The Critical Exponent of Doubly Singular Parabolic Equations 1

Transcript

1 Jounal of Mathematical Analysis and Applications 257, (200) doi:0.006/jmaa , available online at on The Citical Exponent of Doubly Singula Paabolic Equations Xinfeng Liu and Mingxin Wang Depatment of Applied Mathematics, Southeast Univesity, Nanjing 2008, People s Republic of China mxwang@seu.edu.cn Submitted by Howad Levine Received June 24, 999 In this pape we study the Cauchy poblem of doubly singula paabolic equations u t = div u σ u m +t s x θ u p with non-negative initial data. Hee <σ 0, m>max0 σ σ + 2/N satisfying 0 <σ+ m, p>, and s 0. We pove that if θ>max σ + 2, + sn σ m σ + 2, then p c = σ + m+σ + m s +σ s+θ/n > is the citical exponent; i.e, if <p p c then evey non-tivial solution blows up in finite time. But fo p>p c a positive global solution exists. 200 Academic Pess Key Wods: doubly singula paabolic equation; citical exponent; blow up.. INTRODUCTION In this pape we study citical exponent of quasilinea paabolic equations u t = div ( u σ u m) + t s x θ u p x R N t > 0 ux 0 =u 0 x 0 x R N (.) whee <σ 0, m>max0 σ σ + 2/N satisfying 0 <σ+ m, p>, and s 0. u 0 x is a continuous function in R N. The existence, uniqueness, and compaison pinciple fo the solution to (.) had been poved in [] (fo the definition of solution see []). Since 0 <σ+ m, (.) is a doubly singula poblem and does not have finite speed of popagation. Theefoe, ux t > 0 fo all x R N and t>0. This poject was suppoted by PRC Gant NSFC X/0 $35.00 Copyight 200 by Academic Pess All ights of epoduction in any fom eseved. 70

2 doubly singula paabolic equations 7 Because the main inteests of this pape ae to study the lage-time behavio of solution, we assume that the solution u of (.) has vey mild egulaity. In this context, ux t blows up in finite time means that wt = ux t dx + as t T fo some finite time T>0, whee is a bounded domain in R N. Ou main esult eads as follows: Theoem. Assume that s 0, p>, <σ 0, m>max0 σ σ + 2/N satisfying 0 < σ + m. If θ > max σ + 2, + sn σ m σ + 2, then p c =σ + m +σ + m s + σ s+θ/n > is the citical exponent; i.e, if <p p c then evey non-tivial solution of (.) blows up in finite time, wheeas if p>p c then (.) has a small non-tivial global solution. The study of blow-up fo nonlinea paabolic equations pobably oiginates fom Fujita [8], whee he studied the Cauchy poblem of the semilinea heat equation, u t = u + u p x R N t > 0 ux 0 =u 0 x 0 x R N (.2) whee p>, and obtained the following esults: (a) If <p< + 2/N, then evey nontivial solution ux t blows up in finite time. (b) If p> + 2/N and u 0 x δe x2 0 <δ, then (.2) admits a global solution. In the citical case p = + 2/N, it was shown by Hayakawa [0] fo dimensions N=, 2 and by Kobayashi et al. [2] fo all N that (.2) possesses no global solution ux t satisfying u t < fo t 0 Weissle [24] poved that if p = + 2/N, then (.2) possesses no global solution ux t satisfying u t q < fo t>0 and some q +. The value p c = + 2/N is called the citical exponent of (.2). It plays an impotant ole in studying the behavio of the solution to (.2). In the past couple of yeas thee have been a numbe of extensions of Fujita s esults in seveal diections. These include simila esults fo othe geometies (cones and exteio domains) [4, 5, 3, 5, 6], quasilinea paabolic equations, and systems [, 2, 5, 7, 9, 4, 8 20, 22, 23]. In paticula, the authos of [2] consideed degeneate equations on domains with non compact bounday. Thee ae also esults fo nonlinea wave equations and nonlinea Schödinge equations. We efe the eade to the suvey papes by Deng and Levine [5] and Levine [3] fo a detailed account of this aspect.

