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2 J. Math. Anal. Appl. 329 (2007) Abstact On the blow-up ate of lage solutions fo a poous media logistic equation on adial domain Peng Feng Depatment of Mathematics, Michigan State Univesity, East Lansing, MI 48824, USA eceived 16 Novembe 2005 Available online 27 July 2006 Submitted by S. Heikkilä In this pape we establish the exact blow-up ate of the lage solutions of a poous media logistic equation. We conside the caying capacity function with a geneal decay ate at the bounday instead of the usual cases when it can be appoximated by a distant function. Obtaining the accuate blow-up ate allows us to establish the uniqueness esult. Ou esult coves all pevious esults on the ball domain can be futhe adapted in a moe geneal domain Elsevie Inc. All ights eseved. Keywods: Poous media logistic model; Blow-up ate; Lage solutions 1. Intoduction In this wok, we conside the singula bounday value poblem { Δw m = λw a(x)w p in Ω, w = on Ω, Autho's pesonal copy whee Ω is a bounded domain of N, N 1, with bounday Ω of class C 2, λ, m>1, p>1 a(x) C α ( Ω, + ) fo some α (0, 1), + := 0, + ). The solutions ae often (1) * Pesent addess: Depatment of Physical Sciences Mathematics, Floida Gulf Coast Univesity, USA. addess: fengpeng@math.msu.edu X/$ see font matte 2006 Elsevie Inc. All ights eseved. doi: /j.jmaa

3 348 P. Feng / J. Math. Anal. Appl. 329 (2007) known as lage solutions. Moe pecisely, by a lage solution we mean any classical solution w such that w(x) + as d(x) := dist(x, Ω) 0 +. The special case m = 1 is elated to some pescibed cuvatue poblem in iemannian geomety. If λ = 0, a(x) = 1, it is easy to check that f(t)= t p with p>1 satisfies the well-known Kelle Osseman condition { t t 0 0 f(s)ds} 1/2 dt < + fo all t 0 > 0, which is a necessay condition fo the existence of lage solutions. Indeed, the existence of a lage solution can be established by the method of supesolution subsolution povided that the domain Ω is egula enough, see, e.g., 8]. The uniqueness was established by Macus Veon 10,11] in vey geneal domain fo all p>1. Howeve, the existence was not established fo such geneal domain whose bounday is locally epesented as a gaph of a continuous function. The existence fo geneal bounded domain was ecently obtained by Kim 9] unde the estiction n that p (1, n 2 ) fo n 3 p (1, ) fo n = 2. The same autho also established the uniqueness unde the assumption that Ω = Ω. Fo m = 1, λ = 0, a(x) 0 a(x) 0, the existence was established in 6] unde the assumption that thee exist constant C 1,C 2 > 0 ν 2 ν 1 > 2 such that C 2 d(x) ν 2 a(x) C 1 d(x) ν 1, x Ω. Uniqueness can be established fo the case ν 1 = ν 2. Fo m = 1, λ = 0, a(x) a 0 > 0in Ω, diffeent types of equations wee studied, e.g., 2]. In the case a(x) C 0 d(x) ν + o(d ν ) as x Ω, blow-up ate uniqueness wee studied in 5]. Fo the poous media logistic equation, i.e., m>1, p>1, λ, the existence of lage solutions was studied in 3,4]. Thei esult showed that lage solution exist if, only if, the nonlinea diffusion is not too lage. Pecisely, they poved the following theoem. Theoem 1. 2, Theoem 5.2] Suppose λ a(x) C α ( Ω, + ) satisfies the following assumption: The open set Ω + := {x Ω; a(x) > 0} is connected with bounday Ω + of class C 2, the open set Ω 0 := Ω \ Ω + satisfies Ω 0 Ω. Then: (a) When p>m>1, (1) possesses a positive solution. (b) When m p>1, (1) does not admit a positive solution. Howeve, in 5], to pove the uniqueness, it was assumed that Autho's pesonal copy a(x) = βd(x) ν 1 + βd(x) + o(d(x))] as d(x) 0 + fo some constants β>0, ν 0 hence x x 0 a(x) βd(x) ν = 1 unifomly in x 0 Ω. And in 4], to pove the uniqueness, it was assumed that

