Tile Auhor(s) GRADIENT ESTIMATES FOR A SIMPLE PARABOLIC LICHNEROWICZ EQUATION Zhao, Liang Ciaion Osaka Journal of Mahemaics. 51(1) P.45-P.56 Issue Dae 014-01 Tex Version publisher URL hps://doi.org/10.18910/9195 DOI 10.18910/9195 righs
Zhao, L. Osaka J. Mah. 51 (014), 45 56 GRADIENT ESTIMATES FOR A SIMPLE PARABOLIC LICHNEROWICZ EQUATION LIANG ZHAO (Received Augus, 01) Absrac In his paper, we sudy he gradien esimaes for posiive soluions o he following nonlinear parabolic equaion u f u cu «on complee noncompac manifolds wih Bakry Émery Ricci curvaure bounded below, where «, c are wo real consans and «0. 1. Inroducion Le (M n, g) be an n-dimensional complee noncompac Riemannian manifold. For a smooh real-valued funcion f on M, he drifing Laplacian is defined by f Ö f Ö. There is a naurally associaed measure d e f dv on M, which makes he operaor f self-adjoin. The N-Bakry Émery Ricci ensor is defined by Ric N f Ric Ö f 1 N d f Å d f for 0 N ½ and N 0 if and only if f 0. Here Ö is he Hessian and Ric is he Ricci ensor. Recenly, here has been an acive ineres in he sudy of gradien esimaes for he parial differenial equaion. Wu [16] gave a local Li Yau ype gradien esimae for he posiive soluions o a general nonlinear parabolic equaion u u Ö³Öu au log u qu in M [0, ], where a ¾ R, is a C -smooh funcion and q q(x, ) is a funcion, which generalizes many previous well-known gradien esimaes resuls. Zhu [18] 000 Mahemaics Subjec Classificaion. Primary 53J05; Secondary 58J35.
46 L. ZHAO invesigaed he fas diffusion equaion (1.1) u u «and he auhor go he following resuls: Theorem 1.1 (Zhu [18]). Le M be a Riemannian manifold of dimension n wih Ric(M) k for some k 0. Suppose ha Ú («(«1))u «1 is any posiive soluion o he equaion (1.1) in Q R,T B(x 0, R) [ 0 T, 0 ] M ( ½,½). Suppose also ha Ú ÉM in Q R,T. Then here exiss a consan C C(«, M) such ha in Q R,T. ÖÚ Ú 1 C ÉM 1 1 R 1 Ô T Ô k Laer, Huang and Li [5] considered he generalized equaion u f u «on Riemannian manifolds and go some ineresing gradien esimaes. Zhang and Ma [17] considered gradien esimaes for posiive soluions o he following nonlinear equaion (1.) f u cu «0 on complee noncompac manifolds. When N is finie and he N-Bakry Émery Ricci ensor is bounded from below, he auhors in [17] go a gradien esimae for posiive soluions of he above equaion (1.). Theorem 1. (Zhang and Ma [17]). Le (M, g) be a complee noncompac n-dimensional Riemannian manifold wih N-Bakry Émery Ricci ensor bounded from below by he consan K Ï K (R), where R 0 and K (R) 0 in he meric ball B R (p) around p ¾ M. Le u be a posiive soluion of (1.). Then (1) if c 0, we have Öu cu («1) (N n)(n n )c 1 (N n)[(n n 1)c 1 c ] u R R (N n)ô (N n)k c 1 R (N n)k.
