Applied Mathematics, 2011, 2, 11341139 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 Email: 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<<,> m tx t0 t xx . We prove finite time quenching for the solution at the boundary . We also estab uu with a singular boundary condition 0, =0, m x utu =0x lish 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 =m t uu 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 =2m u= t u 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 reactiondiffusion, 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 =m t uu with exponent . The study extends the wellknown 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 (fastdiffusion equation, FDE). >1m <1 =1m 1N m In this paper we study the problem 0 =,>0,>0, 0,=0,,> 0, ,0=,> 0, m txx m x uux t ututt uxu xx (1.1) where >0 , , is a smooth nonnegative >1m0 u ' function satisfying , for 00 m ux 00 '' m u>0x and the compatibility conditions at . =0x The 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 [610] and the references =tT =1u u *Supported by the Fundamental Research Funds for the Central Uni ve sities of China.
Z. C. ZHANG ET AL.1135 therein. The nonlinear parabolic equation 1 =1 mm tx xx m uumu m p u (1.2) with is a mathematical model for many physical problems corresponding to nonlinear diffusion with con vection. The source term on the righthand side of is of convective nature. In the theory of unnatured 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 22 0 1 =1 1, (0,) = 0,1,=1,, ,0=,0,1 , p mmp tx xx m x mm xx m uumuu m mu u ut utmu uxu xx q t (1.3) for ,0,10, tT (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 >1 the blow up rate 12 1qm ,~uxtT t for >>1qm and 11 ,~ m uxtT t for =>1qm and . <2p The 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,. m txx mq x uu xtT ututtT uxu xx , (1.4) They proved that, if 0<1 2qm , then all nonnegative solutions to were global; while for (1.4) >1qm2, 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 , pq txx txx xxxx uuvvv u utvtutvt uxu x vxv xx (1.5) for ,0,10, tT , where mi min,=0, xx vxtv 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]: >0 1) If quenching is nonsimultaneous and, for instance, is the quenching component, then v 0, ~vt Tt for close to T. t 2) If quenching is simultaneous, then for t close to T, a) 11 11 0, ~,0, ~, ,>1,<1; pq pq pq ut TtvtTt pqor pq b) 12 0,,0,~,== 1utvt Ttpq; c) 1 1 1 0,~ log, 0,~log,>= 1. q q q ut Tt vt TtTtqp For the system 00 =,=, ,0,10,, 0, =0,1, =1,,0,, 0,= 0,1,=1,,0,, ,0 =,,0 =,[0,1], txxtxx p xx q xx uu vvxtT utut vttT vtvt uttT uxu 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 nonsimul 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) 22 1, ~, 1, ~, ,>1,<1; ut Ttvt Tt pqor pq 2) 14 14 1,~,1,~,== 1;u tTtvtTtpq 3) 1 1 1 1,~ log, 1,~log, 1=<, q q q ut Tt v tTtTtpq where =1 1ppq and =1 1qpq . And 11 1, ~p vt Tt for nonsimultaneous quenching with quenching only. v In [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, txx q xx uu xtT utut uttT uxu xx (1.7) and obtained that 1 21 1, ~q utTt . In [7], Deng and Xu studied the quenching problem 0 =,0<<1,>0, 0,= 0,1,=1,,> 0, ,0 =,01. xx t xx uu xt ututgutt uxu xx (1.8) For the special case and =m uu = uu , here 0< ,<m , it is well known, 0< corresponds to the porous medium case, refers to the fast diffusion case, and when , the equation in reduces to the heat equation. They obtained that x = 1 is the only quenching point and the quenching rate is <1m >1m =1m (2.8) 121 1, ~m ut 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) and t. We get the same quen ching rate as in [7]. Furthermore, we give the asymptotic profile 1 ,~m uxT 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) t . Lemma 2.1. Suppose that and for , then and 00 ' m ux m x ux 00 '' m ux>0x,>0t ,<0 t uxt in . 