 Applied Mathematics, 2011, 2, 575-578 doi:10.4236/am.2011.25076 Published Online May 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM The Role of Space Dimension on the Blow up for a Reaction-Diffusion Equation Zhilei Liang School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu, China E-mail: zhilei0592@gmail.com Received March 9, 2011; revised March 22, 2011; accepted March 27, 2011 Abstract This paper deals with the doubly degenerate reaction-diffusion equation 11div ,mquuuuut where , 0,1 ,xB0,t1m1, 11qm and B(0,1) denotes a unit ball in RN with the center in origin. We prove that the blow up phenomenon can be restrained if the space dimension N is taken sufficiently large. Moreover, the critical condition guaranteeing the absence (or occurrence) of the blow up is achieved. Keywords: Reaction-Diffusion, Source Term, Space Dimension, Blow up 1. Introduction The blow up phenomenon in parabolic equation has been object of active research in recent years [1-4]. It has at-tracted lots of interests and attentions because of its physical importance and mathematical challenge. Here we formulate the condition that guarantees the absence of the blow up in terms of space dimension. In order to fix ideas, we consider the following problem   110div,0,1 ,0,,0, 0,1,0,,00,0,1 ,mqutuuu uxBtuxtx Btuxu xxB   (1.1) where , 1m1, 11qm, and 0,1B is a unit ball in with the center in origin. The initial function 0 is assumed to be positive bounded. Problem (1.1) is the well known non-Newtonian poly-tropic filtration equation, which co mes up in a wide vari-ety of physical contexts. The local in time existence and the comparison principle of nonnegative weak solutions have been established in , see also . Due to the pos-sible degeneration at the level or NRxu0u0u, we understand the weak solution in distributional sense as follows Definition 1.1 Let 1m. A nonnegative  20,10,0, ;0,1uLBtCtLB with 110,1 0,uLB t  is said to be a weak solu-tion of (1.1), if the integral identity  1100dd,0 d0tmqtBBuuuuu xux xxs is true for all test function . 00,1 0,CB TWhen it comes to blow up, we mean that the solution of (1.1) exists for 0,tT and become unbounded as for some TtT. In this paper, we study the influencing factor of space dimension N on the blow up phenomenon for the Dirich-let problem (1.1). Such question was first investigated by A. Tersenov for a heat equation with a nonlinear source. By analyzing the stationary equation and then using the comparison theorem, the author proved in  that if the space dimension N is chosen large enough, the solutions exist for all positive time. In other word, the high dimen-sion plays a preventive role on the occurrence of the blow up. In this current paper we extend the results ob-tained in  to a doubly degenerate equation. The below theorem is our first result. Theorem 1.1 Given an initial function 0ux,xM and a constant , there is a , such that for every space dimension , the solution 0M**0NNu*NN0,;uxtu of (1.1) satisfies 00,;, 0.uxtuuMt (1.2) In particular, Z. L. LIANG 576 **0000,1,1qNNuMuMmuM u where 1m. Moreover, as *N0u. Just as described in Theorem 1.1, there exists a critical which depends on the given , such that the blow up can be avoided if the space dimension N satis-fies . On the other hand, for a given space di-mension N, there also exists a critical initial datum 0 such that all the non-trivial solutions occur blow up so long as the size of their initial datum is larger than that of 0. To demonstrate this we first present the following proposition *N*u0ux*NNx*uxProposition 1.1 For a given space dimension N, there exist positive constants A, a, which depend on N, as well as ,,mq, such that the function 11_,; ,quxtNTt (1.3) with   111,Aa  xTt and 11qq 0, solves the inequality 11div,,0, .mqNuuuuuxRtTt (1.4) The proof of Proposition 1.1 is available in the final Section 3. Because of 11qq 0, the support of _,;utN falls into the set :xxaTt aT. So, for a fixed N, we choose T so small such that , and therefore, 1aTsupp_,0;0,1 .uNB (1.5) We are ready to state our second result Theorem 1.2 Suppose that (1.5) holds for a given space dimension N, then there exists a critical initial datum 0 *_,0;uxux N, such that for all initial data u0 satisfying  0*0_,0; ,uxuxu x N (1.6) the solution occur blow up before T. More-over, inequality (1.6), along with (3.7) (see Section 3 below), leads to 0,;uxtu 0*_,0;, .uuNA N  Remark 1.1 Theorems 1.1 and 1.2 are also valid for a general convex domain. 2. Proof of the Theorem 1.1 This section is devoted to the proof of Theorem 1.1. First of all, let us analyze the radially symmetric solution Ur of the stationary equation to (1.1), i.e., 111110, 0,Nm qNrUUUUrxr  (2.1) with initial conditions 00,0UKU0. (2.2) The local and unique positive solution of (2.1)-(2.2) follows from the analysis of the equivalent integral equa-tion, by using Banach contraction mapping theorem. Moreover, the solution is decreasing and can be extended whenever it is positive. (See  and .) Let *sup :0,0rrUrr. Integrating (2.1) over 0,r, *0,rr, yields 111 10d,rNm qNrU UUUss (2.3) which implies 11 101d.rNmqNqrUUUssKrN  N i.e., 1qrUr KN  (2.4) Integrating (2.4) once more to get 11011d11 ,1qrqKUrK ssNmKKN  r i.e., 111*11 ,0 .1qmUrKKrr rN (2.5) Clearly, 111*111.1mqrNKm Since the initial function , from (2.5) we have  00, 1ux LB0,0ux UxxB,1, (2.6) provided 110111,1qmUKK uN  Copyright © 2011 SciRes. AM Z. L. LIANG577 which is reduced to *01.1qmKNNKu  (2.7) To guarantee the validity of (2.7), it suffices to take 0KuM with a constant . That is, 0M0*001.1quMmNNuM u  Apparently, (2.7) also ensures  0,,forall 0,1,0Uxuxtx Bt 0. (2.8) In terms of (2.7) and (2.8), the comparison theorem conclude  ,,0,,uxtUxxBR t This deduces that   0,0,0,1,uxtUxUuMx Bt 0. (2.9) This completes the proof of Theorem 1.1. Remark 2.1 In case of 1qmUr, all the solutions of the (1.1) can be bounded by for arbitrary N, only if the is chosen large enoug h (see (2.5)). We also refer to . 0UK 3. Proof of Theorem 1.2 The proof of Theorem 1.2 is a direct consequence of Proposition 1.1. So the remain task is to prove the valid-ity of the Proposition 1.1. In fact, to verify the function 11_,; quxtNTt satisfies (1.4), we turn to prove the following equivalent inequality obtained after relatively computation 1111110, ,1NmNqaq   (3.1) in which 'dd. Due to the explicit form  11a , the derivative of  is non-increasing, i.e., 11111110. 1Aaaa     By this we compute    1111111111211111.NmNNNNII    (3.2) A direct but tedious computation shows 1111111111111,11AIaaAaa     and  12111,1AINaa   where 11a. Hence, (3.2) is equals to 1111111111111111.11NmNAaaANaa         (3.3) Similarly, 11111,11AA    (3.4) 111 .11qqqAAqq 1   (3.5) Summing up (3.3)-(3.5), we receive 1110,qqkl A (3.6) where 11111,11Aka   11 and Copyright © 2011 SciRes. AM Z. L. LIANG Copyright © 2011 SciRes. AM 578 11111111.11AlNaq   10 Firstly, because , we have 0k0 for all 0,kl with 0, 1kl. As to ,1kl, inequality (3.6) holds so long as 11110qqqqkl AkklAl  which is valid if  11 111111111111111111111111111qqqkAAlkNlqaAaANqa            11 (3.7) Clearly, For any fixed , m, q and N, (3.7) is true if we choose A large enough, and at the same vary a such that the ratio 1Aa1 keeps stable. Moreover, A is increasing with respect to N. This completes the proof. 4. References  Z. L. Liang, “Blow up Rate for a Porous Medium Equa-tion with Power Nonlinearity,” Nonlinear Analysis: The-ory, Methods & Applications, Vol. 73, No. 11, 2010, pp. 3507-3512. doi:10.1016/j.na.2010.06.078  A. Samarskii, V. 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