A comparison principle for nonlinear heat Rockland operators on graded groups

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1 Bull. London Mth. Soc doi: /blms A compison pinciple fo nonline het ocklnd opetos on gded goups Michel uzhnsky nd Duvudkhn Sugn Abstct In this note we show compison pinciple fo nonline het ocklnd opetos on gded goups. We give simple poof fo it using puely lgebic eltions. As n ppliction of the estblished compison pinciple we pove the globl in t-boundedness of solutions fo clss of nonline equtions fo the het p-sub-lplcin on sttified goups. 1. Intoduction A connected simply connected Lie goup G is clled gded Lie goup if its Lie lgeb dmits gdtion. Fo exmple, the Eucliden spce, the Heisenbeg goup o ny sttified goup o homogeneous Cnot goup e exmples of gded goups. The min clss of opetos tht we e deling with in this ppe e the so-clled ocklnd opetos, the clss of opetos intoduced in [11] s left-invint diffeentil opetos whose infinitesiml epesenttions e injective on smooth vectos, nd conjectued to coincide with hypoelliptic opetos. In this shot note it will be much simple to dopt nothe equivlent definition following Helffe nd Nouigt s esolution of the ocklnd conjectue in [8]. Nmely, we undestnd by ocklnd opeto ny left-invint homogeneous hypoelliptic diffeentil opeto on G. This will llow us to void the lnguge of the epesenttion theoy, nd we cn efe to [5, Section 4.1] fo detiled exposition of these equivlent notions. The consideed setting is most genel since it is known tht if thee exists left-invint homogeneous hypoelliptic diffeentil opeto on nilpotent Lie goup then the goup must be gded see, fo exmple, [5, Poposition 4.1.3], lso fo histoicl discussion. Such goup cn be lwys identified with Eucliden spce N fo some N, with polynomil goup lw. We efe to the ecent monogph [5] fo definitions nd othe popeties. Note tht the stndd Lebesgue mesue is the H mesue fo G. Let Ω G be bounded set with smooth boundy Ω nd let T>0beel numbe nd =0,T Ω. We denote the Sobolev spce by S,p Ω = S,p Ω, fo >0 nd 1 <p<, defined by the nom 1 u S,p Ω := x p + ux p p dx, 1.1 Ω whee ν is the homogeneous ode of the ocklnd opeto. We lso define the functionl clss S,p 0 Ω to be the completion of C 0 Ω in the nom 1.1. We efe to [5, Chpte 4; 6 fo genel discussion of Sobolev spces on gded goups. eceived 2 August 2017; evised 15 Jnuy Mthemtics Subject Clssifiction 35G20, 22E30 pimy. The uthos wee suppoted in pts by the EPSC gnt EP/003025/1 nd by the Levehulme Gnt PG , s well s by the MESK gnt AP No new dt ws collected o geneted duing the couse of esech. C 2018 London Mthemticl Society. This is n open ccess ticle unde the tems of the Cetive Commons Attibution License, which pemits use, distibution nd epoduction in ny medium, povided the oiginl wok is popely cited.

