International Journal of Differential Equations

Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations

Yusuke Yamauchi

Full-text: Open access

Abstract

Since 1960's, the blow-up phenomena for the Fujita type parabolic equation have been investigated by many researchers. In this survey paper, we discuss various results on the life span of positive solutions for several superlinear parabolic problems. In the last section, we introduce a recent result by the author.

Article information

Source
Int. J. Differ. Equ., Volume 2012, Special Issue (2012), Article ID 417261, 16 pages.

Dates
Received: 15 December 2011
Accepted: 30 January 2012
First available in Project Euclid: 25 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485313353

Digital Object Identifier
doi:10.1155/2012/417261

Mathematical Reviews number (MathSciNet)
MR2909474

Zentralblatt MATH identifier
1250.35123

Citation

Yamauchi, Yusuke. Life Span of Positive Solutions for the Cauchy Problem for the Parabolic Equations. Int. J. Differ. Equ. 2012, Special Issue (2012), Article ID 417261, 16 pages. doi:10.1155/2012/417261. https://projecteuclid.org/euclid.ijde/1485313353


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References

  • H. Fujita, “On the blowing up of solutions of the Cauchy problem for ${u}_{t}=\Delta u+{u}^{1+\alpha }$,” Journal of the Faculty of Science. University of Tokyo. Section IA, vol. 13, pp. 109–124, 1966.
  • K. Hayakawa, “On nonexistence of global solutions of some semilinear parabolic differential equations,” Proceedings of the Japan Academy, vol. 49, pp. 503–505, 1973.
  • K. Kobayashi, T. Sirao, and H. Tanaka, “On the growing up problem for semilinear heat equations,” Journal of the Mathematical Society of Japan, vol. 29, no. 3, pp. 407–424, 1977.
  • F. B. Weissler, “Existence and nonexistence of global solutions for a semilinear heat equation,” Israel Journal of Mathematics, vol. 38, no. 1-2, pp. 29–40, 1981.
  • R. G. Pinsky, “Existence and nonexistence of global solutions for ${u}_{t}=\Delta u+a(x){u}^{p}$ in ${R}^{d}$,” Journal of Differential Equations, vol. 133, no. 1, pp. 152–177, 1997.
  • Y.-W. Qi, “The critical exponents of parabolic equations and blow-up in R$^{n}$,” Proceedings of the Royal Society of Edinburgh A, vol. 128, no. 1, pp. 123–136, 1998.
  • Y.-W. Qi and M.-X. Wang, “Critical exponents of quasilinear parabolic equations,” Journal of Mathe-matical Analysis and Applications, vol. 267, no. 1, pp. 264–280, 2002.
  • J. Aguirre and M. Escobedo, “A Cauchy problem for ${u}_{t}-\Delta u={u}^{p}$ with $0<p<1$. Asymptotic behaviour of solutions,” Annales de la faculté des sciences de Toulouse. Mathématiques, vol. 8, no. 2, pp. 175–203, 1986/87.
  • Y. Aoyagi, K. Tsutaya, and Y. Yamauchi, “Global existence of solutions for a reaction-diffusion sys-tem,” Differential and Integral Equations, vol. 20, no. 12, pp. 1321–1339, 2007.
  • M. Escobedo and M. A. Herrero, “Boundedness and blow up for a semilinear reaction-diffusion system,” Journal of Differential Equations, vol. 89, no. 1, pp. 176–202, 1991.
  • M. Escobedo and H. A. Levine, “Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations,” Archive for Rational Mechanics and Analysis, vol. 129, no. 1, pp. 47–100, 1995.
  • K. Mochizuki and Q. Huang, “Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations,” Methods and Applications of Analysis, vol. 5, no. 2, pp. 109–124, 1998.
  • Y. Yamauchi, “Blow-up results for a reaction-diffusion system,” Methods and Applications of Analysis, vol. 13, no. 4, pp. 337–349, 2006.
  • K. Deng and H. A. Levine, “The role of critical exponents in blow-up theorems: the sequel,” Journal of Mathematical Analysis and Applications, vol. 243, no. 1, pp. 85–126, 2000.
  • H. A. Levine, “The role of critical exponents in blowup theorems,” SIAM Review, vol. 32, no. 2, pp. 262–288, 1990.
  • T.-Y. Lee and W.-M. Ni, “Global existence, large time behavior and life span of solutions of a semi-linear parabolic Cauchy problem,” Transactions of the American Mathematical Society, vol. 333, no. 1, pp. 365–378, 1992.
  • N. Mizoguchi and E. Yanagida, “Blowup and life span of solutions for a semilinear parabolic equa-tion,” SIAM Journal on Mathematical Analysis, vol. 29, no. 6, pp. 1434–1446, 1998.
