Gradient Estimates for a Nonlinear Parabolic Equation with Diffusion on Complete Noncompact Manifolds
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1 Chi. A. Mah. 36B(, 05, DOI: 0.007/s 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. zhaozogliag09@63.com College of Mahemaics Physics ad Iformaio Egieerig, Jiaxig Uiversiy, Jiaxig 3400, Zhejiag, Chia @qq.com This work was suppored by he Jiagsu Provicial Naural Sciece Foudaio of Chia (No. BK ad he Fudameal Research Fuds of he Ceral Uiversiies (No. NS04076.
2 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
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 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,
4 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 <<.
5 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 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
7 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 ( 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 ϕ ( σm w γ ϕ 0 K w. Le y = ϕ w, z = ϕ(be (w + w.
8 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 (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 [ (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 [ (M γ 4( 4 ( ɛ ( ( = (ϕf φ(ϕf 0ψ, ɛ ] 3 ( ( λ(σ ( σm c 0 4( R (ϕf } + K M γy
9 Gradie Esimaes for a Noliear Parabolic Equaio 65 where ad φ = H 0 λ(σ M c ( R + ψ = 3 [ (M γ 4( ( ɛ ] ( 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,
10 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, [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, [6] Wu, J. Y., Li-Yau ype esimaes for a oliear parabolic equaio o complee maifolds, J. Mah. Aal. Appl., 369, 00, [7] Zhag, J. ad Ma, B. Q., Gradie esimaes for a oliear equaio f u + cu α =0ocomplee ocompac maifolds, Comm. Mah., 9, 0, [8] Zhu, X. B., Gradie esimaes ad Liouville heorems for oliear parabolic equaios o ocompac Riemaia maifolds, Noliear Aal., 74, 0,
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