Abstract and Applied Analysis

On the Stability of Heat Equation

Balázs Hegyi and Soon-Mo Jung

Full-text: Open access


We prove the generalized Hyers-Ulam stability of the heat equation, Δ u = u t , in a class of twice continuously differentiable functions under certain conditions.

Article information

Abstr. Appl. Anal., Volume 2013, Special Issue (2013), Article ID 202373, 4 pages.

First available in Project Euclid: 26 February 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Hegyi, Balázs; Jung, Soon-Mo. On the Stability of Heat Equation. Abstr. Appl. Anal. 2013, Special Issue (2013), Article ID 202373, 4 pages. doi:10.1155/2013/202373. https://projecteuclid.org/euclid.aaa/1393448672

Export citation


  • C. Alsina and R. Ger, “On some inequalities and stability results related to the exponential function,” Journal of Inequalities and Applications, vol. 2, no. 4, pp. 373–380, 1998.
  • S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, USA, 2002.
  • D. H. Hyers, “On the stability of the linear functional equation,” Proceedings of the National Academy of Sciences of the United States of America, vol. 27, pp. 222–224, 1941.
  • D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkhäuser, Boston, Mass, USA, 1998.
  • S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, NY, USA, 2011.
  • T. M. Rassias, “On the stability of the linear mapping in Banach spaces,” Proceedings of the American Mathematical Society, vol. 72, no. 2, pp. 297–300, 1978.
  • S. M. Ulam, Problems in Modern Mathematics, John Wiley & Sons, New York, NY, USA, 1964.
  • M. Obłoza, “Hyers stability of the linear differential equation,” Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, no. 13, pp. 259–270, 1993.
  • M. Obłoza, “Connections between Hyers and Lyapunov stability of the ordinary differential equations,” Rocznik Naukowo-Dydaktyczny. Prace Matematyczne, no. 14, pp. 141–146, 1997.
  • T. Miura, S.-M. Jung, and S.-E. Takahasi, “Hyers-Ulam-Rassias stability of the Banach space valued linear differential equations ${y}^{\prime }=\lambda y$,” Journal of the Korean Mathematical Society, vol. 41, no. 6, pp. 995–1005, 2004.
  • S.-E. Takahasi, T. Miura, and S. Miyajima, “On the Hyers-Ulam stability of the Banach space-valued differential equation ${y}^{\prime }=\lambda y$,” Bulletin of the Korean Mathematical Society, vol. 39, no. 2, pp. 309–315, 2002.
  • S.-M. Jung and K.-S. Lee, “Hyers-Ulam stability of first order linear partial differential equations with constant coefficients,” Mathematical Inequalities & Applications, vol. 10, no. 2, pp. 261–266, 2007.
  • A. Prástaro and T. M. Rassias, “Ulam stability in geometry of PDE's,” Nonlinear Functional Analysis and Applications, vol. 8, no. 2, pp. 259–278, 2003.
  • N. Lungu and D. Popa, “Hyers-Ulam stability of a first order partial differential equation,” Journal of Mathematical Analysis and Applications, vol. 385, no. 1, pp. 86–91, 2012.
  • B. Hegyi and S.-M. Jung, “On the stability of Laplace's equation,” Applied Mathematics Letters, vol. 26, no. 5, pp. 549–552, 2013.
  • L. C. Evans, Partial Differential Equations, vol. 19 of Graduate Studies in Mathematics, American Mathematical Society, 1998.
  • S.-M. Jung, “Hyers-Ulam stability of linear differential equations of first order. II,” Applied Mathematics Letters, vol. 19, no. 9, pp. 854–858, 2006.