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 Applied Mathematics, 2011, 2, 1134-1139 doi:10.4236/am.2011.29157 Published Online September 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Quenching Rate for the Porous Medium Equation with a Singular Boundary Condition* Zhengce Zhang, Yanyan Li College of Science, Xi’an Jiaotong University, Xi’an, China E-mail: zhangzc@mail.xjtu.edu.cn, liyan8691 1@126.com Received July 3, 2011; revised July 24, 2011; accepted August 1, 2011 Abstract We study the porous medium equation =,0<<,>mtx t0txx. We prove finite time quenching for the solution at the boundary . We also estab- uu with a singular boundary condition  0, =0,mxutu=0xlish the quenching rate and asymptotic behavior on the quenching point. Keywords: Porous Medium Equation, Quenching Profile, Quenching Rate, Singular Boundary Condition 1. Introduction The nonlinear diffusion equation =mtuu with exponent , is usually called the porous medium equation, written here PME for short. In the particular case , it is called Boussinesq's equation. The PME equation is one of the simplest examples of a nonlinear evolution equation of parabolic type. It appears in the description of different natural phenomena, and its theory and properties depart strongly from the heat equation , its most famous relative. >1m=2mu=tuThere are a number of physical applications where this simple model appears in a natural way, mainly to describe processes involving fluid flow, heat transfer or diffusion. Maybe the best known of them is the descrip- tion of the flow of an isentropic gas through a porous medium, modelled independently by Leibenzon [1] and Muskat [2] around 1930, where represents the den- sity of the gas and is a constant. The most striking manifestation of this nonlinear degeneracy is that in porous medium flow there is a finite speed of propa- gation of disturbances from rest. u2,mOnce the theory for the PME began to be known, a number of applications have been proposed. Some of them concern the fast diffusion equation, the generalized PME and the inhomogeneous versions already com- mented. There are numerous examples with lower order terms, in the areas of reaction-diffusion, where the PME is only responsible for one of the various mechanisms of the equation or system. In [3], it is devoted to present a detailed account of the asymptotic behavior as of the solutions t,uxt of the equation =mtuu with exponent . The study extends the well-known theory of the classical heat equation (HE, the case ) into a nonlinear situation, which needs a whole set of new tools. The space dimension can be any integer . They also present the extension of the results to exponents (fast-diffusion equation, FDE). >1m<1=1m1NmIn this paper we study the problem   0=,>0,>0,0,=0,,> 0,,0=,> 0,mtxxmxuux tututtuxu xx (1.1) where >0, , is a smooth nonnegative >1m0u'function satisfying , for 00mux00''mu>0xand the compatibility conditions at . =0xThe study of quenching (in general the solution is defined up to but some term in the problem ceases to make sense) began with the work of Kawarada [4] appeared in 1975. In that paper he studied the semilinear heat equation as a singular reaction at level . He proved that not only the reaction term, but also the time derivative blows up wherever reaches this value, see also [5]. Quenching problems have been studied by many authors, see [6-10] and the references =tT=1uu*Supported by the Fundamental Research Funds for the Central Uni-versities of China. Z. C. ZHANG ET AL.1135 therein. The nonlinear parabolic equation 1=1pmmtxxxmuumum pu (1.2) with is a mathematical model for many physical problems corresponding to nonlinear diffusion with con- vection. The source term on the right-hand side of is of convective nature. In the theory of un-natured porous medium equation, the convective part may re- present the effect of gravity. Moreover, with is also a Boussinesq equation of hydrology, which is involved in various fields of petroleum technology and ground water hydrology. For instance, in [11], Zhang and Wang studied the following equation: 0m(1.2)=2m   2201=11||,(0,) = 0,1,=1,,,0=,0,1 ,pmmptxxxmxmmxxmuumuummu uut utmuuxu xxqt (1.3) for  ,0,10,xtT(1.3)>>1qm =qm, where and are parameters, and is continuous and satisfies the compatibility conditions. They proved that, every solution of will blow up in finite time for or for and . And they got >1qm<2p<2p0>1ux>1the blow up rate  12 1qm,~uxtT t for >>1qm and  11,~ muxtT t for =>1qmand . <2pThe porous medium equation without convection has been considered extensively in the past few years. For instance, in [12], Galaktionov and Levine studied the following equation:   0=,,0,0,0,=0, ,0,,,0 =,0,.mtxxmqxuu xtTututtTuxu xx, (1.4) They proved that, if 0<1 2qm , then all nonnegative solutions to were global; while for (1.4)>1qm2, the solutions to the equation would blow up in finite time. Moreover, if 12< 1mqm, all nonnegative solutions blow up in finite time; if >1qm, global nontrivial nonnegative solutions existed. Pablo, Quiros and Rossi [13] firstly distinguished non- simultaneous quenching from simultaneous one. They considered a heat system coupled via inner absorptions,    00=,=,0, =0, =1, =1, =0,,0=,,0=,0,1 ,pqtxx txxxxxxuuvvv uutvtutvtuxu x vxv xx (1.5) for ,0,10,xtT , where mi min,=0,xxvxtv t,uv[0,1]n,=0, ,uxtut[0,1] under certain assumptions on the initial data 00 . For the coupled equations (1.5), the following quenching rates were proved in [13]: >01) If quenching is non-simultaneous and, for instance, is the quenching component, then v0, ~vt Tt for close to T. t2) If quenching is simultaneous, then for t close to T, a)   11110, ~,0, ~,,>1,<1;pqpq pqut TtvtTtpqor pq b)  120,,0,~,== 1utvt Ttpq; c) 1110,~ log,0,~log,>= 1.qqqut Ttvt TtTtqp For the system    00=,=, ,0,10,,0, =0,1, =1,,0,,0,= 0,1,=1,,0,,,0 =,,0 =,[0,1],txxtxxpxxqxxuu vvxtTutut vttTvtvt uttTuxu x vxv xx (1.6) the finite time quenching results with the coupled sin- gular nonlinear boundary flux were obtained by Zheng and Song [14], other than the situation in the model of (1.5) with coupled nonlinear absorption terms. The quen- ching in (1.6) may be either simultaneous or non-simul- taneous. This is determined by particular ranges of non- linear exponents and the initial data. They showed that =1x is the only quenching point and there are three kinds of simultaneous quenching rates can be briefly described in the following conclusions: 1)   221, ~, 1, ~,,>1,<1;ut Ttvt Ttpqor pq 2)   14 141,~,1,~,== 1;u tTtvtTtpq 3)  1111,~ log,1,~log, 1=<,qqqut Ttv tTtTtpq where =1 1ppq and =1 1qpq. And 111, ~pvt Tt for non-simultaneous quenching with quenching