GRADIENT ESTIMATES FOR A LICHNEROWICZ EQUATION 47 () if c 0, we have Öu cu («1) (A Ô A)c u «1 inf u B p (R) (N n)c 1 R Ô 1 (n N)K, A n N n N Ô A (N n)[(n n 1)c 1 c ] R (N n)ô (N n)k c 1 R where A (N n)(«1)(«) and c 1, c are absolue posiive consans. For ineresing gradien esimaes in his direcion, we can refer o [1] [] [7] [8] [9]. Recenly, a simple Lichnerowicz equaion u u p 1 u p 1, where p 1, was sudied by Ma [10]. The auhor obained a Liouville ype resul for smooh posiive soluions for he Lichnerowicz equaion in a complee non-compac Riemannian manifold wih he Ricci curvaure bounded from below. Laer, Sun and Zhao [14] sudied a generalized ellipic Lichnerowicz equaion u(x) h(x)u(x) A(x)u p (x) B(x) u q (x) on compac manifold (M, g). The auhors in [14] go he local gradien esimae for he posiive soluions of he above equaion. Moreover, hey considered he following parabolic Lichnerowicz equaion u (x, ) u(x, ) h(x)u(x, ) A(x)u p (x, ) B(x)u q (x, ) on manifold (M, g) and obained he Harnack differenial inequaliy. From he above work, we can see gradien esimaes for posiive soluions o nonlinear hea equaions are ineresing subjecs o researchers. Gradien esimaes ofen lead o Liouville ype heorems and Harnack inequaliies. For nonlinear hea equaions wih drifing Laplacians on manifolds, o ge good conrols of suiable Harnack quaniies (depending on nonlinear erms), one may need he key lower bounds assumpion abou Bakry Émery Ricci curvaures. Wihou he drifing erm, he naure assumpions are abou he Ricci curvaures. These are he main geomeric differences caused by drifing erms. A new research direcion is he nonlinear hea equaion wih negaive power, which has is roo from he Einsein-scalar Lichnerowicz equaion. In his paper, we sudy he following parabolic equaion (1.3) u f u cu «,
48 L. ZHAO where «, c are wo real consans and «0, f is a smooh real-valued funcion on M. We sae our main resuls as follows. Theorem 1.3. Le (M, g) be a complee noncompac n-dimensional Riemannian manifold wih N-Bakry Émery Ricci ensor bounded from below by he consan K Ï K (R), where R 0 and K (R) 0 in he meric ball B R (p) around p ¾ M. Le u be a posiive soluion of (1.3). Then (1) if c 0, we have Öu u cu («1) u u N n (N n)c 1 (1 Æ) 4Æ (1 )R A 1 Á () if c 0 and u («1) ÉM for all (x, ) ¾ B R (p) [0, ½). We have Öu u cu («1) u u N n (1 Æ) (N n)c 1 4Æ (1 )R («) A c ÉM(«1) 1 (1 ) where A ((n 1 Ô nk R)c 1 c c 1 )R, c 1, c, Æ are posiive consans wih 0 Æ 1 and e K. Le R ½, we can ge he following global gradien esimaes for he nonlinear parabolic equaion (1.3). Corollary 1.4. Le (M, g) be a complee noncompac n-dimensional Riemannian manifold wih N-Bakry Émery Ricci ensor bounded from below by he consan K, where K 0. Le u be a posiive soluion of (1.3). Then (1) if c 0, we have Öu u cu («1) u u N n 1 (1 Æ) Á () if c 0 and u («1) ÉM for all (x, ) ¾ M [0, ½). We have Öu cu («1) u u u N n («) c ÉM(«1) 1 (1 Æ) (1 ) here 0 Æ 1 and e K. As an applicaion, we ge he following Harnack inequaliy.,,
