Nonlinear parabolic problems on manifolds, and a nonexistence result for the noncompact Yamabe problem
Author:
Qi S. Zhang
Journal:
Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 4551
MSC (1991):
Primary 35K55; Secondary 58G03
Published electronically:
May 20, 1997
MathSciNet review:
1447067
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Abstract: We study the Cauchy problem for the semilinear parabolic equations on with initial value , where is a Riemannian manifold including the ones with nonnegative Ricci curvature. In the Euclidean case and when , it is well known that is the critical exponent, i.e., if and is smaller than a small Gaussian, then the Cauchy problem has global positive solutions, and if , then all positive solutions blow up in finite time. In this paper, we show that on certain Riemannian manifolds, the above equation with certain conditions on also has a critical exponent. More importantly, we reveal an explicit relation between the size of the critical exponent and geometric properties such as the growth rate of geodesic balls. To achieve the results we introduce a new estimate for related heat kernels. As an application, we show that the wellknown noncompact Yamabe problem (of prescribing constant positive scalar curvature) on a manifold with nonnegative Ricci curvature cannot be solved if the existing scalar curvature decays ``too fast'' and the volume of geodesic balls does not increase ``fast enough''. We also find some complete manifolds with positive scalar curvature, which are conformal to complete manifolds with positive constant and with zero scalar curvatures. This is a new phenomenon which does not happen in the compact case.
 [A]
D.
G. Aronson, Nonnegative solutions of linear parabolic
equations, Ann. Scuola Norm. Sup. Pisa (3) 22 (1968),
607–694. MR 0435594
(55 #8553)
 [Au]
T.
Aubin, The scalar curvature, Differential geometry and
relativity, Reidel, Dordrecht, 1976, pp. 5–18. Mathematical
Phys. and Appl. Math., Vol. 3. MR 0433500
(55 #6476)
 [AM]
Patricio
Aviles and Robert
C. McOwen, Conformal deformation to constant negative scalar
curvature on noncompact Riemannian manifolds, J. Differential Geom.
27 (1988), no. 2, 225–239. MR 925121
(89b:58225)
 [D]
E.
B. Davies, Heat kernels and spectral theory, Cambridge Tracts
in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989.
MR 990239
(90e:35123)
 [Fu]
Hiroshi
Fujita, On the blowing up of solutions of the Cauchy problem for
𝑢_{𝑡}=Δ𝑢+𝑢^{1+𝛼}, J.
Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124
(1966). MR
0214914 (35 #5761)
 [G]
A.
A. Grigor′yan, The heat equation on noncompact Riemannian
manifolds, Mat. Sb. 182 (1991), no. 1,
55–87 (Russian); English transl., Math. USSRSb. 72
(1992), no. 1, 47–77. MR 1098839
(92h:58189)
 [H]
Richard
S. Hamilton, Threemanifolds with positive Ricci curvature, J.
Differential Geom. 17 (1982), no. 2, 255–306.
MR 664497
(84a:53050)
 [K]
Jerry
L. Kazdan, Prescribing the curvature of a Riemannian manifold,
CBMS Regional Conference Series in Mathematics, vol. 57, Published for
the Conference Board of the Mathematical Sciences, Washington, DC; by the
American Mathematical Society, Providence, RI, 1985. MR 787227
(86h:53001)
 [Le]
Howard
A. Levine, The role of critical exponents in blowup theorems,
SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055
(91j:35135), http://dx.doi.org/10.1137/1032046
 [LN]
TzongYow
Lee and WeiMing
Ni, Global existence, large time behavior
and life span of solutions of a semilinear parabolic Cauchy
problem, Trans. Amer. Math. Soc.
333 (1992), no. 1,
365–378. MR 1057781
(92k:35134), http://dx.doi.org/10.1090/S00029947199210577816
 [LSU]
O.