3 72 liu and wang When m =, (.) becomes p-laplacian equations, and the citical exponents wee given by the authos of [9, 2, 22]. When σ = 0, (.) becomes the poous media equations, and the citical exponents wee studied by the authos of [3, 7, 8, 22]. This pape is oganized as follows. In Section 2 we discuss the qualitative behavios and give some estimates of solutions to the homogeneous poblem u t = div ( u σ u m) x R N t > 0 ux 0 =u 0 x 0 x R N (.3) In Section 3, fo convenience, we fist discuss the special case of (.): s = 0, i.e, u t = div ( u σ u m) +x θ u p x R N t > 0 ux 0 =u 0 x 0 x R N (.4) and pove that if < p p c = σ + m +σ θ/n then evey non-tivial solution of (.4) blows up in finite time. In Section 4 we pove Theoem. Remak. We end this section with a simple but vey useful eduction. When we conside the blow-up case, by the compaison pinciple we need only conside that u 0 x is adially symmetic and non-inceasing, i.e, u 0 x =u 0 with =x, and u 0 is non-inceasing in. Theefoe, the solution of (.) is also adially symmetic and non-inceasing in =x. 2. ESTIMATES OF SOLUTIONS TO (.3) In this section we discuss (.3) fo the adially symmetic case; the main esults ae thee popositions. Poposition. Assume that <σ 0 and m> σ σ + 2/N satisfy 0 <σ+ m. (i) If σ + m<, then, fo any c>0, the equation (.3) has a global self-simila solution, ux t =ct α + h ν q whee α = N/Nσ + m +σ + 2β= /Nσ + m +σ + 2ν = σ + 2/σ + q =σ + / σ m =xt β, and h = hc = qν β/+α c σ m/σ+ m /σ+.

4 doubly singula paabolic equations 73 (ii) If σ + m =, then, fo any c>0, the equation (.3) has a global self-simila solution, ux t =ct α exp h ν whee α = N/σ + 2, β = /σ + 2, ν =σ + 2/σ +, =xt β, and h satisfies hν σ+ = β/m. This poposition can be veified diectly. Poposition 2. Assume that <σ 0, m>max0 σ σ + 2/N, such that 0 <σ+ m and u 0 x is a non-tivial andnon-negative continuous function. If u 0 x is a adially symmetic and non-inceasing function, then the solution ux t of (.3) satisfies u t α t u fo all x RN t > 0 (2.) whee α = N/Nσ + m +σ + 2. Poof. Denote k =σ + m/σ +, let f =mk σ+ /kσ+k u when σ + m<, and let f = u when σ + m =. Then (.3) can be ewitten as f t = d div ( f k σ f k) x R N t > 0 f x 0 =f 0 x 0 x R N whee d = when σ + m< and d = mk σ+ when σ + m =. Let g = f k ; then g satisfies the following equation: g /k t = d div ( g σ g ) x R N t > 0 gx 0 =f k 0 x 0 x R N Denote µ = + σ /k/σ + if σ + m<, and let { v = µ gµ if 0 <σ+ m<, ln g if σ + m =. Case. 0 <σ+ m<. In this case, d = and g satisfies g t = kg div ( v σ v ) + g /k g σ+2 kg div ( v σ v ) v t = g /kσ+ g t = kg /kσ+ g /k div ( g σ g ) = kµv div ( v σ v ) + v σ+2 (2.2)