0, 0,T Proof. Let , xxt,= m vxt u and . ,= t wxtu Then and satisfy ,vxt , ,wxt 12 0 =1 >0,>0, 0, =0, =0,,>0, ,0=0,>0, mm txx xx m x ' m vmuv mmuuv xt vtut utt vxu xx (2.1) and 1 11 0 =,>0,>0, 0,=0,,> 0, ,0=0,> 0, m txx m x '' m wmuwx t mu wtutt wx ux (2.2) respectively. From the maximum principle, it follows that , , and hence and >0v<0w ,>0 m x uxt ,<0t t ux in 0,T0, . By the monotone of the on ,uxt , we can find the quenching point in a finite interval 1 0, , where 1 1< < <xK . Let u be a solution of (1.1) with 0 0<uM on 1 0, . Then 0< for all in the existence interval and uMt 1 0, x >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 =0x Proof. By Lemma 2.1, we know and ,>0 m x uxt ,<0 t uxt. Thus, 1 0, min,=0, ,0,. xx uxtu ttT For 1 0 =, xd tuxt x d . , we have 11 00 1 =,d= =, 0, xx 'm txx mm xx tuxtx u uxtut x Since =<0 m t xx uu, we have 1,< 0,. mm xx uxtut There exist >0 small enough such that 1,1 0, mm xx uxtut . Thus 1 =, 0, =0, 'm m xx m x 0, , tu xtut ut ut M and so 11 11 0 0, , FtF tM ut M xx xx t which means that there exists such that >0T lim0,= 0 tT ut . 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 ,= ,, 2 m x hxt uxtx M Copyright © 2011 SciRes. AM
Z. C. ZHANG ET AL.1137 where >0 . Since ,2>0 m x uxT for , there exists >0x 0>0 such that 0>0,2 m x uxT for 0,3 4x. If we take 0 32 9M , then ,2 0hxT, 0,3 4x. We have 12 12 00 1 1>0 ,0,342,, 0, =0, =0, >0, 2, , 9 34, =34,0 32 2,. mm txx xx mm x m x m x hmuh mmuuh mummuux MM xtT T htutut tTT htutM tTT , (2.3) By the maximum principle, in 0h 0,3 42,TT , which means that 2 ,= 2 m x uxt xh M 0, ,0,12 2,. tTT Integrating with respect to , we obtain 3 ,0, 6 mm uxt utx M , ,0,12 2,, tTT and hence for any 00,1 2x, 1 3 0 0 lim inf,> 0. 6 m tT x ux tM We have shown that quenching cannot occur in the interior of 1 0, . 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 ' m ux 00 '' m ux 1 21 12 0, , m CutTt C where and are positive constants. 1 2 Proof. We first present the upper bound. Let C C ,=, , mr x tu uxtuxt , where , <0r>0 and =r . We have 12 11 32 2 32 11 32 1 3 1 , 0,0, 10, 0, 10, ,, 1 10,0,0, mm x xx rr tt mr xxx x rm txx r xt m muu tu utut uutmruuu t u ut umuux tmmuu u utu utut u 21 0,0, 0, r xt utuutut =0 =0 =( 1) tx mr mr mr mr mu ruuu mr ru mr mu ru ut mr ru mr ru 0, >0x>0t 22m x for , , and =0,0,0, =0, 0,=0. mr x r tu tutut utut 0, By the maximum principle, we have ,=,0,0. mr x uuxt ut xt Then 0, xt0 , i.e., 10,0,0,0. mr x rututut 0, xx ut Hence 11 2 0,0,=0, . rm m t rr ut ut ut mm (3.1) Integrating the equality from t to T, we obtain (3.1) 21 2 0, . m utCT 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 t tu tuxt x (3.2) with =0, m tu t , (3.3) where >m . By <0 xx um, we have mm ,< 0, xx uttu .t There exists a >0 small enough such that ,<10,. mm xx uttu t A routine calculation shows () 1 0 1 1 =0,0,(,)d 0,, 0, 0, , 0,0,0,. t ' t mm x m t ytututuxtx ututtu t mu tutt ututItu t Copyright © 2011 SciRes. AM
Z. C. ZHANG ET AL. 1138 Here () 0 =,d 0,,. t m Ituxt x mu tutt Since and in 0 m x u 0 m xx u 0, ,T , we find 0,,,20,utuxtuttut (3.4) for any 0, t t and . By , and , we have ,tT (3.2) (3.3) (3.4) 11 0,20, , mm utytu or equivalently, 11 3 0,0,for ,. m ut ytCuttT (3.5) 0 0 0 0 1 22 =,d ,, =,d ,d 0, 0, , 2 0, t t t t x m m x Itmuxt x muxtutt muxtx muttxt mut mututt dx x where 0< <tt , and >0 x u 2 0d= 2 t tx t . Then it follows that 1 4 0, . 'm ytut Cyt Integrating the above equality from t to T, we obtain 21 1 5. m m tCTt That is, 1 21 1 1, m m yCTt 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 1 c2 c 11 12 ,,0,0 <1. mm cxuxtcxxTt 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 3 c 3 ,,,0,0< x xuxtcu xtxTt1. Proof. Let 2 4 =, mm x , xuxtcxu xt, where 4 2 >1 m cm . Then satisfies 12 11 44 2 4 22 1 44 12 4 2 4 22 4 1 =4 12 122 =4 1 12 1 0, mm txx xx mmm txx mm x x mm 2 () m x mm m x xx mm x x m x JmuJ mm uuJ cmxuumc xuu mmc xuuu cm muumcuu mxuuc mmuu mmc xuuu cm muu and 0,= 0Jt . Then 2 4 ,, mm x, uxtcxuxt and the lemma is proved, where 3 2 >1 cm. Proof of Theorem 4.1. We first present the lower bound. Let :=,, . m x uxtuxt Then satisfies 12 2 3 1 =1 >0, mm txx m x xx muJmmuu J muu and 0, =0,0, =0. m x Jt utut By the maximum principle, we obtain =, , m x Ju xtuxt 0. Then 11. m x uum Integrating the above equality from 0 to x, we obtain that 1 1 ,. m uxt cx We then give the upper bound. Let 5 =, m x ucu Copyright © 2011 SciRes. AM