only. vIn [15], Fila and Levine studied the quenching pro- Copyright © 2011 SciRes. AM Z. C. ZHANG ET AL. 1136 blem for the scalar case    0=, ,0,10,,0, =0,1, =1,,0,,,0=> 0,0,1,txxqxxuu xtTutut uttTuxu xx (1.7) and obtained that  1211, ~qutTt. In [7], Deng and Xu studied the quenching problem   0=,0<<1,>0,0,= 0,1,=1,,> 0,,0 =,01.xxtxxuu xtututguttuxu xx (1.8) For the special case and =muu=guu, here 0< ,1m=1m(2.8)1211, ~mut Tt for the porous medium case. Our main purpose in this paper is to examine the quenching behavior of the solution of the problem , that is, the solution reaches zero in finite time and the quenching rate about (1.1)x and t. We get the same quen- ching rate as in [7]. Furthermore, we give the asymptotic profile 1,~muxT x. The paper is organized as follows: In Section 2, we prove that quenching occurs only at . In Section 3, we derive estimates for the quenching rate. In Section 4, we give the precise asymptotic profile near . =0x=0x 2. Quenching on the Boundary We state a lemma that guarantees that, for certain initial data, the solution of decreases with and increases with (1.1) tx. Lemma 2.1. Suppose that and for , then and 00'muxmxux00''mux>0x,>0t,<0tuxt in . 0, 0,TProof. Let ,xxt,= mvxt u and . ,=twxtuThen and satisfy ,vxt,,wxt 120=1>0,>0,0, =0, =0,,>0,,0=0,>0,mmtxx xxmx'mvmuv mmuuvxtvtut uttvxu xx (2.1) and  1110=,>0,>0,0,=0,,> 0,,0=0,> 0,mtxxmx''mwmuwx tmu wtuttwx ux (2.2) respectively. From the maximum principle, it follows that , , and hence  and >0v<0w,>0mxuxt,<0ttux in 0,T0, . By the monotone of the on ,uxtx, we can find the quenching point in a finite interval 10,x, where 11< < 0. We now present the quenching result for the problem . (1.1)Theorem 2.1. Assume . Then every solution of (1.1 ) quenches in finite time with the only quenching point . u=0xProof. By Lemma 2.1, we know and ,>0mxuxt,<0tuxt. Thus, 10,min,=0, ,0,.xxuxtu ttT For  10=,xdFtuxtxd., we have  11001=,d==, 0,xx'mtxxmmxxFtuxtx uuxtutx Since =<0mtxxuu, we have 1,< 0,.mmxxuxtut There exist >0 small enough such that 1,1 0,mmxxuxtut . Thus 1=,0,=0,'m mxxmx0,,Ftu xtututut M and so 11 1100, ,FtFMtMut Mxx xx t which means that there exists such that >0Tlim0,= 0tTut. To show that is the unique quenching point, it suffices to prove that the quenching cannot occur at any inner point =0x00,1 2x. Define 2,= ,,2mxhxt uxtxM Copyright © 2011 SciRes. AM Z. C. ZHANG ET AL.1137 where >0. Since ,2>0mxuxT for , there exists >0x0>0 such that 0>0,2mxuxT for 0,3 4x. If we take 032 9M, then ,2 0hxT, 0,3 4x. We have  12120011>0,0,342,,0, =0, =0, >0,2, ,934, =34,0322,.mmtxx xxmmxmxmxhmuh mmuuhmummuuxMMxtT ThtututtTThtutMtTT , (2.3) By the maximum principle, in 0h0,3 42,TT, which means that 2,=2mxuxt xhM0, ,0,12 2,.xtTT Integrating with respect to x, we obtain  3,0,6mmuxt utxM, ,0,12 2,,xtTT and hence for any 00,1 2x, 1300lim inf,> 0.6mtTxux tM We have shown that quenching cannot occur in the interior of 10,x. By the monotone, the proof is com- plete. 3. Bounds for the Quenching Rate In this section, we establish bounds on the quenching rate. Theorem 3.1. Suppose that and . Then the solution of (1.1) satisfies 00'mux00''mux121120, ,mCutTt C where and are positive constants. 