4 x x 0 a(x) β(x 0 )d(x) ν(x 0) = 1 P. Feng / J. Math. Anal. Appl. 329 (2007) unifomly in x 0 Ω, fo some β(x) C( Ω; + ) ν(x) C( Ω, + ). The pupose of this wok is to conside moe geneal a(x) Ω a ball B. Indeed, we shall assume a(x) C(Ω) a(x) = a( x )>0inB. Futhemoe, we assume a() satisfies a(s)ds C 1( 0,] ), a() a(s)ds = 0. a() We shall study the exact blow-up ate at the bounday which helps us to establish the uniqueness esult. By a simila pocess as in 4], we can extend the esult to a moe geneal domain Ω. To analyze (1), we shall make the change of vaiable u := w m. (1) becomes { Δu = λu 1/m a(x)u p/m in Ω, (2) u = on Ω. Ou main esult eads as follows. Theoem 2. Conside the adially symmetic semilinea elliptic equation { Δu = λu 1/m a(x)u p/m in B (0), u = on B (0), p>m 1, λ, a C(0,]; 0, )) satisfying a>0 in 0,), A() = 0 whee A() := a() A() a() C1( 0,] ), a(s)ds. Then the poblem (3) admits a solution u satisfying u(x) d(x) 0 M( = 1, A(s) ds) α whee d(x) := dist(x, B (0)) M = α(α + 1)A 0 α ] α m, α= p m. Hee Autho's pesonal copy A 0 = a() A(s) ds. Futhemoe, if λ 0, then (3) admits a unique solution. A() 2 (3) Theoem 2 is a shap impovement of M. Delgado et al. 4, Theoem 1.2]. In the following examples, we shall obtain the pecise blow-up ate fo some special cases compae ou esults with othes.

5 350 P. Feng / J. Math. Anal. Appl. 329 (2007) Example 3. a() = a 0 > 0. An easy calculation shows that A 0 = 2 theefoe u(x) ( 2α 2 + α ) ] α 1 α 2 a 0( ) 2, whee α = p p m. In paticula, when m = 1, the esult agees with 2]. Example 4. a() = a 0 ( ) ν, then A() = 1 ν + 1 a 0( ) ν+1, A 0 = ν + 2 ν + 1, M = α(α + 1) ν + 2 ] α ν + 1 α. Theefoe, ] a α 0 u(x) M ( ) m(ν+2) p m. (ν + 1)(ν + 2) The esult agees with 4, Theoem 1.2]. Example 5. Hee we show an example that coves a moe geneal case which is not included in the pevious esults. Let a() = a 0 exp ( ( ) ν A() ), then a() = 1 ν ( )ν+1 = 0fo ν>0. Futhemoe, ( ) A() = 1 + A() a() a() ν( ) ν 1 0 as. Thus A 0 = 1, M = α 2α with α = p m m. Theefoe, u(x) M A(s) ds] α. This implies that u(x) goes to at the bounday faste than any powe function. We oganize the pape as follows. In Section 2, we establish some peinay esults that will be used late to pove ou main theoem. In Section 3, we study the exact blow-up ate pove the uniqueness based on the blow-up ate. 2. Some peinay esults In this section we include some useful peinay esults. The fist one is an extension of 7, Lemma 4]. Theoem 6. Suppose u ū satisfy Δu λu 1/m a(x)u p/m in Ω, Δū λū 1/m a(x)ū p/m in Ω, u =, d(x) 0 + Autho's pesonal copy ū = d(x) 0 + u ū in Ω. Then (3) possesses a solution u satisfying u u ū.

6 P. Feng / J. Math. Anal. Appl. 329 (2007) The poof of this theoem follows fom 1, Theoem A]. We shall omit hee. We shall also need the following lemmas that povide us infomation on a(x) as d(x) 0 +. Lemma 7. Let f() C(0,], 0, )) f()>0 fo (0,]. We define F()= 0 f(s)ds, G()= If thee exists g C 1 (0,]) such that s g(0) = 0, g (0) 0, then we have (a) (b) (c) F μ () = 0, μ 1; 0 + f() G() 0 + F() = 0; 0 + F 2 () G()f() = C f(t)dt ds. F() g()f() = c>0, Poof. Since F μ () f() = F() g()f() g()f μ 1 () 0, statement (a) follows easily. We can also pove statement (b) using L Hospital ule. To pove statement (c), we note that 0 + F 2 () G()f() = F() F()g() F()g () + F ()g() = c 0 + g()f() G() 0 + F() ( = c g (0) + 1 ) = 1 + cg (0) 1. c An immediate consequence of Lemma 7 is the following: Lemma 8. Let a() C(0,]; 0, )) A() = a(s)ds.if A() a() is diffeentiable in 0,] A() a() = 0, A() a() ] 0, then we have A μ () a() Autho's pesonal copy = 0, μ 1, A() 2 a() A(s) ds = A 0 1. A(s) ds = 0, A()

