| In this paper,we are interested in studying the simultaneous and non-simultaneous quenching for the following system of heat equations coupled on the boundary,where p, q> 0, u0(x), v0(x) > 0,Ω(?)Rn is a bounded domain withsmooth boundary,ηis the unit outward normal, the non-linearboundary flux ((?)u)/((?)η)=-v-p, ((?)v)/((?)η)=-u-q represents some relation of coupled absorbtion.The study of quenching phenomena for parabolic equations was initiated by Kawarada [13] in 1975, in which he consider the singular reaction equationut = uxx +1/(1 -u). He proved that not only the reaction term, but also the time derivative blows up wherever u reaches value u = 1. This phenomena that some derivative (especially ut ) of the solution blows up while the solution itself remains bounded are called quenching by many authors afterwards. Since the article [13], quenching phenomena for various models and problems have attracted a lot of attention. And the definition of quenching has been generalized accordingly, but the essence of these definitions is : as time increases, the solution of the equation approaches the value which generates singularity for the problem.There are many articles contributed to the case of one single equation. Fila and Levine study the quenching phenomenon for the following heat equation with non-linear boundary value in [14] in 1993They prove that as t approaches T, u(1,t)～(T - t)1/2(1+β), and((?)u)/((?)t)(1,t) blows up.Fcrreira,Pablo.Quiros and Rossi [5] study the non-linear diffu- sion problem with non-linear boundary flux in 2004It is a fast diffusion equation. Their interests rest on quenching rates of the solution. And they find that there are two kinds of quenching rates, one of them is natural quenching rate, i.e.the other is superfast quenching rate, i.e.Then, they characterize these two kinds of quenching rates in terms of the parameters m and L:(1)If m < - 1. then quenching is always natural;(2)Let -1≤m < 0. If 0 < L≤(-1/m) then quenching issuperfast; and there exists some L* = L*(m)≥(-1/m)such that ifL > L*. then quenching is superfast.Qiuyi Dai and Yonggeng Gu [6] study the quenching problem for semilinear parabolic equation in 1997whereΩ(?) Rn is a bounded domain, satisfying(G1) g(s) is locally Lipschits continuous on [0,6), g(0) > 0,(G2)They define quenching of the solution u as: if T∈(0, +∞), u is the classical solution on [0, T) andThey study when quenching occurs. Letλ1 be the first eigenvalue of LaplacianΔ,ψ1 the corresponding eigenfunction, i.e.They impose other two hypothese on g(s)and assumeλ1 < c2 + c1/b, then obtain such results as: the solutionu(x.t) must quench in a finite time Tmax i.e. and Tmax can be estimated aswhereThere are many more articles on quenching problem for one single equation, the interested reader may refer to [4]-[14] and the references therein. However, there are not many articles dealing with quenching phenomenon for system of equations. Pablo, Ruiros and Rossi [3] study the system of heat equations coupled in the inner source in 2002where p, q > 0, u0, v0 > 0. Their definition for quenching is: (u, v) is the positive classical solution of the problem in finite time [0, T), and furthermoreTheir interests are that whether the quenching of solution is simultaneous or non-simultaneous, and that how such phenomenon depend on the coupled exponent p, q. By imposing convexity hypothese on the initial data, they gain the results:(1)if p,q≥1, then quenching is always simultaneously;(2)if 0 < p,q < 1, then simultaneous quenching and non-simultaneous quenching are all possible.Ferreira,Pablo,R,uiros and Rossi [1] study the quenching phenomenon for a system of heat equations coupled on the boundary in 2006The definition for quenching and their interests are similar to the above case. By assuming that u0', v0'≤0, u0'', v0'' < 0, they prove the similar results:(1)if p, q≥1, then quenching is always simultaneously;(2)if 0 < p,q < 1, then simultaneous quenching and non-simultaneous quenching are all possible.Zheng and Song [15] show great interests for the simultaneous quenching rates and the non-simultaneous ones of the above problem (2). After imposing convexities and other hypothese on the initial data, they obtain the simultaneous quenching ratesas for non-simultaneous quenching, say, v quenches, thenwhereα= (p -1)/(pq - 1),β= (q -1)/(pq - 1). Moreover they prove : if 00''(x) < 0, v0''(x) < 0 satisfy some other constrains, then v quenches while u does not.The articles [1] by Ferreira and others and [15] by Zheng and Song are both contributed to the case of one dimension, whereinΩ= (0,1). [1] gets the results: if p > 1, q > 1 then for initial data with monotrOnicity u0''(x) < 0. v0''(x) < 0, the solution of (1) is always quenching simultaneously. [6] contains such results: if 0 < p≤q < 1, or 0 < p < 1≤q. then after imposing certain hypotheses on the initial data besides monotonicity u0''(x) < 0,v0''(x) < 0, v quenches while u does not.In this paper, we improve and generalize some results in [1] and [15], mainly including: first, improvement for the case of one dimension, if p≥1,q≥1 then for any positive initial data, the solution of (1) is always quenching simultaneously; second, generalization to the case of radial multi-dimension, if p ≥1,q ≥ 1. then for any positive radially symmetric initial data quenching is always simultaneously; if 00''(r) <0,v0''(r) < 0, v quenches while u does not. Moreover, for any positive initial data, we gain some estimates on the quenching times for the two case mentioned above, and prove that the quenching points are {x = 1} and {x ∈ Rn; |x| = 1}, respectively. As for the case of non-radial multi-dimension, we also get some elementary results.
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