抛物型几何偏微分方程的整体解
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摘要
本文研究了定义在黎曼流形上的抛物型偏微分方程整体解的存在性、不存在性及其渐进性。论文由以下四章组成：
第一章为绪论部分。在这一章中我们回顾了反应扩散方程组和Monge-Ampere方程的研究历史及其一些重要的结果,并叙述了本文的主要结果。
第二章研究了定义在非紧致完备黎曼流形上的反应扩散方程组正整体解的存在性和不存在性,并获得了Fujita?型临界指标。
第三章研究了一类定义在紧致完备的黎曼流形上的抛物型Monge-Ampere方程解的渐进行为,通过能量估计和Gronwall方法,得到了对应于B.Huisken文献[]中渐进性更为一般化的结论。
第四章我们引入一类具有旋转不变量的新的几何流,证明了其整体解的存在性,并研究了在这种特殊情况下,沿着这一几何流,随着时间t趋向于无穷大,欧式空间Rn+1(n≥1)中任一封闭收缩光滑的超曲面会在C∞拓扑意义下趋向于一个圆。
This thesis concerns three kinds of geometric partial differential equations defined on Riemannian manifolds. We investigate the existence of global exis-tence solution, the nonexistence of global solution and asymptotic behavior of solution of these equations. This thesis is organized as follows:
In Chapter1, we recall some research history and well known results on reaction-diffusion system, Monge-Ampere equation and state the main results obtained in this thesis.
In Chapter2, we study the global existence and nonexistence of positive solutions to the following nonlinear reaction-diffusion system where Mn (n≥3) is a non-compact complete Riemannian manifold, and obtain a critical exponent of Fujita type.
Chapter3is devoted to the asymptotic behavior of the parabolic Monge-Ampere equation on a compact complete Riemannian manifold. By using the Gronwall inequality and the energy estimates, we obtains asymptotic result of the corresponding problem which generalizes B.Huisken's convergence result in []
In Chapter4, we introduce a new geometric flow with rotational invariance and prove the existence of global solution.we also show that in a special case, under this kind of flow, an arbitrary smooth closed contractible hypersurface in the Euclidean space Rn+1(n≥1) converges to Sn in the C∞-topology as l goes to the infinity
第一章为绪论部分。在这一章中我们回顾了反应扩散方程组和Monge-Ampere方程的研究历史及其一些重要的结果,并叙述了本文的主要结果。
第二章研究了定义在非紧致完备黎曼流形上的反应扩散方程组正整体解的存在性和不存在性,并获得了Fujita?型临界指标。
第三章研究了一类定义在紧致完备的黎曼流形上的抛物型Monge-Ampere方程解的渐进行为,通过能量估计和Gronwall方法,得到了对应于B.Huisken文献[]中渐进性更为一般化的结论。
第四章我们引入一类具有旋转不变量的新的几何流,证明了其整体解的存在性,并研究了在这种特殊情况下,沿着这一几何流,随着时间t趋向于无穷大,欧式空间Rn+1(n≥1)中任一封闭收缩光滑的超曲面会在C∞拓扑意义下趋向于一个圆。
This thesis concerns three kinds of geometric partial differential equations defined on Riemannian manifolds. We investigate the existence of global exis-tence solution, the nonexistence of global solution and asymptotic behavior of solution of these equations. This thesis is organized as follows:
In Chapter1, we recall some research history and well known results on reaction-diffusion system, Monge-Ampere equation and state the main results obtained in this thesis.
In Chapter2, we study the global existence and nonexistence of positive solutions to the following nonlinear reaction-diffusion system where Mn (n≥3) is a non-compact complete Riemannian manifold, and obtain a critical exponent of Fujita type.
