 Advances in Pure Mathematics, 2011, 1, 280-283 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 P-Laplacian Equation Zhengce Zhang, Yanyan Li College of Science , Xi’an Jiaotong University, Xi’an, China E-mail: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 p-Laplacian equation with absorption by sub- and supersolution method. Special attention is given to the case where the solution of the steady-state problem vanishes in an interior region. Keywords: Parabolic P-Laplacian Equation, Steady State, Dead Core, Decay Estimates 1. Introduction In this paper, we study the following initial-boundary value problem for : ,uxt  200=in=,= 0on=,,0 =,01in,pqtudiv uuuQuxt xux uxux, (1.1) with and . Here ,>1p0< <1qpx is continuous on and 0u can be extended to a func- tion on x, satisfying the compatibility condition  0=,uxx x. The domain is smooth and bounded. >1NNOur purpose is to describe how the solution ,uxt1< <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  where the porous me- dium equation was considered. For the case of and of the problem (1.1), Chen, Qi and Wang  proved the existence of the singular solution. In , they also studied the long time behavior of solutions to the Cauchy pro bl e m of 0:=:= 0xux>1q2=in 0,pqNtudivu uu  with nonnegative initial value in  ,0 =ux xN, where 211q=Ox with =1pq p , they showed that the solution converges as to a self-similar solu- tion. trong abeak absoWe 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,uxtx0 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x is not identically zero in (1.1), the corresponding steady state x does not vanish identically, but it is still true that ,uxt tends to x as . tThe steady-state problem nq 2= in ,= o pdivxx  (1.2) is a special case of the problem xibx2=1 = in ,=1 on Npxiiax uufuDuD (1.3) in , where ND be an arbitrary domain, ,:abD be two continuous functions, and 1pfs is a nondecreasing function with 0=f0. Z. C. ZHANG ET AL.281 From , we know that Problem (1.2) has a unique solution, a dead core exists if  is large enough and 11/0,pFsd0;0=qtzzt z1. (2.1) The solution is given explicitly by   1/ 11/ 1=11 >1= =1=11 0<<1qtqztq tqzt eqzt 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 0min =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. ,>0uxtThe 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 vrRB, with , 0v0< <2rR. We begin by observing that on the positive real line, the function by wx111()= 111pp qpp qppqpwx pqpxqp satisfies 21=pqpwww, ; >0x0=ww0=0. Now we choose  so that =12Rw and consider the function =2Rvwr in the ball in rRN (having extend v to be zero for <2rR). We then obtain   221111111=11= in 2,1=11:= ,p1pppqppqpqqpqpqqNdiv vvpvvvrNvvRRrNpvvrqpvCvCv v   where we use and 0v111qp qp. 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RB on whose boundary 1. Therefore, v and contains a dead core for . For any 0, we can take 0 the distance from 0x ==Rrx to the boundary. Theorem 2.2 shows that for  large enough, 0x belongs to the dead core. This suggests making the following definition. Definition 2. 1 Let . Define 0x*00 0=,=0;=inf infxxWe claim that ,, ,,*020,,NpqN pqpPPr (2.4) where 1,, =1 211pNpq ppPp Npq pq, 00=,rdistx; = inradius of . The proof con- sists in noting that the function 100=ppqxxwr satisfies 2,,0=0pNpq qpPdiv wwwr  in  and 1w . Thus, is a supersolution of (1.2) for any w,,0NpqpPrw. Since vanishes at 0x, 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 0x >, wh ere 0 is defined in (2.3). Then (a) if 0<<11qp, for uxt0,=0001:= ;1tt q (b) if 10ux, then 0uxt,>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 0tt=wz, where ,zt is the solution of the lumped--parameter problem with  to be choosen satisfies the differential inequality for a supersolution and  is the solution of the steady-state problem with 0=. Note that vani- shes at 0wx for 11tq. Since and wzw, it is clear that ,0wx ux0 and x,wt . From the definition of , we have w2200=.pqtpqtqqqwdivwwwzdiv zzz   qw 0.  (2.3) Copyright © 2011 SciRes. APM Z. C. ZHANG ET AL. Copyright © 2011 SciRes. APM 2832211===0pqtpqtqqtqqvdiv vvvdiv vqqBy choosing 0=, we obtain the desired result. .     4. Decay Estimates We consider (1.1) with , where  0ux xx is the solution of the corresponding steady-state 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0ux 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 ,= ,xtuxt 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