Blow-up rates for semi-linear reaction-diffusion systems

Academic Article

Abstract

  • Let Ω⊂Rn (n≥1) be a bounded smooth domain. Consider the following initial-boundary value problem of reaction-diffusion systems(I)∂tu1-δu1-u1p11u2p12=0,in (0,T)×Ω,∂tu2-δu2-u1p21u2p22=0,in (0,T)×Ω,u(t,x)=0,on (0,T)×∂Ω,u(0,x)=Φ(x)≥0,in Ω, where u=(u1, u2)≥0, and Φ(x)=(φ1(x), φ2(x))≥0, and ν is the unit outer normal at ∂Ω and T∈(0, ∞] is the maximum existence time of u (in L∞-norm) and the exponents pij, i, j=1, 2, are non-negative real numbers.Systems of form (I) naturally arise in studying non-linear phenomena in biology, chemistry, medicine and physics. For instance, (I) has been used to model densities and temperatures in chemical reactions, condensate amplitudes in Bose-Einstein condensates, wave amplitudes (or envelops of multiple interacting optical modes) in optical fibers, and pattern formation in ecological systems.Under suitable conditions on pij, i, j=1, 2, we established the following exact blow-up rates for blow-up solutions u of (I)C(T-t)-θi≤supx∈Ωui(t,x)≤C-1(T-t)-θi,i=1,2, where θ1 and θ2>0 are positive exponents depending only on pij, generalizing earlier results in this direction. © 2014 Elsevier Inc.
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    Digital Object Identifier (doi)

    Author List

  • Zou H
  • Start Page

  • 843
  • End Page

  • 867
  • Volume

  • 257
  • Issue

  • 3