ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY

doi:10.1016/S0252-9602(13)60061-2

Abstract

Abstract:

We consider the growth rate and quenching rate of the following problem with singular nonlinearity
u_t = △u − v^−λ, v_t = △v − u^−μ, (x, t) ∈ Rⁿ × (0,∞),
u(x, 0) = u₀(x), v(x, 0) = v₀(x), x ∈Rⁿfor 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≥α₂₁, 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 A_ij , α_ij , b_i, 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 A_i, b_i(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.

Key words: Cauchy problems, singular nonlinearity, growth rate, quenching rate

CLC Number:

35B35

ZHOU Jun. ON THE CAUCHY PROBLEM FOR A REACTION-DIFFUSION SYSTEM WITH SINGULAR NONLINEARITY[J].Acta mathematica scientia,Series B, 2013, 33(4): 1031-1048.

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