Chi. A. Mah. 36B(, 05, 57 66 DOI: 0.007/s40-04-0876- Chiese Aals of Mahemaics, Series B c The Ediorial Office of CAM ad Spriger-Verlag Berli Heidelberg 05 Gradie Esimaes for a Noliear Parabolic Equaio wih Diffusio o Complee Nocompac Maifolds Liag ZHAO Zogwei MA Absrac The auhors obai some gradie esimaes for posiive soluios o he followig oliear parabolic equaio: u = u b(x, uσ o complee ocompac maifolds wih Ricci curvaure bouded from below, where 0 < σ< is a real cosa, ad b(x, is a fucio which is C i he x-variable ad C i he -variable. Keywords Gradie esimaes, Posiive soluios, Harack iequaliy 000 MR Subjec Classificaio 58J05, 58J35 Iroducio I his paper, we sudy he followig oliear parabolic equaio: u = u b(x, uσ (. o complee ocompac maifolds M wih Ricci curvaure bouded from below, where 0 < σ<is a real cosa, ad b(x, is a fucio which is C i he x-variable ad C i he -variable. Gradie esimaes play a impora role i he sudy of he PDE, especially he Laplacia equaio ad he hea equaio. Li ad Yau [5] developed he fudameal gradie esimae, which is ow widely called he Li-Yau esimae, for ay posiive soluio u(x, o he hea equaio o a Riemaia maifold M, ad showed how he classical Harack iequaliy ca be derived from heir gradie esimae. Laer, Hamilo [3] go he marix Harack esimae for he hea equaio. Le (M,gbea-dimesioal complee ocompac Riemaia maifold. For a smooh real-valued fucio f o M, he drifig Laplacia is defied by f = f. Mauscrip received March 6, 03. Revised December, 03. Deparme of Mahemaics, Najig Uiversiy of Aeroauics ad Asroauics, Najig 006, Chia. E-mail: zhaozogliag09@63.com College of Mahemaics Physics ad Iformaio Egieerig, Jiaxig Uiversiy, Jiaxig 3400, Zhejiag, Chia. E-mail: 877856@qq.com This work was suppored by he Jiagsu Provicial Naural Sciece Foudaio of Chia (No. BK040804 ad he Fudameal Research Fuds of he Ceral Uiversiies (No. NS04076.
58 L. Zhao ad Z. W. Ma There is a aurally associaed measure dμ =e f dv o M which makes he operaor f self-adjoi. The N-Bakry-Emery Ricci esor is defied by Ric N f =Ric+ f df df N for 0 N ad N =0ifadolyiff =0. Here is he Hessia operaor ad Ric is he Ricci esor. Huag ad Li [4] cosidered he geeralized equaio u = f u α o Riemaia maifolds ad go some ieresig gradie esimaes. Wu [6] gave a local Li- Yau-ype gradie esimae for he posiive soluios o a geeral oliear parabolic equaio u = f u au log u qu i M [0,τ], where a R, φ is a C -smooh fucio ad q = q(x, is a fucio which geeralizes may previous well-kow resuls abou gradie esimaes. Zhag ad Ma [7] cosidered gradie esimaes o posiive soluios o he followig oliear equaio: f u + cu α =0, α > 0 (. o complee ocompac maifolds, ad he auhors go a gradie esimae for posiive soluios of he above equaio (. whe N is fiie ad he N-Bakry-Emery Ricci esor is bouded from below. Theorem. (see [7] Le (M,g be a complee ocompac -dimesioal Riemaia maifold wih he N-Bakry-Emery Ricci esor bouded from below by he cosa K =: K(R, where R>0 ad K(R > 0 i he meric ball B p (R aroud p M. Le u be a posiive soluio o (.. The ( if c>0, we have u u + cu (α+ (N + (N + +c R + (N + [(N + c + c ] R ( if c<0, we have u u + (N + (N + Kc R +(N + K; + cu (α+ (A + ( α A c if u (N + [(N + c + c ] + B p(r where A =(N + (α +(α +. + (N + ( c R + N ++ + N A ( + + ( + NK, A Recely, Zhu [8] ivesigaed he oliear parabolic equaio R + (N + (N + Kc R u = u + λ(x, u α (x,, (.3