2 2 MICHAEL UZHANSKY AND DUVUDKHAN SUAGAN In this note we conside the initil boundy vlue poblem u t ν u = γ u β 1 u + α u q 2 u, x Ω, t > 0, u t, x =0, x Ω, t > 0, u 0,x=u 0 x, x Ω, 1.2 whee p>1, β>0, q 1, γ 0, α 0, nd the initil dt is u 0 x 0,u 0 x 0, u 0 S,p 0 Ω L Ω. Let { V := v L p 0,T; S,p } 1 0 Ω t v L p 0,T; S,p Ω, p + 1 = p If function u V C0,T; L 2 Ω, with ux =u 0 x fo lmost evey x Ω, stisfies t uϕ + dxdt = γ u β 1 u α u q 2 ϕdxdt 1.4 fo evey nonnegtive test-function ϕ V C0,T; L 2 Ω, then the function u is clled wek solution of 1.2. If function u V C0,T; L 2 Ω, with ux u 0 x fo.e. x Ω, stisfies t uϕ + dxdt γ u β 1 u α u q 2 ϕdxdt 1.5 fo evey nonnegtive test-function ϕ V C0,T; L 2 Ω, then the function u is clled wek sub-solution of 1.2. If function u V C0,T; L 2 Ω, with ux u 0 x fo.e. x Ω, stisfies t uϕ + dxdt γ u β 1 u α u q 2 ϕdxdt 1.6 fo evey nonnegtive test-function ϕ V C0,T; L 2 Ω, then the function u is clled wek sup-solution of 1.2. The gol of this note is to give simple poof of compison pinciple fo the initil boundy vlue poblem fo nonline het ocklnd opetos on gded goups using pue lgebic eltions, inspied by the ecent wok [10]. Fo thoough nlysis of sub-hmonic nlysis in elted settings see, fo exmple, [3], see lso, fo exmple, [1, 2] fo some elted nlysis nd efeences theein. We lso note tht the globl in time well posedness of nonline wve equtions fo ocklnd opetos hve been lso ecently investigted in [15]. In Section 2 we pesent the min esult of this note nd give its shot poof. An ppliction to nonline het equtions fo the p-sub-lplcin is then discussed in Section A compison pinciple on gded goups The following esult is compison pinciple fo solutions of 1.2. Theoem 2.1. Let u nd v be wek sub-solution nd sup-solution of 1.2, espectively. If u nd v e loclly bounded, then u v lmost eveywhee in. Poof. We hve the eltions see, fo exmple, [9] { c c d dc d 4 p d /2 d c /2 c 2 0ifp 2, 2 c c d d 2 dc d =p 1 d c 2 1+ c 2 + d 2 /2 0if2 p 1.

3 A COMPAISON PINCIPLE FO NONLINEA HEAT OCKLAND OPEATOS 3 Setting c = nd d = ν v, we obtin ν v ν v ν v dxdt On the othe hnd, by using the stightfowd inequlities u β 1 u v β 1 v = u β v β > 0, fo u>v>0, u β 1 u v β 1 v = u β + v β > 0, fo u>0 >v, u β 1 u v β 1 v = u β + v β > 0, fo 0 >u>v, we get tht u β 1 u v β 1 v ϕdxdt 0, 2.2 whee ϕ := mx{u v, 0}, thusϕ0,x=0ndϕt, x x Ω = 0. Theefoe, by the definitions of sub- nd sup-solutions, we obtin 1 ϕ 2 1 t, xdx = 2 Ω 2 τ ϕ 2 τ,xdxdτ ϕ τ ϕdxdτ + ν v ν v ν v dxdτ = ϕ τ u v dxdτ + = t uϕ + dxdt γ t vϕ + ν v ν v dxdt ν v ν v dxdτ u β 1 u v β 1 v ϕdxdt+ α u q 2 u v q 2 v ϕdxdτ γ u β 1 u v β 1 v ϕdxdt+ α sup u>v α sup u>v tht is, u q 2 u v q 2 v u v Ω u q 2 u v q 2 v ϕ 2 dxdτ u v ϕ 2 dxdτ, 2.3 ϕ 2 x, tdx L ϕ 2 x, τdxdτ, t [0,T, 2.4 whee L =2α sup u>v u q 2 u v q 2 v/u v. Let us ecll the Gonwll inequlity fo the convenience of the edes: Let g nd f be el-vlued continuous functions defined on [0,T. If f t gtft fo ll t [0,T, then ft f0 exp t 0 gsds, t [0,T. With ft = ϕ 2 τ,xdxdτ, fom the Gonwll inequlity we hve Ω ϕ2 dx = 0. This mens tht ϕ = 0.e. x Ω, tht is, u v.e. t, x.

