Nonlinear parabolic problems on manifolds, and a nonexistence result for the noncompact Yamabe problem

Zhang, Qi¹

¹ Department of Mathematics, University of Missouri, Columbia, MO 65211

Electronic research announcements of the American Mathematical Society, Tome 03 (1997), pp. 45-51

Résumé

We study the Cauchy problem for the semilinear parabolic equations $\Delta u - R u + u^{p} - u_{t} =0$ on $\mathbf {M}^{n} \times (0, \infty )$ with initial value $u_{0} \ge 0$, where $\mathbf {M}^{n}$ is a Riemannian manifold including the ones with nonnegative Ricci curvature. In the Euclidean case and when $R=0$, it is well known that $1+ \frac {2}{n}$ is the critical exponent, i.e., if $p > 1 + \frac {2}{n}$ and $u_{0}$ is smaller than a small Gaussian, then the Cauchy problem has global positive solutions, and if $p1+\frac {2}{n}$, 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 $R$ 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 well-known 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.

Détail

DOI : 10.1090/S1079-6762-97-00022-X

Bibliographie

[1] Aronson, D. G. Non-negative solutions of linear parabolic equations Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 1968 607 694

[2] Aubin, T. The scalar curvature 1976 5 18

[3] Aviles, Patricio, Mcowen, Robert C. Conformal deformation to constant negative scalar curvature on noncompact Riemannian manifolds J. Differential Geom. 1988 225 239

[4] Davies, E. B. Heat kernels and spectral theory 1989

[5] Fujita, Hiroshi On the blowing up of solutions of the Cauchy problem for 𝑢_{𝑡} J. Fac. Sci. Univ. Tokyo Sect. I 1966

[6] Grigor′Yan, A. A. The heat equation on noncompact Riemannian manifolds Mat. Sb. 1991 55 87

[7] Hamilton, Richard S. Three-manifolds with positive Ricci curvature J. Differential Geometry 1982 255 306

[8] Kazdan, Jerry L. Prescribing the curvature of a Riemannian manifold 1985

[9] Levine, Howard A. The role of critical exponents in blowup theorems SIAM Rev. 1990 262 288

[10] Lee, Tzong-Yow, Ni, Wei-Ming Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem Trans. Amer. Math. Soc. 1992 365 378

[11] Ladyženskaja, O. A., Solonnikov, V. A., Ural′Ceva, N. N. Lineĭ nye i kvazilineĭ nye uravneniya parabolicheskogo tipa 1967 736

[12] Li, Peter, Yau, Shing-Tung On the parabolic kernel of the Schrödinger operator Acta Math. 1986 153 201

[13] Meier, Peter On the critical exponent for reaction-diffusion equations Arch. Rational Mech. Anal. 1990 63 71

[14] Moser, Jürgen A Harnack inequality for parabolic differential equations Comm. Pure Appl. Math. 1964 101 134

[15] Ni, Wei Ming On the elliptic equation Δ𝑢+𝐾(𝑥)𝑢^{(𝑛+2)/(𝑛-2)} Indiana Univ. Math. J. 1982 493 529

[16] Saloff-Coste, L. A note on Poincaré, Sobolev, and Harnack inequalities Internat. Math. Res. Notices 1992 27 38

[17] Schoen, Richard Conformal deformation of a Riemannian metric to constant scalar curvature J. Differential Geom. 1984 479 495

[18] Shi, Wan-Xiong Deforming the metric on complete Riemannian manifolds J. Differential Geom. 1989 223 301

[19] Trudinger, Neil S. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 1968 265 274

[20] Yamabe, Hidehiko On a deformation of Riemannian structures on compact manifolds Osaka Math. J. 1960 21 37

[21] Seminar on Differential Geometry 1982

[22] Zhao, Z. On the existence of positive solutions of nonlinear elliptic equations—a probabilistic potential theory approach Duke Math. J. 1993 247 258

[23] Zhang, Qi On a parabolic equation with a singular lower order term Trans. Amer. Math. Soc. 1996 2811 2844