5 74 liu and wang Denote w = div v σ v / = and let z = v; then z>0z > 0, and z t = kµzdiv ( z σ z ) z σ+2 [ = kµz [ w = σ +z σ z + N z ] z σ+ σ+2 =kµzw z σ+2 σ +z σ z + N z ] σ+ w t =σ +z σ z t +σ +σz σ z t z + z t =kµz w+zw σ +2z σ+ z N σ + z σ z t (2.3) z t =kµwz +2w z +w z σ +2 [ z σ+ z +σ +z σ z 2] By a seies of calculation we have [ w t = kµσ + zz σ w + 2z σ+ w +σ + z σ wz + σz σ zz w + N z σ+ w + N ] z σ zw [ σ + σ + 2 z 2σ+ z + + 2σz 2σ z 2 It follows fom (2.3) that + N z 2σ+ z ] (2.4) w = σσ + z σ z 2 +σ + z σ z N z σ+ N σ + + z σ z 2 Denote ε = kµσ + =k + σ /k; substituting the above expession into (2.4) we get w t = εa t w + b tw εw 2 σ + 2 [ ] N z 2σ+2 σ + z 2σ z 2 2 = εa t w + b tw εw 2 +σ + 2 [ σ + wz σ z N z 2σ+2 +σ + N ] z 2σ+ z 2 = εa t w + b tw εw 2 σ + 2 [ w 2 + 2N z σ+ NN w + z 2σ+2 2 ]

6 doubly singula paabolic equations 75 whee a tb t ae functions poduced by z t and z t. Taking into account the Cauchy inequality we have i.e., 2 N z σ+ w N N w2 + NN z 2σ+2 2 w t kσ + /ka t w + b tw [ + kσ + σ + 2 ] w 2 N w t k σ + /ka t w b tw [ ] σ k/k σ + w 2 N Noticing k =σ + m/σ +, we have w t k/k σ + a t w b tw + σ Nσ + m w 2 N Let y t= α/t. It is obvious that y t = k/k σ + a t y b ty + y 2 /α. Since y 0 =, it follows by the compaison pinciple that w α/t (see [3, ]); i.e, div v σ v α/t. By (2.2) we have g t kαg/t. Since g = f k, it follows that f t = αf/t; i.e. u t α t u Case 2. hee. σ + m =. Since this is easy to pove, we omit the details Q.E.D. Remak. Fo the poous media equation, the authos of [3] poved (2.) fo fist time, to ou knowledge. Poposition 3. Unde the assumptions of Popositions and 2, thee exist positive constants δ b such that: (i) When σ + m<, then ux t δt ε α + b ν q x > t>ε>0 (2.5) whee =xt ε β, α, β, ν, and q ae as in Poposition, and b is a positive constant.

7 76 liu and wang (ii) When σ + m =, then ux t δt ε α exp b ν x > t>ε>0 (2.6) whee =xt ε β, α, β, and ν ae as in Poposition, and b is a positive constant. Poof. In view of Popositions and 2, and using a method simila to that of [2], one can pove Poposition 3. Hee we give only the sketch of the poof fo the case σ + m<. Step. By use of the methods of Chap. 6 of [6] we can pove the following compaison lemma: Lemma. Let 0 τ<+ and S =x R N x > τ + Assume that v w ae non-negative functions satisfying Then v t = div v σ v m w t = div w σ w m in S vx t wx t x = τ<t<+ vx τ wx τ x vx t wx t in S Step 2. solutions Fom Poposition we have that poblem (.3) has the similaity U µ x t =µ ρ Uµx t ρ =σ + 2/ σ m whee µ>0 is a paamete, and Ux t =U x t =t α + h ν q =xt β In view of Poposition 2 and the expession of U µ x t we can pove that fo suitably small µ>0, the following holds: U µ t ε u t fo t>ε U µ x t ε =0 ux t fo x t = ε By Lemma we see that (2.5) holds. Q.E.D.