Chapter3is devoted to the asymptotic behavior of the parabolic Monge-Ampere equation on a compact complete Riemannian manifold. By using the Gronwall inequality and the energy estimates, we obtains asymptotic result of the corresponding problem which generalizes B.Huisken's convergence result in []
In Chapter4, we introduce a new geometric flow with rotational invariance and prove the existence of global solution.we also show that in a special case, under this kind of flow, an arbitrary smooth closed contractible hypersurface in the Euclidean space Rn+1(n≥1) converges to Sn in the C∞-topology as l goes to the infinity
引文
[1]D.Aronson and H.F.Weinberger, Multidimensional nonlinear diffusion aris-ing in population genetics, Adv.Math.30 (1978),33-76.
[2]T.Aubin. Nonlinear Analysis on Manifolds, Monge-Ampere Equations, Springer-Verlag, New York,1982.
[3]C.Bandle,H.A.Levine,Q.S.Zhang, Critical exponents of Fujita type for inho-mogeneous parabolic equations and systems, J.Math.Anal.Appl.251 (2000), 624-648.
[4]J.Bebernes and D.Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York,1989.
[5]N.Bedjaoui, Ph.Souplet, Critical blow-up exponents for a system of reaction-diffusion equations with absorption, Z.angew.Math.Phys.53 (2002),197-210.
[6]L.Caffarelli, L.Nirenberg, J.Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I:Monge-Ampere equations, Gomrn. Pure Appl. Math.37 (1984),339-402.
[7]L.Caffarelli, L.Nirenberg, J.Spruck, The Dirichlet problem for nonlinear second-order elliptic equations Ⅲ:Functions of the eigenvalues of the Hes-sian, Acta Math.155 (1985),261-301.
[8]H.D.Cao, Deformation of Kahler metrics to Kahler-Einstein metrics on com-pact Kahler manifold, Invent.Math.,81 (1985),359-372.
[9]M.Chlebik, M.Fila, From critical exponents to blow-up rates for parabolic problems, Rend.Math.Appl. Ser.Ⅶ 19 (1999),449-470.
[10]W.-R. Dai, D.-X. Kong & K.-F. Liu, Hyperbolic geometric flow (Ⅰ):short-time existence and nonlinear stability, Pure and Applied Mathematics Quar- terly (Special Issue:In honor of Michael Atiyah and Isadore Singer) 6 (2010),331-359.
[11]P. Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanni-ennes compactes Ⅰ, J.Fund.Anal.,40 (1980) 358-386.
[12]P. Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanni-ennes compactes Ⅱ, J.Funct.Anal.,41 (1981) 341-353.
[13]P. Delanoe,Equations du type Monge-Ampere sur les Varietes Riemanni-ennes compactes Ⅲ, J.Funct.Anal,45 (1982) 403-430.
[14]K.Deng, Blow-up rates for parabolic systems, Z.angew.Math.Phys.47 (1996),132-143.
[15]S. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom.18 (1983),279-315.
[16]M.Escobedo and M.A.Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J.Differential Equations.89 (1991),176-202.
[17]M.Fila, H.A.Levine, On critical exponents for a semilinear parabolic system coupled with an equation and a boundary condition, J.Math. Anal. App.204 (1996),497-521.
[18]M.Fila, P.Quittner, The blow-up rate for a semilinear parabolic systems, J.Math.Anal.App.238 (1999),468-476.
[19]A.Friedman, J.B.Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ.Math.J,34 (1985),425-447.
[20]A.Friedman, M.A.Herrero, Extinction properties of semilinear heat equa-tions with strong absorption, J.Math.Anal.App.,124 (1987),530-546.
[21]H.Fujita, On the blowing up of solutions of the Cauchy problem for ut= Δu+u1+α, J.Fac.Sci.Univ.Tokyo. Sect.I 13 (1966),109-124.
[22]M. Gage and R.. S. Hamilton, The heat equation shrinking convex plane curves,J. Differential Geom.23 (1986),69-96.
[23]A.A.Grigoryan, The heat equation on noncompact Riemannian manifolds, Math. Sb.72 (1992),47-77.
[24]R.S.Hamilton, Three-manifolds with positive Ricci curvature, J.Differential Geometry.17 (1982),225-306.
[25]R.S.Hamilton, Four-manifolds with positive Ricci curvature operator, J. Differential Geometry.24 (1986),153-179.