4 4 MICHAEL UZHANSKY AND DUVUDKHAN SUAGAN 3. An ppliction to p-sub-lplcin equtions Let us demonstte n ppliction of Theoem 2.1 to sttified Lie goups which is one of impotnt clsses of gded goups nlysed extensively by Follnd [7]. A sttified Lie goup cn be defined in mny diffeent equivlent wys see, fo exmple, [4, 5] fo the Lie goup nd Lie lgeb points of view, espectively. A Lie goup G = N, is clled sttified Lie goup if the following two conditions e stisfied: fo ech λ>0 the diltion δ λ : N N given by δ λ x δ λ x 1,...,x := λx 1,...,λ x is n utomophism of the goup G, whee x k N k fo k =1,..., with N N = N nd N = N1 N. b let X 1,...,X N1 be the left invint vecto fields on G such tht X j 0 = x j 0 fo j =1,...,N 1. Then the Hömnde condition nklie{x 1,...,X N1 }=N holds fo evey x N,thtis,X 1,...,X N1 with thei iteted commuttos spn the whole Lie lgeb of G. The left invint vecto field X j hs n explicit fom: X j = N l + x j l=2 m=1 l j,m x,...,x l 1 x l m, 3.1 see lso [5, Section 3.1.5] fo genel pesenttion. We will lso use the following nottions H := X 1,...,X N1 fo the hoizontl gdient, L p f := H H f H f, 1 <p<, 3.2 fo the hoizontl sub-elliptic p-lplcin o, in shot, p-sub-lplcin, nd x = x x 2 N 1 fo the Eucliden nom on N1. The explicit epesenttion 3.1 llows us to hve, fo instnce, the identity, see fo exmple [12], H x γ = γ x γ 1, 3.3 nd x H x γ = N 1 γ x γ 3.4 fo ll γ, x N1 nd x 0. Hee nd in the sequel x mens hoizontl pt of x, tht is, x =x,x N, whee x is n N 1 -dimensionl vecto nd x is n N-dimensionl vecto. The potentil theoy on sttified goups fom the point of view of lye potentils ws developed in [13]. Let Ω G be bounded open set. Let us conside the following functionl 1 J p u := H u p + u p p dx, Ω nd we define the functionl clss S 1,p 0 Ω to be the completion of C 0 Ω in the nom geneted by J p cf. [14].

5 A COMPAISON PINCIPLE FO NONLINEA HEAT OCKLAND OPEATOS 5 Let us conside the following initil boundy vlue poblem fo the p-sub-lplcin, 1 <p<, u t L p u = u β 1 u + u q 2 u, x Ω, t > 0, ut, x =0, x Ω, t > 0, 3.5 u0,x=u 0 x, x Ω, whee β>0, q 1 nd the initil dt is u 0 x 0,u 0 x 0,u 0 S 1,p 0 Ω L Ω. Coolly 3.1. Let Ω G be bounded open set in sttified goup. Let p q<β+1. Then wek solution of 3.5 is globlly in t-bounded. Poof. Denote := mx x=x,x Ω x, then < since Ω is bounded. Fo ny x =x,x Ω, let x 0 =x 0,x 0 G\Ω be such tht ε x 0 x < + 1. It is cle tht we cn tke ε 0, 1. Let V t, x :=Le σ,= x x 0, x=x,x Ω, 3.6 fo some positive L nd σ to be chosen lte. Define M p v := v t L p v v q 1 + v β. By using 3.3 nd 3.4 let us fist clculte L p V = H H Le σ H Le σ = H L σ e σ x x 0 Lσe σ x x 0 x x 0 x x 0 = H L p 1 σ p 1 e σ x x 0 p 1 x x 0 x x 0 = L p 1 σ p 1 σp 1e σ x x 0 p 1 + L p 1 σ p 1 e σ x x 0 p 1 N 1 1 x x 0 =p 1σp L p 1 e p 1σ + N 1 1 σ p 1 L p 1 e σp 1. Thus, V t, x stisfies M p V = p 1σ p L p 1 e p 1σ N 1 1 σ p 1 L p 1 e p 1σ L q 1 e q 1σ + L β e βσ. Let us find σ nd L such tht M p V 0, tht is, p 1σ p L p 1 e p 1σ + N 1 1 σ p 1 L p 1 e p 1σ + L q 1 e q 1σ L β e βσ. Multiplying both sides of the inequlity by L p+1 e p 1σ,wehve p 1σ p + N 1 1 σ p 1 + L q p e q pσ L β+1 p e β+1 pσ. 3.7 Since ε < +1,topove3.7 it is sufficient to hve p 1σ p + N 1 1 σ p 1 + L q p e q pσ +1 L β+1 p. ε Let us choose 1 σ = q p +1, 3.8 { L = mx 2e 1/β+1 q, 2 p 1σ p + N } 1/β+1 p 1 1 σ p 1 ε