7 352 P. Feng / J. Math. Anal. Appl. 329 (2007) Poof of Theoem 2 Conside the following singula poblem φ N 1 φ = λφ 1/m a()φ p/m in (0,), φ()=, (4) φ (0) = 0, whee >0, λ a C(0,]; 0, )). We claim that fo each ɛ>0, the poblem possesses a positive solution φ ɛ such that φ ɛ () 1 ɛ inf M( φ ɛ () sup A(s) ds) α M( 1 + ɛ, (5) A(s) ds) α whee α = m p m, M = α(α + 1)A 0 α ] α, ( a(s)ds) 2 A 0 = a() s a(t)dt ds. Thus the function u ɛ (x) := φ ɛ (), := x, povides us with a adially symmetic positive lage solution of { Δu = λu 1/m a()u p/m in B (0), (6) u = on B (0), satisfying u ɛ (x) 1 ɛ inf d(x) 0 M( u ɛ (x) sup A(s) ds) α d(x) 0 M( 1 + ɛ, A(s) ds) α whee d(x) := dist(x, B (0)) =. We pove the claim by constucting a supesolution a subsolution with the same blow-up ate. Fist, we claim that, fo each ɛ>0, thee exists a constant A ɛ > 0 such that fo all A + >A ɛ, ( ) 2 α φ ɛ () = A + + B + A(s) ds], whee α = m p m, is a positive supesolution of (4). Fo simplicity, we denote A () := Autho's pesonal copy A(s) ds = B + = (1 + ɛ) α(α + 1)A 0 α ] α s a(t)dt ds.

8 P. Feng / J. Math. Anal. Appl. 329 (2007) An easy calculation shows φ ɛ () = 2B + A 2 () ] ( ) α 2 αb+ A () ] α 1 A () ], φ ɛ () = 2B 1 + A 2 () ] α 4αB+ A 2 () ] α 1 A () ] ( ) 2 + α(α + 1)B + A () ] α 2 A () ] 2 αb + ( ) 2 A () ] α 1 A () ]. By the assumptions of the theoem, A () = 0, we have φ ɛ () = φ ɛ (0) = 0. Thus in ode to show φ ɛ is a supesolution, we only need to show 2N B + A 2 () ] α + (N + 3)αB+ A 2 () ] α 1 A () ] ( ) 2 α(α + 1)B + A () ] α 2 A () ] ( ) αb+ A () ] α 1 A () ] λ A () ] ( ) α/m 2 ] 1/m A + A () α + B + a() A () ] ( ) pα/m 2 ] p/m A + A () α + B +. (7) Multiplying on both sides of inequality (7) by a() 1 A ()] pα/m taking into account that α = m p m, i.e., pα m α = 1, we have 2N B + A () 2 a() + (N + 3)αB A ()] + 2 a() ( ) 2 A () ] 2 ( ) 2 α(α + 1)B + A ()a() + αb A ()] + a() λ 1 A () ] ( ) p 1 p m 2 ] 1/m ( ) 2 ] p/m A + A () α + B + A + A () α + B +. a() (8) By Lemma 8, we have A () a() = 0, Autho's pesonal copy A ()] a() = 0, A () ] 2 A ()a() = A 0

9 354 P. Feng / J. Math. Anal. Appl. 329 (2007) A ()] μ a() whee μ = p 1 A ()] μ A μ () = A μ () a() = 0, p m 1. Thus at =, inequality (8) becomes α(α + 1)B + A 0 + αb + B p/m +. (9) Theefoe, by making the following choice B + = (1 + ɛ) α(α + 1)A 0 α ] α, the inequality (8) is satisfied in ( δ,] fo some δ = δ(ɛ) > 0. Finally, by choosing a sufficiently lage A + A ɛ, (8) is satisfied in the whole inteval 0,] since p>m 1 A () is bounded away fom zeo in 0, δ]. This concludes the claim. Next we constuct a subsolution with the same blow-up ate as the supesolution constucted above. In fact, fo each sufficiently small ɛ>0, thee exists A < 0 fo which the function { ( ) 2 φ ɛ () := max 0; A + B A () ] α} povides us a nonnegative subsolution if B = (1 ɛ) α(α + 1)A 0 α ] α, m whee α = p m. Indeed, it is easy to see that φ ɛ is a subsolution if in the egion whee A + B ( ) 2 A () ] α 0 the following inequality is satisfied: 2N B A () 2 a() + (N + 3)αB A ()] 2 a() ( ) 2 A () ] 2 ( ) 2 α(α + 1)B A ()a() + αb A ()] a() λ 1 A () ] ( ) p 1 p m 2 ] 1/m ( ) 2 ] p/m A A () α + B A A () α + B. a() (10) At =, the above inequality is equivalent to α(α + 1)B A 0 + αb B p/m, i.e., Autho's pesonal copy B α(α + 1)A 0 α ] α. Thus thee exists δ = δ(ɛ) > 0 fo which (10) is satisfied in δ,]. Moeove, fo each A < 0, thee exists a constant z = z(a ) (0,)such that A + B ( ) 2 A () ] α < 0 if 0,z(A ) ),