GRADIENT ESTIMATES FOR A LICHNEROWICZ EQUATION 49 Theorem 1.5. manifold wih Ric N f he equaion Le (M, g) be a complee noncompac n-dimensional Riemannian K, where K 0. Le u(x, ) be a posiive smooh soluion o u f u on M [0, ½). Then for any poins (x 1, 1 ) and (x, ) on M [0, ½) wih 0 1, we have he following Harnack inequaliy: u(x 1, 1 ) u(x, ) 1 (Nn) e (x 1,x, 1, )B, Ê where (x 1, x, 1, ) in f 1 4e K È d, B ((N n))(e K e K 1 ) and is any space ime pah joining (x 1, ) and (x, ). REMARK 1.6. The above Theorem 1.5 has been proved in [6], we can also ge his resul by leing c 0 and Æ 0 in Corollary 1.4. We can refer o [6] for deailed proof.. Proof of Theorem 1.3 Le u be a posiive soluion o (1.3). Se Û ln u, hen Û saisfies he equaion (.1) Û f Û ÖÛ ce Û(«1). Theorem.1. Le (M, g) be a complee noncompac n-dimensional Riemannian manifold wih N-Bakry Émery Ricci ensor bounded from below by he consan K Ï K (R), where R 0 and K (R) 0 in he meric ball B R (p) around p ¾ M. For a smooh funcion Û defined on M [0, ½) saisfies he equaion (.1), we have f F ÖÛ Ö F N n ( 1)ÖÛ F c( «)(«1)e Û(«1) ÖÛ c(«1)e Û(«1) F F, where F ( ÖÛ ce Û(«1) Û ), and e K. Proof. Define F ( ÖÛ ce Û(«1) Û ),
50 L. ZHAO where e K. I is well known ha for he N-Bakry Émery Ricci ensor, we have he Bochner formula: f ÖÛ N n f Û ÖÛÖ( f Û) K ÖÛ. Noicing f Û ( f Û) ÖÛÖÛ c(«1)e Û(«1) Û Û and f Û ÖÛ ce Û(«1) Û 1 1 ( ce Û(«1) Û ) F, we have f F ( f ÖÛ c f e Û(«1) f Û ) and ( f ÖÛ ) c((«1) e Û(«1) ÖÛ («1)e Û(«1) f Û) f Û N n f Û ÖÛÖ( f Û) K ÖÛ c(«1) e Û(«1) ÖÛ ( ce Û(«1) Û ) F N n c(«1)e Û(«1) 1 1 ( ÖÛÖÛ c(«1)e Û(«1) Û Û ) ( 1)ÖÛ F ÖÛÖ F ÖÛÖÛ [( «1)c(«1)e Û(«1) K ]ÖÛ c («1) 1 e Û(«1) 1 c(«1) Û Û c(«1)e Û(«1) F F ( ÖÛ ce Û(«1) Û ) ( ÖÛÖÛ c(«1)e Û(«1) Û Û K ÖÛ ) F ( ÖÛÖÛ c(«1)e Û(«1) Û Û K ÖÛ ).
GRADIENT ESTIMATES FOR A LICHNEROWICZ EQUATION 51 Therefore, i follows ha f F ÖÛÖ F c(«1)e Û(«1) F ÖÛÖ F N n c(«1)e Û(«1) F ÖÛÖ F N n ( 1)ÖÛ F N n [( «1)c(«1)e Û(«1) ]ÖÛ c («1) 1 e Û(«1) 1 c(«1)e Û(«1) Û F ( 1)ÖÛ F (( 1)c(«1)e Û(«1) ) ÖÛ ( «)c(«1)e Û(«1) ÖÛ F ( 1)ÖÛ F ( «)c(«1)e Û(«1) ÖÛ ( 1)c(«1)e Û(«1) F c(«1)e Û(«1) F F ÖÛ Ö F ( 1)ÖÛ F N n ( «)c(«1)e Û(«1) ÖÛ c(«1)e Û(«1) F F. ce Û(«1) 1 Û We complee he proof of Theorem.1.
5 L. ZHAO We ake a C cu-off funcion ɳ defined on [0,½) such ha ɳ(r) 1 for r ¾ [0,1], ɳ(r) 0 for r ¾ [, ½), and 0 ɳ(r) 1. Furhermore ɳ saisfies and ɳ ¼ (r) ɳ 1 (r) c 1 ɳ ¼¼ (r) c for some absolue consans c 1, c 0. Denoe by r(x) he disance beween x and p in M. Se r(x) ³(x) ɳ. R Using an argumen of Cheng and Yau [3], we can assume ³(x) ¾ C (M) wih suppor in B p (R). Direc calculaion shows ha on B p (R) (.) Ö³ ³ c 1 R. I has been shown by Qian [13] ha Hence, we have I follows ha f (r ) n 1 Ö 1 4Kr n f (r) 1 r ( f (r ) Ör ) n r n 1 r n r Ô nk. 1 Ö. 1 4Kr n (.3) f ³ ɳ¼¼ (r)ör R ɳ¼ (r) f r R (n 1 Ô nk R)c 1 c R. For T 0, le (x,s) be a poin in B R (p) [0, T ] a which ³F aains is maximum value P, and we assume ha P is posiive (oherwise he proof is rivial). A he poin (x, s), we have Ö(³F) 0, f (³F) 0, F 0.
GRADIENT ESTIMATES FOR A LICHNEROWICZ EQUATION 53 I follows ha ³ f F F f ³ F³ 1 Ö³ 0. This inequaliy ogeher wih he inequaliies (.) and (.3) yields (.4) ³ f F AF, where A (x, s), by Theorem.1, we have ³ f F ³ÖÛÖ F s³ A (n 1 Ô nk R)c 1 c c 1 R. N n ( 1)ÖÛ F s ( «)c(«1)e Û(«1) ÖÛ where he las inequaliy used c³(«1)e Û(«1) F ³ F s c 1 R ³1 FÖÛ s³ N n c³(«1)e Û(«1) F ³ F s, ( 1)ÖÛ F s ( «)c(«1)e Û(«1) ÖÛ ³ÖÛÖ F FÖÛÖ³ FÖÛ Ö³ c 1 R ³1 FÖÛ. Therefor, by (.4), we obain s³ ( 1)ÖÛ F N n s c 1 R ³1 FÖÛ AF ( «)cs³(«1)e Û(«1) ÖÛ c³(«1)e Û(«1) F ³F s. Following Davies [4] (see also Negrin [1]), we se ÖÛ F.