A. Ladyženskaja, V.
A. Solonnikov, and N.
N. Ural′ceva, Linear and quasilinear equations of parabolic
type, Translated from the Russian by S. Smith. Translations of
Mathematical Monographs, Vol. 23, American Mathematical Society,
Providence, R.I., 1968 (Russian). MR 0241822
(39 #3159b)
 [LY]
Peter
Li and ShingTung
Yau, On the parabolic kernel of the Schrödinger operator,
Acta Math. 156 (1986), no. 34, 153–201. MR 834612
(87f:58156), http://dx.doi.org/10.1007/BF02399203
 [Me]
Peter
Meier, On the critical exponent for reactiondiffusion
equations, Arch. Rational Mech. Anal. 109 (1990),
no. 1, 63–71. MR 1019169
(90k:35038), http://dx.doi.org/10.1007/BF00377979
 [Mo]
Jürgen
Moser, A Harnack inequality for parabolic differential
equations, Comm. Pure Appl. Math. 17 (1964),
101–134. MR 0159139
(28 #2357)
 [Ni]
Wei
Ming Ni, On the elliptic equation
Δ𝑢+𝐾(𝑥)𝑢^{(𝑛+2)/(𝑛2)}=0,
its generalizations, and applications in geometry, Indiana Univ. Math.
J. 31 (1982), no. 4, 493–529. MR 662915
(84e:35049), http://dx.doi.org/10.1512/iumj.1982.31.31040
 [Sa]
L.
SaloffCoste, A note on Poincaré, Sobolev, and Harnack
inequalities, Internat. Math. Res. Notices 2 (1992),
27–38. MR
1150597 (93d:58158), http://dx.doi.org/10.1155/S1073792892000047
 [Sc]
Richard
Schoen, Conformal deformation of a Riemannian metric to constant
scalar curvature, J. Differential Geom. 20 (1984),
no. 2, 479–495. MR 788292
(86i:58137)
 [Sh]
WanXiong
Shi, Deforming the metric on complete Riemannian manifolds, J.
Differential Geom. 30 (1989), no. 1, 223–301.
MR
1001277 (90i:58202)
 [Tr]
Neil
S. Trudinger, Remarks concerning the conformal deformation of
Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup.
Pisa (3) 22 (1968), 265–274. MR 0240748
(39 #2093)
 [Yam]
Hidehiko
Yamabe, On a deformation of Riemannian structures on compact
manifolds, Osaka Math. J. 12 (1960), 21–37. MR 0125546
(23 #A2847)
 [Yau]
Shing
Tung Yau (ed.), Seminar on Differential Geometry, Annals of
Mathematics Studies, vol. 102, Princeton University Press, Princeton,
N.J.; University of Tokyo Press, Tokyo, 1982. Papers presented at seminars
held during the academic year 1979–1980. MR 645728
(83a:53002)
 [Zhao]
Z.
Zhao, On the existence of positive solutions of nonlinear elliptic
equations—a probabilistic potential theory approach, Duke Math.
J. 69 (1993), no. 2, 247–258. MR 1203227
(94c:35090), http://dx.doi.org/10.1215/S001270949306913X
 [Zhang1]
Qi
Zhang, On a parabolic equation with a
singular lower order term, Trans. Amer. Math.
Soc. 348 (1996), no. 7, 2811–2844. MR 1360232
(96k:35073), http://dx.doi.org/10.1090/S0002994796016753
 [Zhang2]
Qi Zhang, On a parabolic equation with a singular lower order term, Part II, Indiana U. Math. J., to appear.
 [Zhang3]
Qi Zhang, Global existence and local continuity of solutions for semilinear parabolic equations, Comm. PDE, to appear.
 [Zhang4]
Qi Zhang, The critical exponent of a reactiondiffusion equation on some Lie groups, Math. Z., to appear.
 [Zhang5]
Qi Zhang, Semilinear parabolic problems on manifolds and applications to the noncompact Yamabe problem, preprint.
