About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 532935, 12 pages
http://dx.doi.org/10.1155/2013/532935
Research Article

Global Existence and Uniform Energy Decay Rates for the Semilinear Parabolic Equation with a Memory Term and Mixed Boundary Condition

School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China

Received 14 July 2013; Accepted 18 September 2013

Academic Editor: Shaoyong Lai

Copyright © 2013 Zhong Bo Fang and Liru Qiu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Z. Tan, “The reaction-diffusion equation with Lewis function and critical Sobolev exponent,” Journal of Mathematical Analysis and Applications, vol. 272, no. 2, pp. 480–495, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  2. G. Da Prato and M. Iannelli, “Existence and regularity for a class of integrodifferential equations of parabolic type,” Journal of Mathematical Analysis and Applications, vol. 112, no. 1, pp. 36–55, 1985. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  3. A. Friedman, The IMA Volumes in Mathematics and Its Applications, vol. 49 of Mathematics in Industrial Problems, Springer, New York, NY, USA, 1992.
  4. J. A. Nohel, “Nonlinear Volterra equations for heat flow in materials with memory,” in Integral and Functional Differential Equations, T. L. Herdman, H. W. Stech, and S. M. Rankin, Eds., vol. 67 of Lecture Notes in Pure and Applied Mathematics, pp. 3–82, Marcel Dekker, New York, NY, USA, 1981. View at Zentralblatt MATH
  5. H. M. Yin, “On parabolic Volterra equations in several space dimensions,” SIAM Journal on Mathematical Analysis, vol. 22, no. 6, pp. 1723–1737, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. H. M. Yin, “Weak and classical solutions of some nonlinear Volterra integro-differential equations,” Communications in Partial Differential Equations, vol. 17, pp. 1369–1385, 1992. View at Publisher · View at Google Scholar
  7. H. A. Levine, “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put=Au+F(u),” Archive for Rational Mechanics and Analysis, vol. 51, no. 5, pp. 371–386, 1973. View at Publisher · View at Google Scholar · View at Scopus
  8. V. K. Kalantarov and O. A. Ladyzhenskaya, “The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types,” Journal of Soviet Mathematics, vol. 10, no. 1, pp. 53–70, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  9. H. A. Levine, S. R. Park, and J. Serrin, “Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type,” Journal of Differential Equations, vol. 142, no. 1, pp. 212–229, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. P. Pucci and J. Serrin, “Asymptotic stability for nonlinear parabolic systems,” in Energy Methods in Continuum Mechanics, pp. 66–74, Kluwer Academic Publishers, Dodrecht, The Netherlands, 1996. View at Zentralblatt MATH
  11. S. Berrimi and S. A. Messaoudi, “A decay result for a quasilinear parabolic system,” Progress in Nonlinear Differential Equations and Their Applications, vol. 53, pp. 43–50, 2005. View at Zentralblatt MATH
  12. S. A. Messaoudi, “Blow-up of solutions of a semilinear heat equation with a Visco-elastic term,” Progress in Nonlinear Differential Equations and Their Applications, vol. 64, pp. 351–356, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Z. B. Fang and L. Sun, “Blow up of solutions with positive initial energy for the nonlocal semilinear heat equation,” The KSIAM Journal, vol. 16, no. 4, pp. 235–242, 2012.
  14. S. A. Messaoudi and B. Tellab, “A general decay result in a quasilinear parabolic system with viscoelastic term,” Applied Mathematics Letters, vol. 25, no. 3, pp. 443–447, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus