A NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p

Transcription

1 A NOTE ON BOUNDARY BLOW-UP PROBLEM OF u = u p SEICK KIM Abstract. Assume that Ω is a bouded domai i R with 2. We study positive solutios to the problem, u = u p i Ω, u(x) as x Ω, where p > 1. Such solutios are called boudary blow-up solutios of u = u p. We show that a boudary blow-up solutio exists i ay bouded domai if 1 < p <. I particular, whe = 2, there exists a boudary blow-up solutio to u = u p for all p (1, ). We also prove the uiqueess uder the additioal assumptio that the domai satisfies the coditio Ω = Ω. 1. Itroductio Let Ω be a bouded domai of R with 2 ad let Ω deote its boudary. I this article we study the problem (1) (2) u(x) = f(u(x)) for x Ω, u(x) + as d(x) := dist(x, Ω) 0, where f(t) = t p + := {max(t, 0)} p with p > 1. Solutios to the problem (1), (2) are called boudary blow-up solutios. I 1957, Keller [5] ad Osserma [11] proved existece of solutios to problem (1), (2) for a rather geeral class of fuctios f; i.e., f : R [0, ) is a locally Lipschitz cotiuous fuctio which is icreasig ad satisfies the followig growth coditio called Keller-Osserma coditio: { t 1/2 (3) f(s) ds} dt < + for all t 0 > 0. t 0 0 It is easy to check that f(t) = t p + with p > 1 satisfies (3). They showed that (3) is a ecessary coditio for the existece of blow-up solutios. Ideed, if the domai Ω is regular eough, say Lipschitz, the the existece of a classical solutio to the problem (1), (2) is established by the method of supersolutios ad subsolutios together with the uiform estimates of Keller [5]. We will briefly review existece results i the ext sectio. The case f(t) = t p + with p > 1 is of special iterest, ad i this article oly this case will be treated. Loewer ad Nireberg [7] studied the case whe p = +2 with > 2, which is related to a problem i differetial geometry. The problem (1), (2) is also related to probability theory. The equatio u = u p +, 1 < p 2, appears i the aalytical theory of a Markov process called superdiffusio; see [2]. By meas of a probabilistic represetatio, a uiqueess result i domais with o-smooth boudary was established by Le Gall [6] i the case whe p = 2. Later, Marcus ad Véro [8, 9] exteded the uiqueess i very geeral domais for all 2000 Mathematics Subject Classificatio. Primary 35J65; Secodary 35B05. Key words ad phrases. Blow-up, semi-liear equatio, existece, uiqueess. 1

2 2 SEICK KIM p > 1, usig purely aalytical method; they proved uiqueess i a domai whose boudary is locally represeted as a graph of a cotiuous fuctio. However, it is ot clear whether a boudary blow-up solutio exists or ot i such a geeral domai. I [10], Matero costructed a boudary blow-up solutio of u = u p + with 1 < p <, i a two-dimesioal domai with fractal boudary called the vo Koch sowflake domai. His approach is based o the compariso with boudary blow-up solutios i a cut-off ope coe. We treat a special case whe p (1, ) for 3 ad p (1, ) for = 2. Some iterestig results are obtaied i that case. We will prove that a boudary blow-up solutio exists i every bouded domai. As a cosequece, it will imply a result of Matero [10] metioed above. We will also show the uiqueess if the domai satisfies a additioal assumptio, Ω = Ω. For example, if Ω ca be locally represeted as a graph of a cotiuous fuctio, the it satisfies the above coditio. I this case, uiqueess was earlier proved by Marcus ad Véro [8, 9]. 2. Prelimiaries I this sectio, we briefly discuss the existece results of Keller [5], Loewer ad Nireberg [7]. We also itroduce some termiology which will be used i the later parts of the paper. We begi with a simple lemma. Lemma 2.1 (Compariso priciple). Let Ω R be a bouded domai. Assume that f is icreasig. Let u, v C 2 (Ω) be solutios of u f(u) ad v f(v) respectively. If lim if x Ω (v u)(x) 0, the v u i Ω. Proof. Suppose, to the cotrary, that there exists x 0 Ω such that u(x 0 ) > v(x 0 ). The for sufficietly small ɛ > 0, Ω ɛ := {u v > ɛ} ad Ω ɛ Ω. Let w := u v ɛ. The w = 0 o Ω ɛ. Sice f is icreasig, Lw f(u) f(v) f(u) f(v + ɛ) 0 i Ω ɛ. The the maximum priciple implies w 0 i Ω ɛ. This cotradictio proves the lemma. Remark 2.2. Let Ω 1, Ω 2 R be bouded domais such that Ω 1 Ω 2, i.e., Ω 1 Ω 2. Suppose u i (i = 1, 2) are solutios to (1), (2) i Ω i. The, it follows from Lemma 2.1 that u 1 u 2 i Ω 1. The ext theorem is quoted from [5]; see also [11]. Theorem 2.3 (Keller [5, pp ]). Let u be a solutio of (1) i a bouded domai Ω. There exist a cotiuous, decreasig fuctio g : (0, ) R determied by f such that lim t 0 g(t) = + ad (4) u(x) g(d(x)), where d(x) := dist(x, Ω). Usig the above estimate (4), Keller proved the existece of a boudary blow-up solutio. Although he claimed the existece i arbitrary domais, his argumet seems to require certai smoothess assumptio o Ω. Let Ω be a regular domai, say a Lipschitz domai. By the method of supersolutios ad subsolutios (see e.g. [3, pp ]), oe ca show that, for each m 1, there exists a uique solutio u m C 0 (Ω) C 2 (Ω) of (1) such that u m = α m o Ω, where α m < α m+1 ad α m as m. The by the compariso priciple, {u m } m=1 is a icreasig sequece of fuctios. By (4), u m (x) g(d(x)) uiformly for m 1. Deote by u(x) the poitwise limit of {u m (x)} m=1. The by the stadard elliptic theory (see

