Advances in Pure Mathematics, 2011, 1, 280283 doi:10.4236/apm.2011.15050 Published Online September 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Solutions with Dead Cores for a Parabolic PLaplacian Equation Zhengce Zhang, Yanyan Li College of Science , Xi’an Jiaotong University, Xi’an, China Email:zhangzc@mail.xjtu.edu.cn, liyan86911@126.com Received June 10, 2011; revised July 15, 2011; acc epted July 25, 2011 Abstract We study the solutions with dead cores and the decay estimates for a parabolic pLaplacian equation with absorption by sub and supersolution method. Special attention is given to the case where the solution of the steadystate problem vanishes in an interior region. Keywords: Parabolic PLaplacian Equation, Steady State, Dead Core, Decay Estimates 1. Introduction In this paper, we study the following initialboundary value problem for : ,uxt 2 00 =in= ,= 0on=, ,0 =,01in, pq t udiv uuuQ uxt x ux uxux , (1.1) with and . Here ,>1p 0< <1qp is continuous on and 0 u can be extended to a func tion on x , satisfying the compatibility condition 0=,uxx x . The domain is smooth and bounded. >1 NN Our purpose is to describe how the solution ,uxt 1< <2p of (1.1) tends to its steady state and the existence of dead cores. A dead core , i.e. a region where the solution van ishes identically may ap pear. Such a region is a waste from the engineering point of view. We concern its existence. Our method is the weak solu tion which is similar to that in [4] where the porous me dium equation was considered. For the case of and of the problem (1.1), Chen, Qi and Wang [3] proved the existence of the singular solution. In [2], they also studied the long time behavior of solutions to the Cauchy pro bl e m of 0:=:= 0xux >1q 2 =in 0, pqN t udivu uu with nonnegative initial value in ,0 =ux x , where 21<NN p<2 and . For initial data of various decay rates, especially the critical decay >1q =Ox with =1pq p , they showed that the solution converges as to a selfsimilar solu tion. t rong ab eak abso We have known the following behavior of the absorp tion near : =0u <1 stsorption,q 1 wrption.q When 0x in (1.1), the steady state vanishes and ,tux tends to zero as . Strong absorption yields extinction in finite time, that is, there is a time such that t T ,uxt 0 for all and t. For small , the absorption is still relatively large in the case of strong absorption and will tend to drive the solution more quickly to zero than in the case of weak absorption. If T u is not identically zero in (1.1), the corresponding steady state does not vanish identically, but it is still true that ,uxt tends to as . t The steadystate problem n q 2= in , = o p div xx (1.2) is a special case of the problem xibx 2 =1 = in , =1 on Np xi i ax uufuD uD (1.3) in [1], where D be an arbitrary domain, ,:abD be two continuous functions, and 1p s is a nondecreasing function with 0=f0 .
Z. C. ZHANG ET AL.281 From [1], we know that Problem (1.2) has a unique solution, a dead core exists if is large enough and 11/ 0, p Fs d<s (1.4) where 0 =d s Fs f . Proofs of existence and uniqueness for Problem (1.1) are based on a suitable notion of weak solution, which we include here for the sake of completeness. Definition 1.1 Let =0, T Q T and let denote the outward unit normal to . A function n 1, , pT uxtCWL QT ;0, is called a weak solution of Problem (1.1) if it satisfies 2 2 0 0 ,,d d QT x s dd =, 0ddd p t QT Tp q uxT xT uu uxt n ux xuxt (1.5) for all 1T CQ with 1, 0 ,, ;0, pT txtCW LQT . Equation (1.5) is obtained easily by multiplying (1.1) by , integrating over , and using the divergence theo rem. T Q A weak supsolution of (1.1) is defined by replacing the equal sign in (1.5) by and restricting to be nonnegative. Similarly, we can define