Blow-up for a degenerate reaction-diffusion system with nonlinear nonlocal sources

This paper investigates the global existence and blow-up of nonnegative solution of the systemu"[email protected]^m+u^p^"^[email protected]!"@Wv^q^"^1dx,v"[email protected]^n+v^p^"^[email protected]!"@Wu^q^"^2dx,(x,t)@[email protected](0,T)with homogeneous Dirichlet boundary conditions, where @[email protected]?R^N is a bounded domain with smooth boundary @[email protected], m, n>1, p"1, p"2, q"1, q"2>0. The results depend crucially on the number p"i, q"i, m, n, the domain @W and the initial data u"0(x), v"0(x). Moreover, we obtain the blow-up rate of the blow-up solution under some appropriate hypotheses.

[1]  Hongwei Chen.  Global Existence and Blow-up for a Nonlinear Reaction-Diffusion System☆ , 1997 .

[2]  Keng Deng,et al.  Global existence for degenerate parabolic equations with a non‐local forcing , 1997 .

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

[4]  Philippe Souplet,et al.  Uniform Blow-Up Profiles and Boundary Behavior for Diffusion Equations with Nonlocal Nonlinear Source , 1999 .

[5]  Mingxin Wang.  Blowup Estimates for a Semilinear Reaction Diffusion System , 2001 .

[6]  Howard A. Levine,et al.  A general approach to critical Fujita exponents in nonlinear parabolic problems , 1998 .

[7]  L. A. Peletier,et al.  Stabilization of solutions of a degenerate nonlinear diffusion problem , 1982 .

[8]  Fucai Li,et al.  Global existence and blow-up for a nonlinear porous medium equation , 2003, Appl. Math. Lett..

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

[10]  Howard A. Levine,et al.  The Role of Critical Exponents in Blowup Theorems , 1990, SIAM Rev..

[11]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[12]  Chunhong Xie,et al.  Global existence and blow-up of solutions to a nonlocal reaction-diffusion system , 2003 .

[13]  Weibing Deng,et al.  Global existence and finite time blow up for a degenerate reaction–diffusion system , 2005 .

[14]  A maximum principle for semilinear parabolic systems , 1979 .

[15]  A. Friedman,et al.  Blow-up of positive solutions of semilinear heat equations , 1985 .

[16]  Chunlai Mu,et al.  Global existence and non-existence for a quasilinear degenerate parabolic system with non-local source , 2005 .

[17]  Paul E. Sacks,et al.  Global Behavior for a Class of Nonlinear Evolution Equations , 1985 .

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

[19]  Zhiwen Duan,et al.  Uniform blow-up profile for a degenerate parabolic system with nonlocal source , 2004 .

[20]  M. Grinfeld,et al.  Local vs. non-local interactions in population dynamics , 1989 .

[21]  C. V. Pao,et al.  Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory , 1992 .

[22]  A. Bressan,et al.  Thermal behavior for a confined reactive gas , 1982 .

[23]  Philippe Souplet,et al.  Blow-up in nonlocal reaction-diffusion equations , 1998 .

[24]  Daniele Andreucci,et al.  Liouville theorems and blow up behaviour in semilinear reaction diffusion systems , 1997 .

[25]  Sining Zheng.  Global existence and global non-existence of solutions to a reaction-diffusion system , 2000 .

[26]  Howard A. Levine,et al.  Some existence and nonexistence theorems for solutions of degenerate parabolic equations , 1984 .

[27]  Keng Deng,et al.  The Role of Critical Exponents in Blow-Up Theorems: The Sequel , 2000 .

[28]  E. DiBenedetto.  Degenerate Parabolic Equations , 1993 .