3 A NOTE ON BOUNDARY BLOW-UP PROBLEM 3 e.g. [4]), u C 2 (Ω) ad u is a solutio of (1). As x approaches Ω, u(x) icreases idefiitely sice u m = α m becomes ifiite o Ω; thus u is a solutio of the problem (1), (2). The solutio u costructed above is called a miimal boudary blow-up solutio. Ideed, if v is a boudary blow-up solutio, the by the compariso priciple, u m v i Ω for all m 1 ad thus, u = lim m u m v follows. Loewer ad Nireberg [7] itroduced aother importat solutio of (1) called a maximal solutio which is ot ecessarily a blow-up solutio but ca be costructed i ay bouded domai Ω. Let {Ω m } m=1 be a exhaustig sequece of smooth subdomais of Ω; i.e., Ω m Ω m+1 Ω ad m=1 Ω m = Ω. Let u m be the miimal blow-up solutio i Ω m for each m 1, ad let v be the miimal blowup solutio i a ball cotaiig Ω. By Remark 2.2, {u m } m=1 is decreasig ad bouded below by v. Hece, the limit fuctio u exists ad by the stadard elliptic theory, it is a solutio to (1). This solutio u is maximal sice if v is a solutio of (1) i Ω, the by the compariso priciple, we see u m v for all m 1. I ext sectio, we will provide a example of maximal solutio which is ot a boudary blow-up solutio; see Remark 3.3 below. 3. Mai Results We cosider the problem (1), (2) with f(t) = t p +. Note that i this case, a solutio to the problem (1), (2) must be positive, which is a simple cosequece of the maximum priciple. Ideed, more geerally, let t 0 := sup {t : f(t) = 0}. If t 0, the by cotiuity, f(t 0 ) = 0 ad thus, u t 0 is a solutio to (1). By Lemma 2.1, we fid that ay blow-up solutio of (1) is bouded below by t 0. Hereafter, we always assume p (1, ) whe = 2, ad p (1, ) whe 3. We will show that i that case, a boudary blow-up solutio exists i ay bouded domai, which obviously iclude the domai cosidered by Matero i [10]. Also, by usig Safoov s iteratio techique i [12], we prove uiqueess provided that Ω satisfies the coditio Ω = Ω. For example, if Ω ca be locally represeted as a graph of a cotiuous fuctio, the it satisfies the above coditio Costructio of a barrier i R \ {0}. We will costruct a solutio of u = u p + i R \ {0} which blows up at the origi. We look for a solutio of the form v(x) = c p x γ, where c p, γ > 0. Sice v is positive ad radially symmetric, v(r) = c p r γ, where r = x, must solve the followig ODE: (5) v (r) + 1 v (r) = v p (r) i (0, ). r Hece, the ukow costats c p, γ should satisfy (6) c p γ(γ + 2 )r γ 2 = c p pr γp. Set γ = 2/(p 1) so that γ + 2 = γp. The assumptio c p > 0 requires a restrictio o p, amely 2/(p 1) > 2. It is satisfied for all p > 1 whe = 2 ad for p (1, ) whe 3. If we choose (7) { } 1/(p 1) c p = {γ(γ + 2 )} γ/2 2 2( 2)p = (p 1) 2, it follows c p p = c p γ(γ + 2 ).

