On the Cauchy Problem for A Reaction-Diffusion System with Singular Nonlinearity

We consider the growth rate and quenching rate of the following problem with singular nonlinearity ut  = Δu−u−λ,  ut  = Δu − u−μ, (x,t) ∈ℝn × (0, ∞),u(x,0) = u0(x), u(x,0) = u0(x), x∈ℝnu(x,0) = u0(x), u(x,0) = u0(x), x∈ℝn for any n≥ 1, where λ,μ 0 are constants. More precisely, for any u0(x), v0(x) satisfying A11(1+|x|2)α11 ≤u0≤A12 (1+|x|2)α12, A21(1+|x|2)α21 ≤ u0 ≤ A22 (1+|x|2)α22 for some constants α12 ≥ α11, α22 ≥ α21, A12 ≥ A11, A22 ≥ A21, the global solution (u,v) exists and satisfies A11(1+|x|2+b1 t)α11 ≤ u ≤ A12 (1+|x|2+b2 t)α12, A21 (1+|x|2+b1 t)a21 ≤ u ≤ A22 (1+|x|2+b2 t)α22 for some positive constants b1,b2 (see Theorem 3.3 for the parameters Aij, αij, bi, i,j = 1,2). When (1-λ)(1-λμ) > 0, (1-λ)(1-λμ) > 0 and 0< u0≤ A1 (b1 T + |x|2)1−λ1−λμ , 0 < u0≤ A2 (b2 T + |x|2) 1−μ1−λμ in ℝn for some constants Ai, bi (i = 1,2) satisfying A2  >  2nA11−λ1−λμ, A1−μ > 2nA2 1−μ1−λμ and 0 < b1 ≤ (1−λμ)A2−λ−(1−λ)2nA1 (1−λ)A1 , 0 < b2 ≤ (1−λμ)A1−μ−(1−μ)2nA2 (1−μ)A2 , we prove that u(x,t) ≤ A1 (b1 (T−t)+|x|2)1−λ1−λμ, u(x,t) ≤ A2 (b2 (T−t)+|x|2)1−μ1−λμ in ℝn× (0,T). Hence, the solution (u,v) quenches at the origin x = 0 at the same time T (see Theorem 4.3). We also find various other conditions for the solution to quench in a finite time and obtain the corresponding decay rate of the solution near the quenching time.