8 doubly singula paabolic equations THE SPECIAL CASE s = 0, <p p c In this section we study poblem (.4) and pove a blow-up esult. Theoem 2. Let σ m p θ be as in Theoem. If <p p c = σ + m+ σ θ/n, then evey non-tivial solution of (.4) blows up in finite time. Let φx be a smooth, adially symmetic, and non-inceasing function which satisfies 0 φx, φx fo x, and φx 0 fo x 2. It follows that fo l>φ l x =φx/l is a smooth, adially symmetic, and non-inceasing function which satisfies 0 φ l x, φ l x fo x l and φ l x 0 fo x 2l. It is easy to see that φ l C/l, φ l C/l 2 Let w l t = uφ l dx whee = R N \B, with B being the unit ball with cente at the oigin. We divide the agument into two cases. Case. m. Let q =m + σ/σ + and v = u q ; then the equation (.4) can be witten as Theefoe, dw l dt v /q t = m q σ+ div( v σ v ) +x θ v p/q = m div q ( v σ v ) φ σ+ l dx + x θ v p/q φ l dx m 2l q ω σ+ N v σ+ φ l N d + x θ v p/q φ l dx By diect computation we have 2l ( 2l v σ+ φ l N d 2l v φ l N d = ω n ) σ+ ( 2l ) σ v φ l N d N φ l d v φ l dx ω N v φ l dx ( ) q/p v φ l dx x θ v p/q φ l dx ( p q/p φ l p φ q l x θq dx) /p q ( φ l p φ q p q/p l x θq dx) /p q = C l Np q θq 2p/p ( 2l ) σ N φ l d = C 2 l N σ In view of m, we have q, and hence p/q >.

9 78liu and wang Case 2. m>. In this case one has dw l = div ( u σ u m) φ dt l dx + x θ u p φ l dx = u σ um η φ l ds u σ u m φ l dx + u σ u m φ l dx + x θ u p φ l dx 2l mω N u σ+ u m φ l N d + x θ u p φ l dx By diect computation and using Hölde s inequality one has 2l ( 2l ) σ+ u σ+ u m φ l N d u N φ l d 2l ( ( φ l u m /σ N d = 2l ( 2l σ φ l u m /σ d) N ( φ l u m /σ dx u N φ l d = ( u φ l dx { x θm φ l pσ φ m l x θ u p φ l dx ) m /pσ ( x θ u p { φ l dx x θm } ) /m +pσ m +pσ/pσ dx φ l pσ φ m l ω N ( u φ l dx x θ u p φ l dx ) /p ω N u φ l dx { x θ φ l p φ } ) /p p /p l dx } /m +pσ dx ) m +pσ/pσ = C lθm +Nm +pσ pσ/pσ { x θ φ l p φ } ) /p p /p l dx = C 2 l Np 2p θ/p In view of m>, 0 <m+ σ, it follows that 0 < m /σ. Fo the above two cases we always have dw ( ) l σ+m/pl C dt 3 x θ u p θm+σ/p 2 σ+n Nσ+m/p φ l dx + x θ u p φ l dx

10 i.e., dw l dt doubly singula paabolic equations 79 { ( ) } p σ m/p C 3 l θσ+m/p 2 σ+n Nσ+m/p + x θ u p φ l dx ( σ+m/p x θ u p φ l dx) (3.) By Hölde s inequality we have ( ) p ( p x θ u p φ l dx uφ l dx x θ/p φ l dx) Hence cw p l l θ Np x θ u p φ l dx cw p l ln l p if cw p l We now pove Theoem 2. if θ<np, θ = Np, if θ>np. (3.2) (i) Fist we conside the case θ<np. It follows fom (3.) and (3.2) that has (a) dw l dt { C 3 l θσ+m/p 2 σ+n Nσ+m/p + C 4 w p σ+m l l θ Np p σ+m/p} ( σ+m/p x θ u p φ l dx) (3.3) p< p c = σ + m +σ θ/n. Unde this assumption, one θ Np p σ +m/p>n 2 σ Nσ +m+θm+σ/p and consequently l θ Np p σ+m/p /l N 2 σ Nσ+m+θm+σ/p + as l + (3.4) Using the fact that w l is an inceasing function of l, we find fom (3.3) and (3.4) that thee exist δ>0, l such that dw l dt δ x θ u p φ l dx δw p l tl θ Np t>0 Thus w l, and consequently u, blows up in finite time, since p>.