[26]K.Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad.49 (1973),503-525.
[27]G.Huisken, Flow by mean curvature of convex surface into sphere, J.Differential Geometry.20 (1980),237-266.
[28]J.N. Huang and Z.W. Duan, Existence of the global solution for the parabolic Monge-Ampere equations on compact Riemannian manifold, J.Math.Anal.Appl.,389 (2012) 597-607.
[29]W.Huang, J.Yin, Y.Wang,On critical Fujita exponents for the porus medium equation with a nonlinear boundary condition, J. Math. Anal. App.286 (2003), 369-377.
[30]B.Huisken, Parabolic Monge-Ampere Equations on Riemannian Manifolds J.Funct.Anal,147 (1997),140-163.
[31]A.G.Kartsatos, V.V.Kurta, On a Liouville-type theorem and the Fujita blow-up phenomenon, Proc.Amer.Math.Soc.132 (2003),807-813.
[32]K.Kobayashi,T.Sirao and H.Tanaka, On the blowing up problem for semi-linear heat equations, J.Math.Soc. Japan.29 (1977),407-424.
[33]D.-X. Kong, Hyperbolic geometric flow, the Proceedings of ICCM 2007, Vol. II, Higher Educationial Press, Beijing,2007,95-110.
[34]D.-X. Kong k. K.-F. Liu, Wave character of metrics and hyperbolic geometric flow, J. Math. Phys.48 (2007),103508.
[35]D.-X. Kong, K.-F. Liu & Y.-Z. Wang, Life-span of classical solutions to hyperbolic geometric flow in two space variables with slow decay initial data, Communications in Partial Differential Equations 36 (2011),162-184.
[36]D.-X. Kong, K.-F. Liu & D.-L. Xu, The hyperbolic geometric flow on Rie-mann surfaces, Communications in Partial Differential Equations 34 (2009), 553-580.
[37]D.-X. Kong, Q. Ru, A new geometric flow with rotational invariance, arXiv.1109.0811v1[math.AP].
[38]H.A.Levine, A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equation, Z.angew.Math.Phys., 42 (1992),408-430.
[39]H.A.Levine, The role of critical exponents in blow-up theorems, SIAM Rev, 32 (1990),262-288.
[40]H.A.Levine, A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations, Z.angew.Math.Phys. 42 (1992),408-430.
[41]P.Li,S.T.Yau, On the parabolic kernel of the Schrodinger operator, Acta Math.156 (1986),153-201.
[42]F.C.Li, S.X.Huang,C.H.Xie, Global and blow-up solutions to a nonlocal reaction-diffusion system, Dist.Cont.Dyn.Systems 9 (2003),1519-1532.
[43]L. Songzhe, Existence of solution to initial value problem for a parabolic Monge-Ampere equation and application, Nonlinear Anal,65 (2006) 59-78.
[44]P.Meier, On the critical exponent for reaction-diffusion equations, Arch. Rat. Mech. Anal.109 (1990),63-71.
[45]G. Perelman, The entropy formula for the Ricci flow and its geometric ap-plications, arXiv.math.DG/0211159.
[46]G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109.
[47]G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math.DG/0307245.
[48]Q.Ru, Z.W.Duan, Existence and nonexistence of the global solution for the semilinear parabolic system on Riemannian manifold, Nonlinear Analysis 72 (2010),856-866.
[49]Q.Ru, A critical exponent of Fujita type for a nonlinear reaction-diffusion system on a Riemannian manifold, J. Math. Anal. Appl.401 (2013),488-500.
[50]Q. Ru, A note on the asymptotic behavior of parabolic Monge-Ampere equa-tions on Riemannian manifolds, Journal of Applied Mathematics, vol.2013, Article ID 304864,4 pages,2013. doi:10.1155/2013/304864.
[51]Rios C., Sawyer E.T., Wheeden R.L., Regularity of subelliptic Monge-Ampere equations, Adv. in Math.217(3) (2008),967-1026.
[52]R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys.65 (1979),45-76.