Gradie Esimaes for a Noliear Parabolic Equaio 59 where 0 < α <, ad λ(x, is a fucio defied o M (, 0], which is C i he firs variable ad C 0 i he secod variable. The auhor go a Hamilo-ype esimae ad a Liouville-ype heorem for posiive soluios o (.3. Theorem. (see [8] Le (M,g be a Riemaia maifold of dimesio wih Ric(M k for some k 0. Suppose ha u is a posiive soluio o (.3 i Q R,T B(x 0,R [ 0 T, 0 ] M (,. Suppose also ha u M ad λ λ θ i Q R,T. The here exiss a cosa C = C(α, M, such ha u u CM α( R + + k + Cθ 4 M 3 ( α T i Q R, T. I his paper, we will sudy he ieresig Li-Yau ype esimae for he posiive soluios o (.. Moivaed by he above work, we prese our mai resuls abou (. as follows. Theorem.3 Le (M,g be a complee ocompac -dimesioal Riemaia maifold wih Ricci curvaure bouded from below by he cosa K =: K(R, where R>0 ad K(R > 0 ihemericballb p (R aroud p M. Assume ha b λ(r, b θ(r ad b γ(r i B p (R [0,T for some cosas λ(r,θ(r ad γ(r. Le u be a posiive soluio o (. wih u M. The for ay cosa 0 <<, if <σ<, we have u u bu u u ( ( ( + KRc +c R 3 + + c 4( R + 3 8( ( ɛ + + [M θ + λ( σm ] where c ad c are posiive cosas ad ɛ (0,. λ(σ M ( M 4σ 4 γ 4 ( 3 4ɛ 4 8 ( M λ(σ ( σ }, + K Le R, ad we ca ge he global Li-Yau ype gradie esimaes for he oliear parabolic equaio (.. Corollary. Le (M,g be a complee ocompac -dimesioal Riemaia maifold wih Ricci curvaure bouded from below by he cosa K, where K > 0. Assume ha b λ(m, b θ(m ad b γ(m i M [0,T for some cosas λ, θ ad γ. Le u be a posiive soluio o (. wih u M. The for ay cosa 0 <<, if <σ<, we have u u bu u u ( λ(σ M + + N,
60 L. Zhao ad Z. W. Ma where 3 ( M 4σ 4 γ 4 ( 3 3 ( M λ(σ ( σ N = 8 4ɛ 4 + 8( + K ( ɛ + [M θ + λ( σm ] } c ad c are posiive cosas ad ɛ (0,. As a applicaio, we ge he followig Harack iequaliy., Theorem.4 Le (M,g be a complee ocompac -dimesioal Riemaia maifold wih Ric(M K, where K>0. Assume ha b is a oposiive cosa ad b λ(m. Le u(x, be a posiive smooh soluio o he equaio u = u bu σ o M [0, +. The if <σ<, for ay pois (x, ad (x, o M [0, + wih 0 < <, we have he followig Harack iequaliy: ( u(x, u(x, e φ(x,x,,+ñ(, where φ(x,x,, =if γ 4 γ d ad Ñ = 3 8( ( ɛ + [λ( σm ] ( M λ(σ ( σ } + λ(σ M. + K Proof of Theorem.3 Le u be a posiive soluio o (.. Se w =lu. Thew saisfies he equaio w = w + w be (w. (. Lemma. Le (M,g be a complee ocompac -dimesioal Riemaia maifold wih Ricci curvaure bouded from below by he cosa K =: K(R, where R>0 ad K>0 ihemericballb p (R aroud p M. Le w be a posiive soluio o (.. The ( F w F + ( w be (w w + b(σ ( σe (w w +( σe (w w b } e (w b + b(σ e (w F F, where F = ( w be (w w, ad is ay cosa saisfyig 0 <<.