11 80 liu and wang (b) p = p c = σ + m +σ θ/n. In this case, θ Np p σ + m/p = N 2 σ Nσ + m+θm + σ/p < 0. If we can pove that uφ l dx is a unbounded function of t fo some l, then it can be shown that, as in the above case, w l, and hence u, blows up in finite time. Othewise, u t L fo all t>0 and thee exists an M>0 such that ut L M fo all t>0 (3.5) We will pove (3.5) is impossible. Suppose the contay; it is clea fom (3.) that, fo the lage l, if xθ u p dx < + then dw l /dt 2 xθ u p φ l dx and if xθ u p dx =+ then w l t. Theefoe, { w l t k lt = min } x θ u p φ 2 l dx l w l t w l 0 t 0 k l τdτ Let wt = ux t dx and take l + in the above inequality. We obtain wt w0 kτdτ (3.6) 0 whee kt =min 2 R x θ u p dx. When σ + m<, using (2.5) and by N diect computation we have x θ u p dx δ p t ε y t ε y θ + by ν qp dy β ct ε t t When σ + m =, using (2.6) and by diect computation we have x θ u p dx δ p t ε y t ε y θ exp by ν dy β In view of (3.6) it yields ct ε t lim wt = + t + i.e., lim ux t dx =+ t + This shows that (3.5) is impossible. And hence ux t blows up in finite time.

12 doubly singula paabolic equations 8 (ii) Next we conside the case θ Np. Since m> σ σ + 2/N, it follows that N 2 σ Nσ + m +θm + σ/p < 0. Combining (3.2) and (3.) we find that, fo the case θ = Np, dw l dt ( ) C 3 l N 2 σ Nσ+m+θm+σ/p + Cw p σ+m l ln l σ+m pp p ( σ+m/p x θ u p φ l dx) and fo the case θ>np dw l dt ( C ) 3 l N 2 σ Nσ+m+θm+σ/p p σ+m + Cw l ( σ+m/p x θ u p φ l dx) Simila to the aguments of (i) one can pove that w l, and consequently u, blows up in finite time. Remak 23. The eason fo using = R N \B athe than R N itself is that if θ>0, then B x θ/p dx may not convege. 4. PROOF OF THEOREM (i) If p p c = σ + m +σ + m +σ s+θ/n, using the methods simila to those of the last section and the papes [9, 2], it can be poved that evey non-tivial solution of (.) blows up in finite time. We omit the details. (ii) If p>p c = σ + m +σ + m s +σ s+θ/n, we shall pove that (.) has global positive solutions fo the small initial data. By the compaison pinciple, it is enough to pove this conclusion fo the poblem (since s 0) u t = div u σ u m + + t s x θ u p x R N t > 0 ux 0 =u 0 x 0 x R N (4.) whee the constants m σ s θ p ae as in poblem (.). We shall deal with the global solutions of (4.) by using the similaity solutions which take the fom ux t = + t α w with =x + t β

13 82 liu and wang θ σ+2 whee α = + s + p /p σ m σ+2 σ m /p θβ = σ m + s+ σ+2 θσ + 2, and w satisfies the following ODE: mσ +w σ w w m +mm w m 2 w σ+2 +m N w σ w w m +αw+βw + θ w p =0 >0 w0=η>0 w σ w { 0= lim 0 + θ+ w p+ m /N m } (4.2) We call w a solution of (4.2) in 0 Rη fo some Rη > 0ifw > 0 in 0 Rη w C 2 0 Rη, and w satisfies the initial condition of (4.2). Unde ou assumptions it follows that p> + σ mθ/σ + 2, α>0, β>0. We obseve that a function ūx t = + t α vx + t β is an uppe solution of the equation (4.) if and only if v satisfies the following inequality: mσ + v σ v v m + mm v m 2 v σ+2 + m N v σ v v m + αv + βv + θ v p 0 > 0 (4.3) () We fist discuss the case θ 0. In this case, we ty to find an uppe solution of (4.), i.e., the solution of (4.3). When σ + m<, let v =ε + b k q, whee k =σ + 2/σ +, q =σ + / σ m, and ε and b ae positive constants to be detemined late. By diect computation we have v = εqbk k + b k q v = εqq + b 2 k 2 2k 2 + b k q 2 εqbkk k 2 + b k q v satisfies (4.3) if and only if ε + b k q[ α mnε σ+m bqk σ+] + εqbk k + b k q [ mε σ+m bqk σ+ β ] + ε p θ + b k qp 0 (4.4) Unde ou assumptions it follows that θ + q pk = θ + pσ + 2/ σ m < 0. Thee exists a>0, such that θ + b k q p a fo all 0 since θ 0 (4.5) Choose b = bε such that i.e., β = mε σ+m bqk σ+ b =qk ( βm ε σ m) /σ+

14 doubly singula paabolic equations 83 Fo this choice of b, (4.4) is equivalent to α Nβ + θ ε p + b k q p 0 (4.6) By (4.5) we see that (4.6) is tue if the following inequality holds: α Nβ + aε p 0 (4.7) In view of p>p c = σ + m +σ + m s +σ s+θ/n, it follows that α<nβ. Hence, thee exists ε 0 > 0 such that (4.7) holds fo all 0 <ε ε 0. These aguments show that v =ε + b k q satisfies (4.3) fo all 0 <ε ε 0. Using the compaison pinciple we get that the solution ux t of (4.) exists globally povided that ux 0 vx = ε + bx k q. And hence, so does the solution of (.). When σ + m =, let v =ε exp b k, whee k =σ + 2/σ +, and ε and b ae positive constants to be detemined late. By diect computation we know that v satisfies (4.3) if and only if ε [ α mnbk σ+] e bk + εbk [ mbk σ+ β ] k e bk + ε p θ e pbk 0 (4.8) Since θ 0, thee exists a>0 such that θ exp { p b k} a fo all 0 Choose b such that β = mbk σ+. Then (4.8) holds povided that α Nβ + aε p 0 Simila to the case σ + m<, we have that the solution ux t of (4.) exists globally povided that ε and ux 0 vx = ε exp bx k. And hence, so does the solution of (.). (2) Next we conside the case θ<0. If m =, this poblem was discussed by [9] fo σ = 0, and by [2] fo σ<0. In the following we always assume that m. Ou main pupose is to pove that (4.2) has gound state fo the small η>0. By the standad aguments one can pove that fo any given η>0, thee exists a unique solution w of (4.2), which is twice continuously diffeentiable in whee w 0. Denote Rη =maxr w > 0 0R. So0<Rη +, and wrη = 0 when Rη < We divide the poof into seveal lemmas. Lemma 2. The solution w of (4.2) satisfies w < 0 in 0Rη. In addition, if Rη =+ then w 0 as +.

15 84 liu and wang Poof. We fist pove that w < 0 fo 0 <<Rη when θ + 0. Since w σ w 0 = lim 0 + θ+ w p+ m /N m < 0, one has w < 0 fo. If thee exists 0 0 < 0 <Rη such that w < 0 in 0 0 and w 0 =0, then w σ w w m 0 0. But by the equation (4.2) we see that mw σ w w m 0 = ( αw θ w p 0 ) < 0 a contadiction. When θ + > 0, it follows that w 0 =0. Using the equation (4.2) one has mnw σ w =0 = ( αw 2 m 0+ lim 0 + θ w p+ m ) < 0 Hence w σ w < 0, and consequently w < 0 fo all. Simila to the case of θ + 0 it follows that w < 0 fo all 0 <<Rη. If Rη = +, since w < 0 and w > 0 in 0 + one has lim + w =L. IfL>0, an integation of (4.2) gives N ( mw σ w w m + βw ) = whee mw σ w w m lim = α + N L A N A = 0 { α Nβ + s θ w p s } s N ws ds { L p if θ = 0, 0 if θ<0, + if θ>0. It follows that lim + w =, a contadiction. Thus w 0as + Q.E.D. Lemma 3. Let w be the solution of (4.2). Then fo any given small η>0 thee exists R 0 η > 0 which satisfies lim η 0 + R 0 η =+ andsuch that w > 0 mw σ w w m + βw > 0 R 0 η (4.9) Poof. satisfies Let z = η w; then z = w > 0 0 <z <η, and z mσ +z σ z η z m mm η z m 2 z σ+2 + m N z σ+ η z m =αη z βz + θ η z p >0 z0=0 z σ z 0= lim θ+ η z p+ m /N m (4.0) 0 +

16 doubly singula paabolic equations 85 Since p>p c, one has Nβ>α. An integation of (4.0) gives m N z σ+ η z m + β N z [ = Nβ αs N z + αηs N + s N+θ η z p] ds 0 αη N N +β α N N z+ N + θ ηp N+θ (4.) Since m and <σ 0, we know that if σ + m = then σ<0 and <m<2. Denote R 0 η =minr zr =η η a, whee a = 2 min σ p+ if σ + m<and m>, a =p + /2 ifσ + m< m and m<, and a = p+2m 3 min p+ if σ + m =. Then R 2 m 0 η > 0 and z η η a <ηfo all 0 < R 0 η. We fist conside the case σ + m<. Fom (4.) it follows that fo 0 < R 0 η m N z σ+ η z m < αη N N +β α N ηn + N + θ ηp N+θ = βη N + N + θ ηp θ+n Denote b = a when m>, and b = when m<. Using η a η z η one has that N z σ+ < { } βη + mb + η p+ mb m N + θ θ+ Since σ +, it follows that z < { } β /σ+ m η+ mb + mn + θ ηp+ mb θ+ C {η + mb /σ+ +η p+ mb +θ /σ+} Integating this inequality fom 0 to R 0 η we have η η a + C 2 { η + mb/σ+( R 0 η ) σ+2/σ+ + η p+ mb/σ+( R 0 η ) σ+θ+2/σ+ } In view of a> and p + mb/σ + > + mb/σ + >, it follows that R 0 η + as η 0 +.

17 86 liu and wang Using wr 0 η = η a and w η a fo all 0 R 0 η, an integation of (4.2) gives, fo 0 <R 0 η, m N w σ w w m + β N w = 0 Nβ αs N ws ds 0 s N+θ w p s ds Nβ αw ( R 0 η ) s N ds η p s N+θ ds 0 0 ( = η a N β α N ) N + θ ηp a θ Since θ<0nβ>αand p>a, it follows that m N w σ w w m + β N w > 0 ( R 0 η ) (4.2) Second, we conside the case σ + m =. Fom (4.) it follows that, fo 0 < R 0 η, m N z σ+ η z m < α N N η z+ N + θ ηp N+θ Using σ + = 2 m and <m<2 we have that z C { η z /σ+ + η p+ ma/σ+ +θ/σ+} (4.3) Denote γ =p + ma/σ +. Integating (4.3) fom 0 to R 0 η we have { η η a C η a σ + ( R0 η ) σ+2/σ+ +η γ σ + ( R0 η ) σ+2+θ/σ+ σ +2 σ +2+θ + σ + σ +2 R0 η 0 σ+2/σ+ z } (4.4) Substituting (4.3) into (4.4) and using the inductive method we have that + η η a η a n! An +Cσ +R 0 η σ+2+θ/σ+ η γ n= + ( ) A n (4.5) n=0 n+σ +2+θ n! whee A=C σ+ σ+2 R 0η σ+2/σ+. In view of a> and γ =p+ ma/ σ +>, it follows fom (4.5) that R 0 η + as η 0 +. Simila to the case σ +m<, we have that (4.2) holds. The poof of Lemma 2 is completed. Q.E.D.