4 4 SEICK KIM The, v(x) = c p x γ is a solutio of v = v p + o R \{0} such that v(x) + as x 0. We summarize the above result as a lemma. Lemma 3.1. Let p > 1 whe = 2, ad let p (1, ) whe 3. The, v(x) := c p x γ is a solutio of v = v p + i R \ {0} such that v(x) + as { } 1/(p 1) 2 2( 2)p x 0. Here, γ = 2/(p 1) ad c p = (p 1) Existece ad uiqueess of boudary blow-up solutio. Theorem 3.2. Let Ω R be a bouded domai. The, there exists a solutio u to the problem (1), (2). Proof. Let {Ω m } m=1 be a exhaustig sequece of smooth subdomais of Ω, ad let u m be the miimal blow-up solutio of (1) i Ω m. The, the limit u := lim m u m is a maximal solutio; see Sectio 2. We eed to show that u is ideed a boudary blow-up solutio. For ay y Ω, choose a poit y 0 Ω such that d(y) = y y 0. Let v(x) := c p x y 0 γ with c p, γ defied as i Lemma 3.1. Sice y 0 / Ω m for each m 1, wee fid v(x) < + for all x Ω m. Hece, we coclude by Lemma 2.1 that (8) u m (y) v(y) = c p d γ (y) provided m is large eough so that y Ω m. Therefore, by passig to the limit, we fid u(y) c p d γ (y) for ay y Ω. Clearly, u(y) + as d(y) 0, ad thus, u is a desired solutio. Remark 3.3. I Theorem 3.2, the restrictio that p < whe > 2 is essetial. Let Ω := {x R : 0 < x < 1}, where > 2. Brézis ad Véro [1] showed that if p, the ay positive solutio u of u = up i Ω satisfies lim x 0 u(x) < +. Cosequetly, there is o solutio of the problem (1), (2) i Ω. This also shows that i geeral, a maximal solutio is ot ecessarily a boudary blow-up solutio. Theorem 3.4. I additio, assume that Ω satisfies Ω = Ω. The, the solutio of the problem (1), (2) is uique. Proof. Let u 1, u 2 be two boudary blow-up solutios i Ω. followig estimate holds: (9) N 1 d γ (x) u i (x) N 2 d γ (x), for all x Ω; i = 1, 2, We claim that the where N 1, N 2 > 0 are costats depedig oly o ad p. Fix x 0 Ω ad deote r := d(x 0 ). Choose z 0 Ω such that x 0 z 0 = r. From the assumptio that Ω = Ω, there exists a poit y 0 B r (z 0 ) \ Ω. Note that r x 0 y 0 2r. Let v(x) := c p x y 0 γ. Sice Ω is bouded ad y 0 / Ω, we fid, by see Lemma 2.1, that u i (x) v(x), where i = 1, 2. I particular, (10) u i (x 0 ) c p x 0 y 0 γ c p 2 γ d γ (x 0 ); i = 1, 2. Also, by cosiderig a ball B r (x 0 ) ad the miimal boudary blow-up solutio i that ball as a compariso fuctio, it is ot hard to see u i (x 0 ) N 2 d(x 0 ) γ, i = 1, 2, for some costat N 2 > 0 depedig oly o ad p; see e.g. [5]. Therefore, we coclude that the estimate (9) holds. Oce we obtai the estimate (9), u 1 u 2 will follow from the iteratio techique of Safoov i [12]. For the reader s coveiece, we will reproduce his techique here.