Gradie Esimaes for a Noliear Parabolic Equaio 6 Proof Defie F = ( w be (w w. ByheBocherformula,wehave Noicig ha w w + w ( w K w. (. w =( w = w w + b e (w + b(σ e (w w + w (.3 ad w = w + be (w + w =( (be(w + w F =( w F, we kow F = ( w (be (w w. ad By (. (.3, we obai w ( ( w F + w ( w K w = ( ( w F [( + w (be (w + w F ] K w = ( ( w F +( e (w w b +b( (σ e (w w +( w w w F K w (be (w =e (w b +(σ e (w w b + b(σ e (w w + b(σ e (w w =e (w b +(σ e (w w b + b(σ e (w w + b(σ e (w[ ( w F ].
6 L. Zhao ad Z. W. Ma So, we have ad F ( w be (w w +( e (w w b +b( (σ e (w w +( w w w F K w e (w b (σ e (w w b b(σ e (w w b(σ e (w[ ( w F ] } ( w w + b e (w + b(σ e (w w + w F = F + ( w w b e (w b(σ e (w w w. This implies ha ( F w F + ( w be (w w + b(σ ( σe (w w +( σe (w w b We complee he proof of Lemma.. e (w b K w } + b(σ e (w F F. Proof of Theorem.3 We ake a C cu-off fucio ϕ defied o [0,, such ha ϕ(r =forr [0, ], ϕ(r =0forr [,, ad 0 ϕ(r. Furhermore, ϕ saisfies ϕ (r ϕ (r c ad ϕ (r c for some absolue cosas c,c > 0. Deoe by r(x he disace bewee x ad p i M. Se ( r(x ϕ(x = ϕ. R Usig a argume of Cheg ad Yau [], we ca assume ϕ(x C (M wihsuppori B p (R. Direc calculaio shows ha o B p (R, ϕ ϕ c R. (.4 By he Laplacia compariso heorem i [], ϕ ( ( + KRc + c R. (.5 For T 0, le (x 0, 0 beapoiib R (p [0,]awhichϕF aais is maximum value P. We assume ha P is posiive (oherwise he proof is rivial. A he poi (x 0, 0, we have (ϕf =0, (ϕf 0, F 0. (.6
Gradie Esimaes for a Noliear Parabolic Equaio 63 I follows ha ϕ F + F ϕ Fϕ ϕ 0. where This iequaliy ogeher wih he iequaliies (.4 (.5 yields ϕ F HF, H = ( ( + KRc + c +c R. A (x 0, 0, by Lemma. ad (.6, we have 0 ϕ F HF HF + ϕ F + b(σ e (w F 0 + 0 ( w be (w w w F + b 0 (σ ( σe (w w + 0 ( σe (w w b 0 e (w b + 0 b(σ e (w 0 K w } HF ϕ 0 F +F w ϕ + 0 ϕ( w be (w w + b(σ e (w ϕf + b 0 ϕ(σ ( σe (w w 0 ϕe (w b + 0 ϕb(σ e (w + 0 ϕ( σe (w w b ϕ 0 K w HF ϕ 0 F +F w ϕ + 0 ϕ( w be (w w + λ(σ M ϕf λ 0 ϕ(σ ( σm w 0 ϕm θ + 0 ϕλ(σ M + 0 ϕ( σm w γ ϕ 0 K w. Muliplyig boh sides of he above iequaliy by 0 ϕ, ad oig he fac ha 0 <ϕ<, we have 0 H 0 ϕf ϕf + 0 ϕf w ϕ + λ(σ M 0 ϕf + 0 ϕ ( w be (w w λ 0 ϕ (σ ( σm w M θ 0 λ( σm 0 + 0ϕ ( σm w γ ϕ 0K w H 0 ϕf ϕf c R 0ϕF w ϕ 3 + λ(σ M 0 ϕf + 0 ϕ[ ( w be (w w λ(σ ( σm w ] M θ 0 λ( σm 0 + 0 ϕ ( σm w γ ϕ 0 K w. Le y = ϕ w, z = ϕ(be (w + w.