1 2Proof. We first present the upper bound. Let C C,=, ,mrxxtu uxtuxt , where , <0r>0 and =r. We have     121132 2321132 131, 0,0,10, 0,10,,, 110,0,0,mmx xxrrttmrxxxxrmtxxrxtm muutu ututuutmruuu tu utumuux tmmuuu utu ututu  210,0, 0,rxtutuutut =0=0=(1)txmrmrmrmrmuruuumr rumr muru utmr rumr ru 0,>0x>0t22mx for , , and   =0,0,0,=0, 0,=0.mrxrtu tutututut0, By the maximum principle, we have ,=,0,0.mrxuuxt utxt Then 0,xt0, i.e., 10,0,0,0.mrxrututut0,xxut Hence  11 20,0,=0, .rm mtrrut ut utmm     (3.1) Integrating the equality from t to T, we obtain (3.1)2120, .mutCT t Thus we prove the desired upper bound. We then give the lower bound. We use a modification of an argument from [16]. For ,tT with some  such that 0,< 1ut , set  0=0, ,dtytu tuxtx (3.2) with =0,mtu t, (3.3) where >m. By <0xxum, we have mm,< 0,xxuttu .t There exists a >0 small enough such that ,<10,.mmxxuttut A routine calculation shows   ()1011=0,0,(,)d0,, 0,0, , 0,0,0,.t'tmmxmtytututuxtxututtu tmu tuttututItu t Copyright © 2011 SciRes. AM Z. C. ZHANG ET AL. 1138 Here  ()0=,d0,,.tmItuxt xmu tutt Since and in 0mxu0mxxu0, ,T , we find  0,,,20,utuxtuttut  (3.4) for any 0,xt t and . By , and , we have ,tT(3.2) (3.3)(3.4) 110,20, ,mmutytu   or equivalently,  1130,0,for ,.mut ytCuttT (3.5)    0000122=,d,,=,d,d0,0, ,20,ttttxmmxItmuxt xmuxtuttmuxtxmuttxtmutmututt   dxx where 0< 0xu20d= 2txtx t. Then it follows that 140, .'mytut Cyt   Integrating the above equality from t to T, we obtain  2115.mmytCTt  That is, 12111,mmyCTt which in conjunction with yields the desired lower bound. (3.5) 4. Asymptotic Profile In this section, we shall derive the following precise asymptotic profile near x = 0. Theorem 4.1. Suppose that is the solution of (1.1) and assume that the quenching occurs at a finite time , then there exist , such that u>0=tT 1c2c1112,,0,0 <1.mmcxuxtcxxTt  We first prove a lemma as follows. Lemma 4.1. Assume that is the solution of (1.1) and assume that the quenching occurs at a finite time , then there exists a such that u>0=tT 3c3,,,0,0<xxuxtcu xtxTt1. Proof. Let 24=,mmx,Jxuxtcxu xt, where 42>1mcm. Then J satisfies    1211442422 144124242241=412122=4 11210,mmtxx xxmmmtxxmmxxmm2()mxxmm mxxxmmxxmxJmuJ mm uuJcmxuumc xuummc xuuucm muumcuumxuuc mmuummc xuuucm muu  and 0,= 0Jt . Then 24,,mmx,xuxtcxuxt and the lemma is proved, where 32>1cm. Proof of Theorem 4.1. We first present the lower bound. Let :=,, .mxJuxtuxt Then J satisfies 12231=1 >0,mmtxxmxxxJmuJmmuu Jmuu and 0, =0,0, =0.mxJt utut By the maximum principle, we obtain =, ,mxJu xtuxt0. Then 11.mxuum  Integrating the above equality from 0 to x, we obtain that 11,.muxt cx We then give the upper bound. Let 5=,mxxucu Copyright © 2011 SciRes. AM Z. C. ZHANG ET AL. Copyright © 2011 SciRes. AM 1139where 210< =<12m, 1=2 and . 5>0c1211215232552123522 121521 3512=111122=11mmtxx xxmmtxmmxxxmmxxmxmm mxxmm mxxmum muumxucu umxuucmuucm uummuxu ucm muumxu ucmuxumxuucmuu         22122121521212325522322 15521122=1112=1112mm mxxmxmmxmmxxmmxmmxxmmxmmmuxu umxuucmux umxuummuxu ucmuucmux umx uuumxucmuucmux umx u      3322 15511120.mxmmxucmcmuucmux u   1m1mxxxx.x On the other hand,  50,=0,< 0.tcut By the maximum principle, we have 5=0mxxu cu Then 15.mxuucx  Integrating the above equality, we obtain 1122,=.mmuxtcxcx Remark 4.1. Let , we can get tT1112,,mmcxuxTcxx 0. 5. References [1] L. S. Leibenzon, “The Motion of a Gas in a Porous Me-dium,” Russian Academy of Sciences, Moscow, 1930. [2] M. 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