article, the authors establish conditions for the extinction of solutions, in finite time, of a class of fast diffusion system u t = Δ u m + v p , v t = Δ v n + u q with 0 m , n 1 , p , q > 0 , in a bounded domain Ω ⊂ R N with N > 2 . More precisely speaking, it is shown that if p q > m n and the initial values u and v are “comparable”, then any solution vanishes in finite time, and if p q = m n and λ 1 (the first eigenvalue of − Δ in Ω with homogeneous boundary condition) is large enough, then there exists a solution vanishing in finite time for small initial data.

[1]  Lili Du,et al.  Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources , 2007 .

[2]  Avner Friedman,et al.  Extinction and positivity for a system of semilinear parabolic variational inequalities , 1992 .

[3]  Mingxin Wang,et al.  BLOW-UP PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS AND SYSTEMS OF PARABOLIC TYPE , 2009 .

[4]  Mingxin Wang,et al.  Fast-slow diffusion systems with nonlinear boundary conditions , 2001 .

[5]  C. Xie,et al.  Blow-up and global existence for a nonlocal degenerate parabolic system , 2003 .

[6]  M. A. Herrero,et al.  A semilinear parabolic system in a bounded domain , 1993 .

[7]  Howard A. Levine,et al.  The critical exponent of degenerate parabolic systems , 1993 .

[8]  Jeffrey R. Anderson.  Local Existence And Uniqueness of Solutions of Degenerate Parabolic Equations , 1991 .

[9]  Chunlai Mu,et al.  Extinction and non-extinction for a p-Laplacian equation with nonlinear source ☆ , 2008 .

[10]  A. S. Kalashnikov.  The propagation of disturbances in problems of non-linear heat conduction with absorption , 1974 .

[11]  M. Ughi,et al.  The fast diffusion equation with strong absorption: the instantaneous shrinking phenomenon , 1994 .

[12]  Chunhua Jin,et al.  Critical extinction and blow-up exponents for fast diffusive polytropic filtration equation with sources , 2009, Proceedings of the Edinburgh Mathematical Society.

[13]  Wenjie Gao,et al.  Extinction for a fast diffusion equation with a nonlinear nonlocal source , 2011 .

[14]  Mingxin Wang,et al.  Quasilinear Reaction–Diffusion Systems with Nonlinear Boundary Conditions☆☆☆ , 1999 .

[15]  Jingxue Yin,et al.  Non-extinction and critical exponent for a polytropic filtration equation , 2009 .

[16]  M. A. Herrero.  On a class of nonlinear degenerate parabolic equations , 1982 .

[17]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[18]  Shijie Dong,et al.  Periodic solutions for a delayed predator-prey system with dispersal and impulses , 2005 .