64 L. Zhao ad Z. W. Ma I follows ha 0 ϕf ( H 0 + λ(σ M 0 c + 0 ϕ ( w be (w w + σ + 0 M w γϕ ϕf ( H 0 + λ(σ M 0 (y z + σ M γy R 0F w ϕ 3 ( λ(σ ( σm } M θ 0 λ( σm 0 ( λ(σ ( σm } M θ 0 λ( σm 0. Followig he mehod i [5, pp. 6 6], we kow + K ϕ w + K y c R y (y z (y z c R y (y αz Ky (α γy α (y αz 8 c α (α R (y αz 3 4 4 4 3 3 (γ 4 (α α ɛ 3 4 ( ɛ (α α K for ay 0 <ɛ<. Therefore, i our case, we have ( λ(σ ( σm (y z (y z c R y (y z (y z + K (y z c R y ( (y z ( R 8 ( ( c y z 3 [ 4 4 4 3 3 (M γ 4( ɛ ] 3 ( 4 ( ɛ ( ( ( λ(σ ( σm Noicig ha y z = ϕf, we obai y c R y σ (y z+ M γy ( λ(σ ( σm + K y σ M γy ( λ(σ ( σm ( + K y + K. 0 ϕf ( H 0 + λ(σ M 0 + (ϕf + 0 3 [ 4 4 4 3 3 (M γ 4( 4 ( ɛ ( ( = (ϕf φ(ϕf 0ψ, ɛ ] 3 ( ( λ(σ ( σm c 0 4( R (ϕf } + K M γy
Gradie Esimaes for a Noliear Parabolic Equaio 65 where ad φ = H 0 λ(σ M c 0 + 0 4( R + ψ = 3 [ 4 4 4 3 3 (M γ 4( ( ɛ ] 3 + 4 ( ɛ ( ( ( λ(σ ( σm + M θ + λ( σm. + K From he iequaliy Ax Bx C, we have x B A + C A. We ca ge Noice ha for all [0,T], We complee he proof of Theorem.3. (ϕf (x 0, 0 φ + ( ψ sup T [ w be (w w ] (ϕf (x 0, 0. B p(r ProofofTheorem.4 For ay pois (x, ad(x, om [0, + wih0< <, we ake a curve γ( parameerized wih γ( =x ad γ( =x. Oe ges from Corollary. ha log u(x, log u(x, = which meas ha ( ((log u + log u, γ d ( log u ( 4 γ + 4 γ d +log. bu Ñ log u γ d + bu + Ñ d ( + Ñ( log u(x, u(x, ( 4 γ d +log + Ñ(. Therefore, ( u(x, u(x, e φ(x,x,,+ñ(, where φ(x,x,, =if γ 4 γ d ad Ñ = 3 8( ( ɛ + [λ( σm ] ( M λ(σ ( σ } + λ(σ M. + K,
66 L. Zhao ad Z. W. Ma Ackowledgemes The auhors would like o hak he ediors ad he aoymous referees for heir valuable commes ad helpful suggesios ha helped o improve he qualiy of his paper. Moreover, he auhors would like o hak heir supervisor Professor Kefeg Liu for his cosa ecourageme ad help. Refereces [] Aubi, T., Noliear Aalysis o Maifolds, Spriger-Verlag, New York, 98. [] Cheg, S. Y. ad Yau, S. T., Differeial equaios o Riemaia maifolds ad heir geomeric applicaios, Commu. Pure Appl. Mah., 8, 975, 333 354. [3] Hamilo, R., A marix Harack esimae for he hea equaio, Comm. Aal. Geom.,, 993, 3 6. [4] Huag, G. Y. ad Li, H. Z., Gradie esimaes ad eropy formulae of porous medium ad fas diffusio equaios for he Wie Laplacia, 0. arxiv: mah.dg/03.548v [5] Li, P. ad Yau, S. T., O he parabolic kerel of he Schrodiger operaor, Aca Mah., 56, 986, 53 0. [6] Wu, J. Y., Li-Yau ype esimaes for a oliear parabolic equaio o complee maifolds, J. Mah. Aal. Appl., 369, 00, 400 407. [7] Zhag, J. ad Ma, B. Q., Gradie esimaes for a oliear equaio f u + cu α =0ocomplee ocompac maifolds, Comm. Mah., 9, 0, 73 84. [8] Zhu, X. B., Gradie esimaes ad Liouville heorems for oliear parabolic equaios o ocompac Riemaia maifolds, Noliear Aal., 74, 0, 54 546.