6 6 MICHAEL UZHANSKY AND DUVUDKHAN SUAGAN if q>p,nd σ =1, { L = mx 2 1/β+1 q, 2 p 1+ N } 1/β+1 p 1 1 ε 3.9 if q = p. Then we hve M p V 0. We cn ssume tht L u 0 L Ω, othewise we cn lwys multiply L by sufficiently lge numbe. Then V 0,x=Le σ u 0 x, tht is, V t, x is sup-solution of 3.5. Theefoe, ccoding to Theoem 2.1, we hve ut, x Le σ +1 <, = mx x=x,x Ω x The ight-hnd side of 3.10 is independent of t, sothtut, x is globlly in t-bounded. efeences 1. S. Bigi nd A. Bonfiglioli, The existence of globl fundmentl solution fo homogeneous Hömnde opetos vi globl lifting method, Poc.Lond.Mth.Soc A. Bonfiglioli nd A. Kogoj, Weighted L p -Liouville theoems fo hypoelliptic ptil diffeentil opetos on Lie goups, J. Evol. Equ A. Bonfiglioli nd E. Lnconelli, Subhmonic functions in sub-iemnnin settings, J. Eu. Mth. Soc. JEMS A. Bonfiglioli, E. Lnconelli nd F. Uguzzoni, Sttified Lie goups nd potentil theoy fo thei sub-lplcins Spinge, Belin/Heidelbeg, V. Fische nd M. uzhnsky, Quntiztion on nilpotent Lie goups, Pogess in Mthemtics 314 Bikhäuse, Bsel, open ccess book 6. V. Fische nd M. uzhnsky, Sobolev spces on gded goups, Ann. Inst. Fouie Genoble G. B. Follnd, Subelliptic estimtes nd function spces on nilpotent Lie goups, Ak. Mt B. Helffe nd J. Nouigt, Ccteistion des opéteus hypoelliptiques homogènes invints à guche su un goupe de Lie nilpotent gdué, Comm. Ptil Diffeentil Equtions P. Lindqvist, Notes on the p-lplce eqution, 2006, Y. Li, Z. Zhng nd L. Zhu, Clssifiction of cetin qulittive popeties of solutions fo the qusiline pbolic equtions, Sci. Chin Mth C. ocklnd, Hypoellipticity on the Heisenbeg goup-epesenttion-theoetic citei, Tns. Ame. Mth. Soc M. uzhnsky nd D. Sugn, On hoizontl Hdy, ellich, Cffelli-Kohn-Nienbeg nd p-sub- Lplcin inequlities on sttified goups, J. Diffeentil Equtions M. uzhnsky nd D. Sugn, Lye potentils, Kc s poblem, nd efined Hdy inequlity on homogeneous Cnot goups, Adv. Mth M. uzhnsky nd D. Sugn, Geen s identities, compison pinciple nd uniqueness of positive solutions fo nonline p-sub-lplcin equtions on sttified Lie goups, Pepint, 2017, Xiv: M. uzhnsky nd N. Tokmgmbetov, Nonline dmped wve equtions fo the sub-lplcin on the Heisenbeg goup nd fo ocklnd opetos on gded Lie goups, J. Diffeentil Equtions, Michel uzhnsky Deptment of Mthemtics Impeil College London 180 Queen s Gte London SW7 2AZ United Kingdom m.uzhnsky@impeil.c.uk Duvudkhn Sugn Deptment of Mthemtics School of Science nd Technology Nzbyev Univesity 53 Kbnby Bty Ave Astn Kzkhstn duvudkhn.sugn@nu.edu.kz