10 P. Feng / J. Math. Anal. Appl. 329 (2007) while A + B ( ) 2 A () ] α 0 if z(a ), ]. The claim above follows fom the obsevation below ( ) 2 A + B A () ] α =, ( ) 2 A + B A () ] α = A < 0 0 ( ) 2 A + B A () ] ] α < 0 in(0,), whee epesents the deivative with espect to. In fact, z(a ) is deceasing in A z(a ) =, z(a ) = 0. A A 0 Then by choosing A such that z(a ) = δ, it follows that φ ɛ povides us a subsolution of (4). Finally, since φ ɛ () φ 1 ɛ inf M( ɛ () sup A(s) ds) α M( 1 + ɛ, A(s) ds) α it follows fom Theoem 6 that thee exists a solution of (4), denoted by φ ɛ, satisfying (5). Letting ɛ 0 + we obtain φ() M( = 1. A(s) ds) α Fo any two abitay lage solutions u 1 (x) = φ 1 (), u 2 (x) = φ 2 (), it eadily follows that u 1 (x) d(x) 0 + u 2 (x) = 1. Hence fo any ɛ>0thee exists δ = δ(ɛ) such that (1 ɛ)u 2 u 1 (1 + ɛ)u 2 fo any x Ω with 0 <d(x) δ. It is clea that u 1 is a positive solution of the bounday value poblem { Δφ = λφ 1/m a(x)φ p/m in Ω δ, (11) φ = u 1 on Ω δ, whee Ω δ := {x Ω d(x) > δ}. It is easy to see that φ = (1 ɛ)u 2 povides us with a positive sub-solution φ + = (1 + ɛ)u 2 povides us with a supe-solution of (4). Thus the unique solution of (4) u 1 satisfies φ u 1 φ + in Ω δ. Thus Autho's pesonal copy (1 ɛ)u 2 u 1 (1 + ɛ)u 2 fo any x Ω. Passing to the it ɛ 0 +, we conclude u 1 u 2.

11 356 P. Feng / J. Math. Anal. Appl. 329 (2007) efeences 1] H. Amann, On the existence of positive solution of nonlinea elliptic bounday value poblems, Indiana Univ. Math. J. 21 (1971) ] C. Ble, M. Macus, On second ode effects in the bounday behavio of lage solutions of semilinea elliptic poblems, Diffeential Integal Equations 11 (1998) ] M. Delgado, J. López-Gómez, A. Suáez, Chaacteizing the existence of lage solutions fo a class of sublinea poblems with nonlinea diffusion, Adv. Diffeential Equations 7 (2002) ] M. Delgado, J. López-Gómez, A. Suáez, Singula bounday value poblem of a poous media logistic equation, Hioshima Math. J. 34 (2004) ] Y. Du, Q. Huang, Blow-up solutions fo a class of semilinea elliptic paabolic equations, SIAM J. Math. Anal. 31 (1999) ] J. Gacía-Melián, A emak on the existence of lage solutions via sub supesolutions, Electon. J. Diffeential Equations 110 (2003) ] J. Gacía-Melián,. Letelie-Albonoz, J.C. Sabina de Lis, Uniqueness asymptotic behavio fo solutions of semilinea poblems with bounday blow-up, Poc. Ame. Math. Soc. 129 (2001) ] J.B. Kelle, On solutions of Δu = f(u), Comm. Pue Appl. Math. 10 (1957) ] S. Kim, A note on bounday blow-up poblem of Δu = u p, IMA pepint No. 1820, ] M. Macus, L. Véon, Uniqueness of solutions with blowup at the bounday fo a class of nonlinea elliptic equations, C.. Acad. Sci. Pais Se. I Math. 317 (6) (1993) ] M. Macus, L. Véon, Uniqueness asymptotic behavio of solutions with bounday blow-up fo a class of nonlinea elliptic equations, Ann. Inst. H. Poincaé Anal. Non Linéaie 14 (2) (1997) Autho's pesonal copy