[53]R.Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J.Differential Geometry.20 (1984),479-495.
[54]S.T.Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equations, I, Comm. Pure Appl. Math.,31 (1978) 339-411.
[55]S.T. Yau, Seminar on Differential Geometry, in:Annals of Math. Studies,vol.102. Princeton Univ.Press,Princeton,NJ.1982.
[56]S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A.74 (1977),1798-1799.
[57]S.-T. Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equations I, Comm. Pure Appl. Math.31 (1978),339-411.
[58]Q.S.Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J.97 (1999),515-539.
[59]Q.S.Zhang, A new critical phenomenon for semilinear parabolic problems, J.Math.Anal.Appl 219 (1998),125-139.
[60]Q.S.Zhang, Nonlinear parabolic problems on manifolds and non-existence result of the noncompact Yamabe problem, Elec-tron.Res.Announ.Amer.Math.Soc.3(1997),45-51.
[61]Q.S.Zhang, An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Rice≥0,Math.Ann.316 (2000),703-731.
[62]Q.S.Zhang, Semilinear parabolic problems on manifolds and applications to the non-compact Yamabe problem, Electron. J. Differential Equations 46 (2000),1-30.
[63]Z.T. Zhang and K.L. Wang, Existence and non-existence of solutions for a class of Monge-Ampere equations, Journal of Differential Equations,246 (2009) 2849-2875.
[64]S.Zheng, X.F.Song, Z.X.Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math.Anal. App. 298 (2004),308-324.
[65]陈维桓,微分流形初步,高等教育出版社.2004.
[66]虞言林,指标定理与热方程方法,上海科学技术出版社.1996.
[67]丘成桐,孙理察,微分几何讲义,高等教育出版社.2004.
[68]叶其孝,李正元,反应扩散方程引论,北京：科学出版社.1999.
[69]王明新,非线性抛物型方程,北京：科学出版社.1993.
[2]T.Aubin. Nonlinear Analysis on Manifolds, Monge-Ampere Equations, Springer-Verlag, New York,1982.
[3]C.Bandle,H.A.Levine,Q.S.Zhang, Critical exponents of Fujita type for inho-mogeneous parabolic equations and systems, J.Math.Anal.Appl.251 (2000), 624-648.
[4]J.Bebernes and D.Eberly, Mathematical Problems from Combustion Theory, Springer-Verlag, New York,1989.
[5]N.Bedjaoui, Ph.Souplet, Critical blow-up exponents for a system of reaction-diffusion equations with absorption, Z.angew.Math.Phys.53 (2002),197-210.
[6]L.Caffarelli, L.Nirenberg, J.Spruck, The Dirichlet problem for nonlinear second-order elliptic equations I:Monge-Ampere equations, Gomrn. Pure Appl. Math.37 (1984),339-402.
[7]L.Caffarelli, L.Nirenberg, J.Spruck, The Dirichlet problem for nonlinear second-order elliptic equations Ⅲ:Functions of the eigenvalues of the Hes-sian, Acta Math.155 (1985),261-301.
[8]H.D.Cao, Deformation of Kahler metrics to Kahler-Einstein metrics on com-pact Kahler manifold, Invent.Math.,81 (1985),359-372.
[9]M.Chlebik, M.Fila, From critical exponents to blow-up rates for parabolic problems, Rend.Math.Appl. Ser.Ⅶ 19 (1999),449-470.
[10]W.-R. Dai, D.-X. Kong & K.-F. Liu, Hyperbolic geometric flow (Ⅰ):short-time existence and nonlinear stability, Pure and Applied Mathematics Quar- terly (Special Issue:In honor of Michael Atiyah and Isadore Singer) 6 (2010),331-359.
[11]P. Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanni-ennes compactes Ⅰ, J.Fund.Anal.,40 (1980) 358-386.
[12]P. Delanoe, Equations du type Monge-Ampere sur les Varietes Riemanni-ennes compactes Ⅱ, J.Funct.Anal.,41 (1981) 341-353.