  • T. Cazenave, F. Dickstein, and F. B. Weissler, “Global existence and blowup for sign-changing solu-tions of the nonlinear heat equation,” Journal of Differential Equations, vol. 246, no. 7, pp. 2669–2680, 2009.
  • F. Dickstein, “Blowup stability of solutions of the nonlinear heat equation with a large life span,” Journal of Differential Equations, vol. 223, no. 2, pp. 303–328, 2006.
  • H. A. Levine, “Some nonexistence and instability theorems for solutions of formally parabolic equa-tions of the form $P{u}_{t}=-Au+f(u)$,” Archive for Rational Mechanics and Analysis, vol. 51, pp. 371–386, 1973.
  • N. Mizoguchi and E. Yanagida, “Critical exponents for the blow-up of solutions with sign changes in a semilinear parabolic equation,” Mathematische Annalen, vol. 307, no. 4, pp. 663–675, 1997.
  • N. Mizoguchi and E. Yanagida, “Critical exponents for the blowup of solutions with sign changes in a semilinear parabolic equation. II,” Journal of Differential Equations, vol. 145, no. 2, pp. 295–331, 1998.
  • S. Kaplan, “On the growth of solutions of quasi-linear parabolic equations,” Communications on Pure and Applied Mathematics, vol. 16, pp. 305–330, 1963.
  • Y. Fujishima and K. Ishige, “Blow-up for a semilinear parabolic equation with large diffusion on R$^{N}$,” Journal of Differential Equations, vol. 250, no. 5, pp. 2508–2543, 2011.
  • Y. Fujishima and K. Ishige, “Blow-up for a semilinear parabolic equation with large diffusion on R$^{N}$. II,” Journal of Differential Equations, vol. 252, no. 2, pp. 1835–1861, 2012.
  • Y. Giga, Y. Seki, and N. Umeda, “Blow-up at space infinity for nonlinear heat equations,” in Recent Advances in Nonlinear Analysis, pp. 77–94, World Scientific, Hackensack, NJ, USA, 2008.
  • Y. Giga and N. Umeda, “On blow-up at space infinity for semilinear heat equations,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 538–555, 2006.
  • Y. Giga and N. Umeda, “Blow-up directions at space infinity for solutions of semilinear heat equa-tions,” Boletim da Sociedade Paranaense de Matemática. 3rd Série, vol. 23, no. 1-2, pp. 9–28, 2005.
  • C. Gui and X. Wang, “Life spans of solutions of the Cauchy problem for a semilinear heat equation,” Journal of Differential Equations, vol. 115, no. 1, pp. 166–172, 1995.
  • Y. Kobayashi, “Behavior of the life span for solutions to the system of reaction-diffusion equations,” Hiroshima Mathematical Journal, vol. 33, no. 2, pp. 167–187, 2003.
  • K. Mukai, K. Mochizuki, and Q. Huang, “Large time behavior and life span for a quasilinear parabolic equation with slowly decaying initial values,” Nonlinear Analysis: Theory, Methods & Applications, vol. 39, no. 1, pp. 33–45, 2000.
  • K. Mochizuki and R. Suzuki, “Blow-up sets and asymptotic behavior of interfaces for quasilinear degenerate parabolic equations in R$^{N}$,” Journal of the Mathematical Society of Japan, vol. 44, no. 3, pp. 485–504, 1992.
  • T. Ozawa and Y. Yamauchi, “Life span of positive solutions for a semilinear heat equation with general non-decaying initial data,” Journal of Mathematical Analysis and Applications, vol. 379, no. 2, pp. 518–523, 2011.
  • R. G. Pinsky, “The behavior of the life span for solutions to ${u}_{t}=\Delta u+a(x){u}^{p}$ in ${R}^{d}$,” Journal of Differential Equations, vol. 147, no. 1, pp. 30–57, 1998.
  • Y. Seki, “On directional blow-up for quasilinear parabolic equations with fast diffusion,” Journal of Mathematical Analysis and Applications, vol. 338, no. 1, pp. 572–587, 2008.
  • Y. Seki, N. Umeda, and R. Suzuki, “Blow-up directions for quasilinear parabolic equations,” Proceedings of the Royal Society of Edinburgh A, vol. 138, no. 2, pp. 379–405, 2008.
  • Y. Yamauchi, “Life span of solutions for a semilinear heat equation with initial data having positive limit inferior at infinity,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 5008–5014, 2011.
  • M. Yamaguchi and Y. Yamauchi, “Life span of positive solutions for a semilinear heat equation with non-decaying initial data,” Differential and Integral Equations, vol. 23, no. 11-12, pp. 1151–1157, 2010.
  • N. Mizoguchi and E. Yanagida, “Life span of solutions with large initial data in a semilinear parabolic equation,” Indiana University Mathematics Journal, vol. 50, no. 1, pp. 591–610, 2001.