5 A NOTE ON BOUNDARY BLOW-UP PROBLEM 5 Assume, to the cotrary, that u 2 (x 1 )/u 1 (x 1 ) > k > 1 for some x 1 Ω. Let Ω 0 := {u 2 > ku 1 } B r (x 1 ), where r = 1 2 d(x 1). The, we fid (u 2 ku 1 ) = u p 2 kup 1 > (kp k)u p 1 c 1kr γp, where c 1 = 2 γp N p 1 (kp 1 1). Therefore, (u 2 ku 1 + w) 0 i Ω 0, where w = c 1 2 kr γp (r 2 x x 1 2 ). By the maximum priciple w(x 1 ) < (u 2 ku 1 + w) (x 1 ) sup Ω 0 (u 2 ku 1 + w). Note that the maximum must be achieved o B r (x 1 ) Ω 0 Ω 0 ; otherwise, it is achieved o {u 2 = ku 1 } B r (x 1 ), where we have u 2 ku 1 + w w(x 1 ). Hece, w(x 1 ) < (u 2 ku 1 )(x 2 ), where x 2 B r (x 1 ) Ω 0 Ω. O the other had, by (9), we fid (recall x 2 B r (x 1 ) so that d(x 2 ) r) w(x 1 ) = c 1 2 kr γp r 2 = c 1 2 kr γ c 2 ku 1 (x 2 ), where c 2 = c 1 N p 1 = 2N 2 2 γp+1 (k p 1 1). Therefore, we coclude u 2 (x 2 )/u 1 (x 2 ) > N 2 (1 + c 2 )k. By iteratig the above process (both c 1 ad c 2 are mootoe icreasig i k), we obtai a sequece of poits {x j } j=1 i Ω satisfyig u 2(x j )/u 1 (x j ) > (1 + c 2 ) j k, which teds to ifiity as j. O the other had, by (9), u 2 (x) u 1 (x) < N 2d γ (x) N 1 d γ (x) = N 2 x Ω. N 1 This cotradictio proves the uiqueess. Remark 3.5. If Ω = { x R 2 : 0 < x < 1 }, the Ω Ω. The uiqueess fails i this case; see [13]. Ackowledgmet. This paper is based o a presetatio by the author at the 2001 AMS sectioal meetig, Williamstow, MA, ad was supported i part by NSF Grat No. DMS Refereces [1] Brézis, H. ad Véro, L.: Removable sigularities for some oliear elliptic equatios. Arch. Ratioal Mech. Aal. 75 (1980/81), o. 1, 1 6. [2] Dyki, E. B.: Diffusios, superdiffusios ad partial differetial equatios. America Mathematical Society Colloquium Publicatios, Vol. 50, America Mathematical Society, Providece, RI, [3] Evas, L. C.: Partial differetial equatios. America Mathematical Society, Providece, RI, 1998 [4] Gilbarg, D. ad Trudiger, N. S.: Elliptic partial differetial equatios of secod order. Secod editio (1983), Spriger-Verlag, Berli. [5] Keller, J. B.: O solutios of u = f(u). Comm. Pure Appl. Math. 10 (1957), [6] Le Gall, J. F.: A path-valued Markov process ad its coectios with partial differetial equatios. I First Europea Cogress of Mathematics, Vol. II (Paris, 1992), , Progr. Math., Vol. 120, Birkhäuser, Basel, [7] Loewer, C. ad Nireberg, L.: Partial differetial equatios ivariat uder coformal or projective trasformatios. I Cotributios to aalysis, Academic Press, New York, 1974, , [8] Marcus, M. ad Véro, L: Uiqueess of solutios with blowup at the boudary for a class of oliear elliptic equatios. C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), o. 6,

6 6 SEICK KIM [9] Marcus, M ad Véro, L: Uiqueess ad asymptotic behavior of solutios with boudary blow-up for a class of oliear elliptic equatios. A. Ist. H. Poicaré Aal. No Liéaire 14 (1997), o. 2, [10] Matero, J.: Boudary-blow up problems i a fractal domai. Z. Aal. Aweduge 15 (1996), o. 2, [11] Osserma, R.: O the iequality u f(u). Pacific J. Math. 7 (1957) [12] Safoov, M. V.: O the uiqueess of blowup solutios for semiliear elliptic equatios. Preprit. [13] Véro, L.: Solutios sigulières d équatios elliptiques semiliéaires (Frech). C. R. Acad. Sci. Paris Sér. A-B 288 (1979), o. 18, A867 A869. School of Mathematics, Uiversity of Miesota, Mieapolis, Miesota address: skim@math.um.edu