a weak subsolu tion. For our purposes, it suffices to consider the usual super and subsolu tions defined as follows. We say that 0u is a supersolution of (1.1) if 2 0 0, ,0 ,. pq t udiv uu uxu xx  u u (1.6) Similarly, 0u is a subsolution if all the inequa lities in (1.6) are reversed. If uu , then the unique weak solution of (1.1) satisfies uuuu, almost every where in . T It follows from the maximum principle that the solutions of (1.1) and of (1.2) satisfy in , in . Q ,uxt1Q 1x To show monotonicity of ,uxt in time, we need to impose a natural condition on the initial value 0 ux: 2 000 <0, . pq divuxuxuax (1.7) This condition holds automatically if is a posi tive constant. If satisfies (1.7), it is an upper solution to (1.1) so that 0 ux 0 ux ,uxt 0 u x for any . Now let t ,=vxt ,uxt ; then satisfies ,vxt 2=, ,0 =,,,=. pq t vdivvvv vxuxt vtx Since 0 ,uxu x , is subsolution of (1.1), and hence v t,uxt u ,x. Hence, is monotoni cally decreasing. Standard theorems can be used to show that (, )ut ,uxt tends to the steadystate as t. 2. Monotonicity and Other Comparison Theorems Consider problem (1.1) when only one part of the data is changed. We then have the following monotonicity pro perties: 1) Let 1 and 2 u be the solutions corresponding to u 1 , 2 , respectively, with 12 ; then in Q. 21 2) If the initial or the boundary value is decreased so is the solution. uu 3) Let 01ux , and consider two domains 1x 12 . Then on . 21 1 These all are easy to prove using super and subsolu tion techniques. Here we omit the proofs. uuQ Next we look at the “lumpedparameter” problem and the steadystate problem with a view to using them as comparison problems for (1.1). The lumped parameter problem has no diffusion term. It can be obtained as a special case of (1.1) with initial value 01ux and boundary condition of vanishing normal derivative. We can then seek a solution independent of ()zt : =,>0;0= q t zzt z 1. (2.1) The solution is given explicitly by 1/ 1 1/ 1 =11 >1 = =1 =11 0<<1 q t q ztq tq zt eq zt qtq (2.2) Therefore, extinction occurs in finite time if and only if . Then 0< <1q =1 1t q and for >0zt <1 1tq . Comparison with (2.1) leads immediately to two re sults for (1.1). Theorem 2.1 (a) If =0x ux , then is a super solution of (1.1), so that and, if the absorption is strong, there is extinction in finite time for zt zt ,t ,uxt. (b) If 0 min =ux > 0 , then the solution ,zt of (2.1) with initial value is a subsolution of (1.1) so that ,,uxt zt . If the absorption is weak, then ,>0 zt for all t, and hence in Q. ,>0uxt The following theorem shows that (1.4) is necessary and sufficient for the existence of a dead core for Copyright © 2011 SciRes. APM
Z. C. ZHANG ET AL. 282 sufficiently large . Theorem 2.2 For , , (1.4) is satisfied. Then a dead core exists for sufficiently large 0< <1qp>1p . Proof. We shall construct, for sufficiently large, an upper solution to (1.2) for a ball vr B, with , 0v0< <2rR. We begin by observing that on the positive real line, the function by wx 1 1 1 ()= 1 11 pp q pp q q p wx pq px qp satisfies 2 1= pq pwww , ; >0x 0=ww0=0 . Now we choose so that =1 2 R w and consider the function =2 R vwr in the ball in rR (having extend v to be zero for <2rR). We then obtain 22 1 1 1 1 11 1 =1 1 = in 2, 1 =11 := , p1 p p q p p qp q qp qp qq N div vvpvvv r N vvRR r Np vv rqp vCv Cv v where we use and 0v1 11qp q p . Since 2=0vR , 2=0vR , the last inequality can be ex tended to . Furthermore, so that 0,R =1vR vr is an upper solution of the elliptic problem (1.2) for . Since vanishes for v<2rR, so does . Now consider (1.2) on an arbitrary domain with . Then contains a ball 1 B on whose boundary 1 . Therefore, v and contains a dead core for . For any 0, we can take 0 the distance from 0 x ==Rr to the boundary. Theorem 2.2 shows that for large enough, 0 belongs to the dead core. This suggests making the following definition. Definition 2. 