[13]P. Delanoe,Equations du type Monge-Ampere sur les Varietes Riemanni-ennes compactes Ⅲ, J.Funct.Anal,45 (1982) 403-430.
[14]K.Deng, Blow-up rates for parabolic systems, Z.angew.Math.Phys.47 (1996),132-143.
[15]S. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom.18 (1983),279-315.
[16]M.Escobedo and M.A.Herrero, Boundedness and blow up for a semilinear reaction-diffusion system, J.Differential Equations.89 (1991),176-202.
[17]M.Fila, H.A.Levine, On critical exponents for a semilinear parabolic system coupled with an equation and a boundary condition, J.Math. Anal. App.204 (1996),497-521.
[18]M.Fila, P.Quittner, The blow-up rate for a semilinear parabolic systems, J.Math.Anal.App.238 (1999),468-476.
[19]A.Friedman, J.B.Mcleod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ.Math.J,34 (1985),425-447.
[20]A.Friedman, M.A.Herrero, Extinction properties of semilinear heat equa-tions with strong absorption, J.Math.Anal.App.,124 (1987),530-546.
[21]H.Fujita, On the blowing up of solutions of the Cauchy problem for ut= Δu+u1+α, J.Fac.Sci.Univ.Tokyo. Sect.I 13 (1966),109-124.
[22]M. Gage and R.. S. Hamilton, The heat equation shrinking convex plane curves,J. Differential Geom.23 (1986),69-96.
[23]A.A.Grigoryan, The heat equation on noncompact Riemannian manifolds, Math. Sb.72 (1992),47-77.
[24]R.S.Hamilton, Three-manifolds with positive Ricci curvature, J.Differential Geometry.17 (1982),225-306.
[25]R.S.Hamilton, Four-manifolds with positive Ricci curvature operator, J. Differential Geometry.24 (1986),153-179.
[26]K.Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad.49 (1973),503-525.
[27]G.Huisken, Flow by mean curvature of convex surface into sphere, J.Differential Geometry.20 (1980),237-266.
[28]J.N. Huang and Z.W. Duan, Existence of the global solution for the parabolic Monge-Ampere equations on compact Riemannian manifold, J.Math.Anal.Appl.,389 (2012) 597-607.
[29]W.Huang, J.Yin, Y.Wang,On critical Fujita exponents for the porus medium equation with a nonlinear boundary condition, J. Math. Anal. App.286 (2003), 369-377.
[30]B.Huisken, Parabolic Monge-Ampere Equations on Riemannian Manifolds J.Funct.Anal,147 (1997),140-163.
[31]A.G.Kartsatos, V.V.Kurta, On a Liouville-type theorem and the Fujita blow-up phenomenon, Proc.Amer.Math.Soc.132 (2003),807-813.
[32]K.Kobayashi,T.Sirao and H.Tanaka, On the blowing up problem for semi-linear heat equations, J.Math.Soc. Japan.29 (1977),407-424.
[33]D.-X. Kong, Hyperbolic geometric flow, the Proceedings of ICCM 2007, Vol. II, Higher Educationial Press, Beijing,2007,95-110.
[34]D.-X. Kong k. K.-F. Liu, Wave character of metrics and hyperbolic geometric flow, J. Math. Phys.48 (2007),103508.
[35]D.-X. Kong, K.-F. Liu & Y.-Z. Wang, Life-span of classical solutions to hyperbolic geometric flow in two space variables with slow decay initial data, Communications in Partial Differential Equations 36 (2011),162-184.
[36]D.-X. Kong, K.-F. Liu & D.-L. Xu, The hyperbolic geometric flow on Rie-mann surfaces, Communications in Partial Differential Equations 34 (2009), 553-580.
[37]D.-X. Kong, Q. Ru, A new geometric flow with rotational invariance, arXiv.1109.0811v1[math.AP].
[38]H.A.Levine, A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equation, Z.angew.Math.Phys., 42 (1992),408-430.
[39]H.A.Levine, The role of critical exponents in blow-up theorems, SIAM Rev, 32 (1990),262-288.