18 doubly singula paabolic equations 87 Now we pove that, fo the case θ<0, (4.2) has gound state fo small η. Choose η 0 η p 0 <Nβ α such that (4.9) holds fo all 0<η η 0. Since p>p c, which implies Nβ>α, using θ<0 R 0 η>ws<η and integating (4.2) fom R 0 η to R 0 η<<rη we have m N w σ w w m +β N w = ( m N w σ w w m +β N w ) =R0 η +Nβ α R 0 η s N ws [ Nβ α s θ w p s ] ds s N ws [ Nβ α η p ] ds 0 (4.6) R 0 η In view of w>0 and w <0 fo 0<<Rη, it follows that Rη=+ by (4.6). Theefoe (4.2) has a gound state. ACKNOWLEDGMENTS The authos thank Pofesso H. A. Levine fo diecting thei attention to [2] and the efeees fo thei helpful comments and suggestions. REFERENCES. D. Andeucci, New esults on the Cauchy poblem fo paabolic systems and equations with stongly nonlinea souces, Manuscipt Math. 77 (992), D. Andeucci and A. F. Tedeev, A Fujita type esults fo a degeneate Neumann poblem in domains with noncompact bounday, J. Math. Anal. Appl. 23 (999), D. G. Aonson and Ph. Benilan, Regulaite des solutions de l equation des milieux poeux dans R N, C. R. Acad. Sci. Pais 288 (979), C. Bandle and H. A. Levine, On the existence and nonexistence of global solutions of eaction diffusion equations in sectoial domains, Tans. Ame. Math. Soc. 655 (989), K. Deng and H. A. Levine, The ole of citical exponents in blow-up theoems: The sequel, J. Math. Anal. Appl. 243 (2000), E. DiBenedetto, Degeneate Paabolic Equations, Spinge-Velag, New Yok, M. Escobedo and M. A. Heeo, Boundedness and blow up fo a semilinea eaction diffusion system, J. Diffeential Equations 89 (989), H. Fujita, On the blowing up of solutions of the Cauchy poblem fo u t = u+u +α, J. Fac. Sci. Univ. Tokyo Sect. 3 (996), V. A. Galaktionov and H. A. Levine, A geneal appoach to citical Fujita exponents in nonlinea paabolic poblems, Nonlinea Anal. 34, No. 7 (998), K. Hayakawa, On nonexistence of global solutions of some semilinea paabolic equations, Poc. Japan Acad. 49 (973), A. S. Kalashnikov, Some poblems of the qualitative theoy of non-linea degeneate second-ode paabolic equations, Uspekhi Mat. Nauk 42, No. 2 (987), (in Russian). English tanslation: Russian Math. Suveys 42, No. 2 (987),

19 88 liu and wang 2. K. Kobayashi, T. Siao, and H. Tanaka, On the blowing up poblem fo semilinea heat equations, J. Math. Soc. Japan 29 (977), H. A. Levine, The ole of citical exponents in blowup theoems, SIAM Rev. 32 (990), H. A. Levine, A Fujita type global existence-global nonexistence theoem fo a weakly coupled system of eaction diffusion equations, Z. Angew. Math. Phys. 42 (990), H. A. Levine and P. Meie, A blow up esult fo the citical exponent in cones, Isael J. Math. 67 (989), H. A. Levine and P. Meie, The value of the citical exponent fo eaction diffusion equations in cones, Ach. Rational Mech. Anal. 09 (989), Y. W. Qi, On the equation u t = u α +u β, Poc. Roy. Soc. Edinbugh Sect. A 23 (993), Y. W. Qi, Citical exponents of degeneate paabolic equations, Sci. China Se. A 38, No. 0 (995), Y. W. Qi, The citical exponents of paabolic equations and blow-up in R N, Poc. Roy. Soc. Edinbugh Sect. A 28 (998), Y. W. Qi and H. A. Levine, The citical exponent of degeneate paabolic systems, Z. Angew. Math. Phys. 44 (993), Y. W. Qi and M. X. Wang, Citical exponents of quasilinea paabolic equations, pepint. 22. A. A. Samaskii, V. A. Galaktionov, S. P. Kudynumov, and A. P. Mikhailov, Blow-up in Quasilinea Paabolic Equations, Nauka, Moscow, 987 (in Russian); English tanslation: de Guyte, Belin, P. Souplet, Finite time blow-up fo a nonlinea paabolic equation with a gadient tem and applications, Math. Methods Appl. Sci. 9 (996), F. B. Weissle, Existence and nonexistence of global solutions fo a semilinea heat equation, Isael J. Math. 38 (98),