1 Let . Define 0 x * 00 0 =,=0;= inf inf x x We claim that ,, ,, * 02 0 ,, pqN pq p PP r (2.4) where 1 ,, =1 2 11 p Npq pp Pp N pq pq , 00 =,rdistx ; = inradius of . The proof con sists in noting that the function 1 0 0 = pq xx wr satisfies 2,, 0 =0 pNpq q p P div www r in and 1w . Thus, is a supersolution of (1.2) for any w ,, 0 pq p P r w. Since vanishes at 0 , so does . This proves the first part of (2.4) and the second part follows at once. 3. The Corresponding Evolution Problem Since (1.4) is satisfied with large enough, the steady state has a dead core. Does the corresponding evolu tion problem have a dead core and, if so, how does it behave for large t? The answer is given by the fol lowing theorem. Theorem 3.1 For fixed 0, choose 0 x > , wh ere 0 is defined in (2.3). Then (a) if 0<<11qp , for uxt 0,=0 00 1 := ; 1 tt q (b) if 1<qp1 and 0 min>0ux , then 0 uxt,>0 for all t. Proof. Part (b) is equivalent to Theorem 2.1(b). To prove part (a), it suffices to exhibit a supersolution ,wxt such that 0,=0wx t for . We try a func tion 0 tt =wz , where ,zt is the solution of the lumpedparameter problem with to be choosen satisfies the differential inequality for a supersolution and is the solution of the steadystate problem with 0 = . Note that vani shes at 0 w for 11tq . Since and wzw , it is clear that ,0wx u x 0 and x ,wt . From the definition of , we have w 2 2 00 = . pq t pq t q qq wdivwww zdiv z zz q w 0 . (2.3) Copyright © 2011 SciRes. APM
Z. C. ZHANG ET AL. Copyright © 2011 SciRes. APM 283 2 2 1 1 = = =0 pq t pq t qqt q q vdiv vvv div v q q By choosing 0 = , we obtain the desired result. . 4. Decay Estimates We consider (1.1) with , where 0 ux x is the solution of the corresponding steadystate problem (1.2). It then follows, since is a lower solution, that . If we assume in addition that , then is bounded a.e. in . ,uxt x 0 ux L uQ Our estimates hold for almost all . If the data are smooth, the solution is continuous and the estimates hold pointwise . We see k deca y estim a tes for x ,= , tuxt x Because we also have v and v , we see that is a supersolution of (1.1). Then v ==uv . 1 . Our principal results are con tained in the following theorem. Theorem 4.1 is proved. Theorem 4.1 (a) if , then 1<qp5. Acknowledgments 0, , tzt where zt 0< is the solution of (2.1). The authors are grateful to the Fundamental Research Funds for the Central Universities of China. (b) if , then <1 1qp 0,, tt 6. References where is the solution of t [1] C. Bandle and S. VernierPiro, “Estimates for Solutions of Quasilinear Problems with Dead Cores,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 54, No. 5, 2003, pp. 815821. 1 =,0 =1,=2. q tq (4.1) Proof. (a) We choose ,=wxt ztx , where satisfies (2.1) and zt satisfies (1.2). Then, we have [2] X. F. Chen, Y. W. Qi and M. X. Wang, “Long Time Be havior of Solutions to PLaplacian Equation with Ab sorption,” SIAM Journal on Mathematical Analysis, Vol. 35, No. 1, 2003, pp. 123134. doi:10.1137/S0036141002407727 2 2 = = =0, pq t pq t qq t qq qq wdivwww zdiv w zw zuz [3] X. F. Chen, Y. W. Qi and M. X. Wang, “Singular Solu tions of Parabolic PLapacian with Absorption,” Trans actions of the American Mathemarical Society, Vol. 359 2007, pp. 56535668. [4] C. Bandle, T.Nanbu, I. Stakgold, “Porous Medium Equa tion with Absorption,” SIAM Journal on Mathematical Analysis, Vol. 29, 1998, pp. 12681278. doi:10.1137/S0036141096311423 where we use . Because we also have and 1qwz w , we see that is a supersolution of (1.1). Then w ==uw .z (b) We choose ,=vxtt x , where t satisfies (4.1) and satisfies (1.2). Then, we have