 [A]
 D.G. Aronson, Nonnegative solutions of linear parabolic equations, Annali della Scuola Norm. Sup. Pisa 22 (1968), 607694. MR 55:8553
 [Au]
 T. Aubin, The scalar curvature, Differential geometry and relativity, edited by Cahen and Flato, Reidel, Dordrecht, 1976, pp. 518. MR 55:6476
 [AM]
 P. Aviles and R. C. McOwen, Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds, J. Diff. Geometry 27 (1988), 225239. MR 89b:58225
 [D]
 E. B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press, Cambridge and New York, 1989. MR 90e:35123
 [Fu]
 H. Fujita, On the blowing up of solutions of the Cauchy problem for , J. Fac. Sci. Univ. Tokyo, Sect. I 13 (1966), 109124. MR 35:5761
 [G]
 A. A. Grigoryan, The heat equation on noncompact Riemannian manifolds, Mat. Sb. 182 (1991), 5587; English transl., Math. USSR Sb. 72 (1992), 4777. MR 92h:58189
 [H]
 R. Hamilton, Threemanifolds with positive Ricci curvature, J. Diff. Geom. 17 (1982), 255306. MR 84a:53050
 [K]
 J. Kazdan, Prescribing the curvature of a Riemannian manifold, Amer. Math. Soc., Providence, RI, 1985. MR 86h:53001
 [Le]
 H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review 32 (1990), 269288. MR 91j:35135
 [LN]
 TzongYow Lee and WeiMing Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Transactions AMS 333 (1992), 365378. MR 92k:35134
 [LSU]
 O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Uraltseva, Linear and quasilinear equations of parabolic type, Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, RI, 1968. MR 39:3159b
 [LY]
 P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153201. MR 87f:58156
 [Me]
 P. Meier, On the critical exponent for reactiondiffusion equations, Arch. Rat. Mech. Anal. 109 (1990), 6371. MR 90k:35038
 [Mo]
 J. Moser, A Harnack inequality for parabolic differential equations, Comm. Pure and Appl. Math. 17 (1964), 101134. MR 28:2357
 [Ni]
 W. M. Ni, On the elliptic equation , its generalizations and applications in geometry, Indiana Univ. Math. J. 31 (1982), 493529. MR 84e:35049
 [Sa]
 L. SaloffCoste, A note on Poincaré, Sobolev and Harnack inequality, IMRN, Duke Math. J. 2 (1992), 2738. MR 93d:58158
 [Sc]
 R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geometry 20 (1984), 479495. MR 86i:58137
 [Sh]
 W.X. Shi, Deforming the metric on complete Riemannian manifolds, J. Diff. Geometry 30 (1989), 225301. MR 90i:58202
 [Tr]
 N. Trudinger, Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1968), 265274. MR 39:2093
 [Yam]
 H. Yamabe, On a deformation of Riemannian structures on compact manifolds, Osaka Math. J. 12 (1960), 2137. MR 23:A2847
 [Yau]
 S. T. Yau, Seminar on differential geometry, Annals of Math. Studies, vol. 102, Princeton Univ. Press, Princeton, NJ 1982. MR 83a:53002
 [Zhao]
 Z. Zhao, On the existence of positive solutions of nonlinear elliptic equationsA probabilistic potential theory approach, Duke Math. J. 69 (1993), 247258. MR 94c:35090
 [Zhang1]
 Qi Zhang, On a parabolic equation with a singular lower order term, Transactions of AMS 348 (1996), 28112844. MR 96k:35073
 [Zhang2]
 Qi Zhang, On a parabolic equation with a singular lower order term, Part II, Indiana U. Math. J., to appear.
 [Zhang3]
 Qi Zhang, Global existence and local continuity of solutions for semilinear parabolic equations, Comm. PDE, to appear.
 [Zhang4]
 Qi Zhang, The critical exponent of a reactiondiffusion equation on some Lie groups, Math. Z., to appear.
 [Zhang5]
 Qi Zhang, Semilinear parabolic problems on manifolds and applications to the noncompact Yamabe problem, preprint.
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Additional Information
Qi S. Zhang
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211
Email:
sz@mumathnx6.math.missouri.edu
DOI:
http://dx.doi.org/10.1090/S107967629700022X
PII:
S 10796762(97)00022X
Received by editor(s):
February 19, 1997
Published electronically:
May 20, 1997
Communicated by:
Richard Schoen
Article copyright:
© Copyright 1997
American Mathematical Society
