## International Journal of Differential Equations

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

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

**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

#### 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.Zentralblatt MATH: 0163.34002 - K. Hayakawa, “On nonexistence of global solutions of some semilinear parabolic differential equations,”
*Proceedings of the Japan Academy*, vol. 49, pp. 503–505, 1973.Zentralblatt MATH: 0281.35039 - 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.Zentralblatt MATH: 0876.35048 - 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.Zentralblatt MATH: 0892.35088 - 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.Zentralblatt MATH: 1010.35005 - 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.Zentralblatt MATH: 1212.35225 - 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.Zentralblatt MATH: 0735.35013 - 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.Zentralblatt MATH: 0822.35068 - 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.Zentralblatt MATH: 0913.35065 - Y. Yamauchi, “Blow-up results for a reaction-diffusion system,”
*Methods and Applications of Analysis*, vol. 13, no. 4, pp. 337–349, 2006.Zentralblatt MATH: 1157.35054

Mathematical Reviews (MathSciNet): MR2384258

Digital Object Identifier: doi:10.4310/MAA.2006.v13.n4.a2

Project Euclid: euclid.maa/1202219967 - 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.Zentralblatt MATH: 0942.35025 - H. A. Levine, “The role of critical exponents in blowup theorems,”
*SIAM Review*, vol. 32, no. 2, pp. 262–288, 1990.Zentralblatt MATH: 0706.35008

Mathematical Reviews (MathSciNet): MR1056055

Digital Object Identifier: doi:10.1137/1032046 - 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.Zentralblatt MATH: 0785.35011 - 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.Zentralblatt MATH: 0909.35056

Mathematical Reviews (MathSciNet): MR1638070

Digital Object Identifier: doi:10.1137/S0036141097324934 - 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.Zentralblatt MATH: 1176.35010

Mathematical Reviews (MathSciNet): MR2503017

Digital Object Identifier: doi:10.1016/j.jde.2009.01.035 - 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.Zentralblatt MATH: 1100.35044

Mathematical Reviews (MathSciNet): MR2214937

Digital Object Identifier: doi:10.1016/j.jde.2005.08.009 - 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.Zentralblatt MATH: 0278.35052 - 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.Zentralblatt MATH: 0872.35046 - 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.Zentralblatt MATH: 0924.35055 - S. Kaplan, “On the growth of solutions of quasi-linear parabolic equations,”
*Communications on Pure and Applied Mathematics*, vol. 16, pp. 305–330, 1963.Zentralblatt MATH: 0156.33503 - 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.Zentralblatt MATH: 1225.35034

Mathematical Reviews (MathSciNet): MR2756074

Digital Object Identifier: doi:10.1016/j.jde.2010.12.008 - 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.Zentralblatt MATH: 1106.35029 - 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.Zentralblatt MATH: 1050.35031 - 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.Zentralblatt MATH: 0936.35034 - 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.Zentralblatt MATH: 0805.35065 - 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.Zentralblatt MATH: 1215.35091 - 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.Zentralblatt MATH: 0914.35056 - 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.Zentralblatt MATH: 1144.35030

Mathematical Reviews (MathSciNet): MR2386440

Digital Object Identifier: doi:10.1016/j.jmaa.2007.05.033 - 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.Zentralblatt MATH: 1167.35393 - 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.Zentralblatt MATH: 1221.35076

Mathematical Reviews (MathSciNet): MR2810683

Digital Object Identifier: doi:10.1016/j.na.2011.04.064 - 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.Zentralblatt MATH: 0996.35006

### More like this

- Fujita type results for a class of degenerate parabolic operators

Pascucci, Andrea, Advances in Differential Equations, 1999 - Quasilinear parabolic equations with localized reaction

Fukuda, Isamu and Suzuki, Ryuichi, Advances in Differential Equations, 2005 - BLOW-UP SOLUTIONS TO THE NONLINEAR SECOND ORDER DIFFERENTIAL
EQUATION u

Li, Meng-Rong, Taiwanese Journal of Mathematics, 2008

- Fujita type results for a class of degenerate parabolic operators

Pascucci, Andrea, Advances in Differential Equations, 1999 - Quasilinear parabolic equations with localized reaction

Fukuda, Isamu and Suzuki, Ryuichi, Advances in Differential Equations, 2005 - BLOW-UP SOLUTIONS TO THE NONLINEAR SECOND ORDER DIFFERENTIAL
EQUATION u

Li, Meng-Rong, Taiwanese Journal of Mathematics, 2008 - ON THE DIFFERENTIAL EQUATION $u'' - u^{p} = 0$

Li, Meng-Rong, Taiwanese Journal of Mathematics, 2005 - Local solvability of a fully nonlinear parabolic equation

Akagi, Goro, Kodai Mathematical Journal, 2014 - Life span of solutions for a quasilinear parabolic equation with initial data having positive limit inferior at infinity

Igarashi, Takefumi, Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2013 - Blow-Up Analysis for a Quasilinear Degenerate Parabolic Equation with Strongly Nonlinear Source

Zheng, Pan, Mu, Chunlai, Liu, Dengming, Yao, Xianzhong, and Zhou, Shouming, Abstract and Applied Analysis, 2012 - Global solutions of higher-order semilinear parabolic equations in the supercritical
range

Egorov, Yu. V., Galaktionov, V. A., Kondratiev, V. A., and Pohozaev, S. I., Advances in Differential Equations, 2004 - Numerical Blow-Up Time for a Semilinear Parabolic Equation with Nonlinear Boundary Conditions

Assalé, Louis A., Boni, Théodore K., and Nabongo, Diabate, Journal of Applied Mathematics, 2008 - Critical Fujita exponents for a coupled non-Newtonian filtration system

Sining Zheng and Miaoqing Tian, Sining Zheng and Miaoqing Tian, Advances in Differential Equations, 2010