[40]H.A.Levine, A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations, Z.angew.Math.Phys. 42 (1992),408-430.
[41]P.Li,S.T.Yau, On the parabolic kernel of the Schrodinger operator, Acta Math.156 (1986),153-201.
[42]F.C.Li, S.X.Huang,C.H.Xie, Global and blow-up solutions to a nonlocal reaction-diffusion system, Dist.Cont.Dyn.Systems 9 (2003),1519-1532.
[43]L. Songzhe, Existence of solution to initial value problem for a parabolic Monge-Ampere equation and application, Nonlinear Anal,65 (2006) 59-78.
[44]P.Meier, On the critical exponent for reaction-diffusion equations, Arch. Rat. Mech. Anal.109 (1990),63-71.
[45]G. Perelman, The entropy formula for the Ricci flow and its geometric ap-plications, arXiv.math.DG/0211159.
[46]G. Perelman, Ricci flow with surgery on three-manifolds, arXiv:math.DG/0303109.
[47]G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, arXiv:math.DG/0307245.
[48]Q.Ru, Z.W.Duan, Existence and nonexistence of the global solution for the semilinear parabolic system on Riemannian manifold, Nonlinear Analysis 72 (2010),856-866.
[49]Q.Ru, A critical exponent of Fujita type for a nonlinear reaction-diffusion system on a Riemannian manifold, J. Math. Anal. Appl.401 (2013),488-500.
[50]Q. Ru, A note on the asymptotic behavior of parabolic Monge-Ampere equa-tions on Riemannian manifolds, Journal of Applied Mathematics, vol.2013, Article ID 304864,4 pages,2013. doi:10.1155/2013/304864.
[51]Rios C., Sawyer E.T., Wheeden R.L., Regularity of subelliptic Monge-Ampere equations, Adv. in Math.217(3) (2008),967-1026.
[52]R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys.65 (1979),45-76.
[53]R.Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J.Differential Geometry.20 (1984),479-495.
[54]S.T.Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equations, I, Comm. Pure Appl. Math.,31 (1978) 339-411.
[55]S.T. Yau, Seminar on Differential Geometry, in:Annals of Math. Studies,vol.102. Princeton Univ.Press,Princeton,NJ.1982.
[56]S.-T. Yau, Calabi's conjecture and some new results in algebraic geometry, Proc. Nat. Acad. Sci. U.S.A.74 (1977),1798-1799.
[57]S.-T. Yau, On the Ricci curvature of a compact Kahler manifold and the complex Monge-Ampere equations I, Comm. Pure Appl. Math.31 (1978),339-411.
[58]Q.S.Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J.97 (1999),515-539.
[59]Q.S.Zhang, A new critical phenomenon for semilinear parabolic problems, J.Math.Anal.Appl 219 (1998),125-139.
[60]Q.S.Zhang, Nonlinear parabolic problems on manifolds and non-existence result of the noncompact Yamabe problem, Elec-tron.Res.Announ.Amer.Math.Soc.3(1997),45-51.
[61]Q.S.Zhang, An optimal parabolic estimate and its applications in prescribing scalar curvature on some open manifolds with Rice≥0,Math.Ann.316 (2000),703-731.
[62]Q.S.Zhang, Semilinear parabolic problems on manifolds and applications to the non-compact Yamabe problem, Electron. J. Differential Equations 46 (2000),1-30.
[63]Z.T. Zhang and K.L. Wang, Existence and non-existence of solutions for a class of Monge-Ampere equations, Journal of Differential Equations,246 (2009) 2849-2875.
[64]S.Zheng, X.F.Song, Z.X.Jiang, Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math.Anal. App. 298 (2004),308-324.
[65]陈维桓,微分流形初步,高等教育出版社.2004.
[66]虞言林,指标定理与热方程方法,上海科学技术出版社.1996.
[67]丘成桐,孙理察,微分几何讲义,高等教育出版社.2004.
[68]叶其孝,李正元,反应扩散方程引论,北京：科学出版社.1999.
[69]王明新,非线性抛物型方程,北京：科学出版社.1993.
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