We consider the growth rate and quenching rate of the following problem with singular nonlinearity
ut = △u − v−λ, vt = △v − u−μ, (x, t) ∈ Rn × (0,∞),
u(x, 0) = u0(x), v(x, 0) = v0(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 ≤ v0 ≤ A22(1+|x|2) α22 for some constants α12 ≥ α11, α22 ≥α 21, A12 ≥ A11, A22 ≥ A21, the global solution (u, v) exists and satisfies A11(1+|x|2+b1t) α11 ≤ u ≤ A12(1+|x|2+b2t) α12 , A21(1+|x|2+b1t) α21 ≤ v ≤ A22(1+|x|2+b2t) 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(b1T +|x|2)1−λ/1−λμ , 0 < v0 ≤ A2(b2T +|x|2)1−μ/1−λμ in Rn for some constants Ai, bi (i = 1, 2)satisfying A−λ2 > 2nA11−λ/1−λμ , A−μ1 > 2nA21−μ/1−λμ and 0 < b1 ≤ (1−λμ)A−λ2−(1−λ)2nA1/(1−λ)A1, 0 < b2 ≤ (1−λμ)A−μ1−(1−μ)2nA2/(1−μ)A2, we prove that u(x, t) ≤ A1(b1(T −t)+|x|2)1−λ/1−λμ , v(x, t) ≤A2(b2(T − t) + |x|2)1−μ/1−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.