54 L. ZHAO Then we have ³ (( 1)s 1) F (N n)s c 1 R ³1 1 F 3 AF ( «)cs³(«1)e Û(«1) F c³(«1)e Û(«1) F ³F s. Nex, we consider he following wo cases: (1) c 0; () c 0. (1) When c 0, hen we have ³ (( 1)s 1) F (N n)s c 1 R ³1 1 F 3 AF ³F s, muliplying boh sides of he above inequaliy by s³, we have (( 1)s 1) N n So, i follows ha (³F) c 1 R ³1 1 (³F) 3 As³F ³F Æ (( 1)s 1) (³F) (N n)c1 s N n Æ (( 1)s 1) R ³F As³F ³F. N n (N n)c1 P s (1 Æ) (( 1)s 1) Æ (( 1)s 1) R As 1. Since we ge (( 1)s 1) (1 )s 1 (1 )s, P N n (N n)c 1 s (1 Æ) 4Æ (1 )R As 1. Now, (1) of Theorem 1.3 can be easily deduced from he inequaliy above. () When c 0, hen we have ³ (( 1)s 1) F (N n)s c 1 R ³1 1 F 3 AF ( «)cs³(«1) ÉMF c ÉM(«1)³F ³F s,
GRADIENT ESTIMATES FOR A LICHNEROWICZ EQUATION 55 muliplying boh sides of he above inequaliy by s³, we have (( 1)s 1) (³F) N n c 1 R ³1 1 (³F) 3 As³F ( «)cs ³ («1) ÉMF c ÉM(«1)³s F ³F Æ (( 1)s 1) (³F) (N n)c1 s N n Æ (( 1)s 1) R ³F As³F ( «)cs ³(«1) ÉMF c ÉM(«1)³s F ³F. So, i follows ha P N n (N n)c 1 s (1 Æ) 4Æ (1 )R («) As (1 ) Similarly, we can obain () of Theorem 1.3. c ÉM(«1)s 1. ACKNOWLEDGEMENT. The auhor would like o hank his supervisor Professor Kefeng Liu for his consan encouragemen and help. This work is suppored by he Posdocoral Science Foundaion of China (013M53134), Naional Naural Science Foundaion of China (Gran No. 11101085), Naional Naural Science Foundaion of China (No. 116069) and Naural Science Foundaion of Zhejiang Province of China (LQ13A010018). References [1] L. Chen and W. Chen: Gradien esimaes for a nonlinear parabolic equaion on complee non-compac Riemannian manifolds, Ann. Global Anal. Geom. 35 (009), 397 404. [] L. Chen and W. Chen: Gradien esimaes for posiive smooh f -harmonic funcions, Aca Mah. Sci. Ser. B, Engl. Ed. 30 (010), 1614 1618. [3] S.Y. Cheng and S.T. Yau: Differenial equaions on Riemannian manifolds and heir geomeric applicaions, Comm. Pure Appl. Mah. 8 (1975), 333 354. [4] E.B. Davies: Hea Kernels and Specral Theory, Cambridge Univ. Press, Cambridge, 1989. [5] G.-Y. Huang, H.-Z. Li: Gradien esimaes and enropy formulae of porous medium and fas diffusion equaions for he Wien Laplacian, (01), arxiv:mah.dg/103.548v1. [6] G.-Y. Huang and B.-Q. Ma: Gradien esimaes for a nonlinear parabolic equaion on Riemannian manifolds, Arch. Mah. (Basel) 94 (010), 65 75. [7] X.-D. Li: Liouville heorems for symmeric diffusion operaors on complee Riemannian manifolds, J. Mah. Pures Appl. (9) 84 (005), 195 1361. [8] L. Ma: Gradien esimaes for a simple ellipic equaion on complee non-compac Riemannian manifolds, J. Func. Anal. 41 (006), 374 38. [9] L. Ma: Hamilon ype esimaes for hea equaions on manifolds, (010), arxiv:mah.dg/ 1009.0603v1.
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