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

Let Ω⊂RnΩ⊂Rn (n≥1n≥1) be a bounded smooth domain. Consider the following initial–boundary value problem of reaction–diffusion systemsequation(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)≥0u=(u1,u2)≥0, and Φ(x)=(φ1(x),φ2(x))≥0Φ(x)=(φ1(x),φ2(x))≥0, and ν is the unit outer normal at ∂Ω   and T∈(0,∞]T∈(0,∞] is the maximum existence time of u (in L∞L∞-norm) and the exponents pijpij, i,j=1,2i,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 pijpij, i,j=1,2i,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 θ1θ1 and θ2>0θ2>0 are positive exponents depending only on pijpij, generalizing earlier results in this direction.