## Duke Mathematical Journal

### Blow-up results for nonlinear parabolic equations on manifolds

Qi S. Zhang

#### Article information

Source
Duke Math. J. Volume 97, Number 3 (1999), 515-539.

Dates
First available in Project Euclid: 19 February 2004

http://projecteuclid.org/euclid.dmj/1077228801

Digital Object Identifier
doi:10.1215/S0012-7094-99-09719-3

Mathematical Reviews number (MathSciNet)
MR1682987

Zentralblatt MATH identifier
0954.35029

#### Citation

Zhang, Qi S. Blow-up results for nonlinear parabolic equations on manifolds. Duke Math. J. 97 (1999), no. 3, 515--539. doi:10.1215/S0012-7094-99-09719-3. http://projecteuclid.org/euclid.dmj/1077228801.

#### References

• [AW] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math. 30 (1978), no. 1, 33–76.
• [AH] Thierry Aubin, Nonlinear analysis on manifolds. Monge-Ampère equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 252, Springer-Verlag, New York, 1982.
• [BP] Pierre Baras and Michel Pierre, Critère d'existence de solutions positives pour des équations semi-linéaires non monotones, Ann. Inst. H. Poincaré Anal. Non Linéaire 2 (1985), no. 3, 185–212.
• [BCN] H. Berestycki, I. Capuzzo-Dolcetta, and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal. 4 (1994), no. 1, 59–78.
• [BCC] I. Birindelli, I. Capuzzo Dolcetta, and A. Cutrì, Liouville theorems for semilinear equations on the Heisenberg group, Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), no. 3, 295–308.
• [DK] Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of the initial-Dirichlet problem for generalized porous medium equations in cylinders, J. Amer. Math. Soc. 1 (1988), no. 2, 401–412.
• [D] E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1990.
• [F] Hiroshi Fujita, On the blowing up of solutions of the Cauchy problem for $u\sbt=\Delta u+u\sp1+\alpha$, J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124 (1966).
• [Ga1] V. A. Galaktionov, Conditions for the absence of global solutions of a class of quasilinear parabolic equations, Zh. Vychisl. Mat. i Mat. Fiz. 22 (1982), no. 2, 322–338, 492.
• [Ga2] Victor A. Galaktionov, Blow-up for quasilinear heat equations with critical Fujita's exponents, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 3, 517–525.
• [GKMS] V. A. Galaktionov, S. P. Kurdjumov, A. P. Mihaĭ lov, and A. A. Samarskiĭ, On unbounded solutions of the Cauchy problem for the parabolic equation $u\sbt=\nabla (u\sp\sigma \nabla u)+u\sp\beta$, Dokl. Akad. Nauk SSSR 252 (1980), no. 6, 1362–1364.
• [GL] Victor A. Galaktionov and Howard A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal. 34 (1998), no. 7, 1005–1027.
• [Gr] A. A. Grigoryan, The heat equation on noncompact Riemannian manifolds, Mat. Sb. 182 (1991), no. 1, 55–87.
• [H] Kantaro Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad. 49 (1973), 503–505.
• [KV] N. J. Kalton and I. E. Verbitsky, Nonliner equations and weighted norm inequality, to appear in Trans. Amer. Math. Soc.
• [Ka] Tadashi Kawanago, Existence and behaviour of solutions for $u\sb t=\Delta(u\sp m)+u\sp l$, Adv. Math. Sci. Appl. 7 (1997), no. 1, 367–400.
• [Ki] Seongtag Kim, The Yamabe problem and applications on noncompact complete Riemannian manifolds, Geom. Dedicata 64 (1997), no. 3, 373–381.
• [KST] Kusuo Kobayashi, Tunekiti Sirao, and Hiroshi Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan 29 (1977), no. 3, 407–424.
• [LN] Tzong-Yow Lee and Wei-Ming 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.
• [Le] Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288.
• [LS] Howard A. Levine and Paul E. Sacks, Some existence and nonexistence theorems for solutions of degenerate parabolic equations, J. Differential Equations 52 (1984), no. 2, 135–161.
• [LY] Peter Li and Shing-Tung Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), no. 3-4, 153–201.
• [Li] Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co. Inc., River Edge, NJ, 1996.
• [Me] Peter Meier, On the critical exponent for reaction-diffusion equations, Arch. Rational Mech. Anal. 109 (1990), no. 1, 63–71.
• [MM] Kiyoshi Mochizuki and Kentaro Mukai, Existence and nonexistence of global solutions to fast diffusions with source, Methods Appl. Anal. 2 (1995), no. 1, 92–102.
• [MS] Kiyoshi Mochizuki and Ryuichi Suzuki, Critical exponent and critical blow-up for quasilinear parabolic equations, Israel J. Math. 98 (1997), 141–156.
• [Ni] Wei Ming Ni, On the elliptic equation $\Delta u+K(x)u\sp(n+2)/(n-2)=0$, its generalizations, and applications in geometry, Indiana Univ. Math. J. 31 (1982), no. 4, 493–529.
• [P] Ross G. Pinsky, Existence and nonexistence of global solutions for $u\sb t=\Delta u+a(x)u\sp p$ in $\bf R\sp d$, J. Differential Equations 133 (1997), no. 1, 152–177.
• [Q] Yuan-Wei Qi, On the equation $u\sb t=\Delta u\sp \alpha+u\sp \beta$, Proc. Roy. Soc. Edinburgh Sect. A 123 (1993), no. 2, 373–390.
• [Sa] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices (1992), no. 2, 27–38.
• [SGKM] A. A. Samarskii, V. A. Galaktionov, S. P. Kurdyumov, and A. P. Mikhailov, Blow-up in Quasilinear Parabolic Equations (in Russian), Nauka, Moscow, 1987.
• [SY] R. Schoen and S.-T. Yau, Lectures on differential geometry, Conference Proceedings and Lecture Notes in Geometry and Topology, I, International Press, Cambridge, MA, 1994.
• [Z1] Qi S. Zhang, Nonlinear parabolic problems on manifolds, and a nonexistence result for the noncompact Yamabe problem, Electron. Res. Announc. Amer. Math. Soc. 3 (1997), 45–51 (electronic).
• [Z2] Qi S. Zhang, A new critical phenomenon for semilinear parabolic problems, J. Math. Anal. Appl. 219 (1998), no. 1, 125–139.