Z. C. ZHANG ET AL. Copyright © 2011 SciRes. AM 1139 where 21 0< =<1 2m , 1 =2 and . 5>0c 121 121 5 2 32 55 21 2 3 5 22 121 5 21 3 5 12 =1 1 1 1 22 =11 mm txx xx mm tx mm xxx mm xx m x mm m x x mm m x x mum muumxu cu umxuu cmuucm uu mmuxu u cm muu mxu ucmuxu mxuucmuu 2 21221 21 5 21 21 2 32 55 22 322 1 55 21 12 2 =1 1 12 =11 12 mm m xx m x mm x mm xx mm x mm x x mm x m mmuxu umxuu cmux u mxuu mmuxu u cmuucmux u mx uuumxu cmuucmux u mx u 3 322 1 55 11 12 0. m x mm x ucm cmuucmux u 1m 1 m x x x x . x On the other hand, 5 0,=0,< 0.tcut By the maximum principle, we have 5 =0 m x xu cu Then 1 5. m x uucx Integrating the above equality, we obtain 1 1 22 ,=. m m uxtcxcx Remark 4.1. Let , we can get tT 11 12 ,, mm cxuxTcxx 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. Muskat, “The Flow of Homogeneous Fluids through Porous Media,” McGrawHill, New York, 1937. 3] J. L. Vázquez, “Asymptotic Behaviour for the Porous Medium Equation Posed in the Whole Space,” Journal of Evolution Equations, Vol. 3, No. 1, 2003, pp. 67118. doi:10.1007/s000280300004 [4] H. Kawarada, “On Solutions of Initial Boundary Value Problem for ut = uxx = 1/(1–u),” Publications of the Research Institute for Mathematical Sciences, Vol. 10, 1975, pp. 729736. doi:10.2977/prims/1195191889 [5] C. Y. Chan and M. K. Kwong, “Quenching Phenomena for Singular Nonlinear Parabolic Equations,” Nonlinear Analysis, Vol. 12, No. 2, 1998, pp. 13771383. doi:10.1016/0362546X(88)900855 [6] C. Y. Chan, “New Results in Quenching,” Proceeding of 1st World Congress of Nonlinear Analysts, Tampa, Vol. 1, 1926 August 1992, pp. 427434. [7] K. Deng and M. X. Xu, “Quenching for a Nonlinear Dif fusion Equation with a Singular Boundary Condition,” Zeitschrift fur Angewandte Mathematik und Physik, Vol. 50, No. 4, 1999, pp. 574584. doi:10.1007/s000330050167 [8] K. Deng and M. X. Xu, “On Solutions of a Singular Dif fusion Equation,” Nonlinear Analysis, Vol. 41, No. 34, 2000, pp. 489500. doi:10.1016/S0362546X(98)002922 [9] H. A. Levine, “Advances in Quenching,” Proceeding of International Conference on ReactionDiffusion Equa tions and Their Equilibrium States, Vol. 7, 1992, pp. 319346. [10] H. A. Levine and G. M. Lieberman, “Quenching of Solu tions of Parabolic Equations with Nonlinear Boundary Conditions in Several Dimensions,” Journal für Die Reine und Angewandte Mathematik, Vol. 1983, No. 345, 1983, pp. 2338. [11] Z. C. Zhang and B. Wang, “Blowup Rate Estimate Para bolic Equation with Nonlinear Gradient Term,” Applied Mathematics and Mechanics, Vol. 31, No. 6, 2010, pp. 787796. doi:10.1007/s1048301013136 [12] V. A. Galaktionov and H. A. Levine, “On Critical Fujita Exponents for Heat Equations with Nonlinear Flux Con ditions on the Boundary,” Israel Journal of Mathematics, Vol. 94, No. 1, 1996, pp. 125146. [13] A. De Pablo, F. Quiros and J. D. Rossi, “Nonsimultane ous Quenching,” Applied Mathematics Letters, Vol. 15, No. 3, 2002, pp. 265269. doi:10.1016/S08939659(01)001288 [14] S. N. Zheng and X. F. Song, “Quenching Rates for the Heat Equatons with Coupled Singular Nonlinear Bound ary Flux,” Science in China Series AMathematics, Vol. 51, No. 9, 2008, pp. 16311643. [15] M. Fila and H. A. Levine, “Quenching on the Boundary,” Nonlinear Analysis, Vol. 21, No. 10, 1993, pp. 795802. doi:10.1016/0362546X(93)90124B [16] J. Filo, “Difusivity versus Absorption through the Bound ary,” Journal of Differential Equations, Vol. 99, No. 2, 1992, pp. 281305. [
