Blow-up rates for parabolic systems

  • Keng Deng
Original Papers

Abstract

Let Ω ⊂ ℝ n be a bounded domain andB R be a ball in ℝ n of radiusR. We consider two parabolic systems: utu +f(υ), υi=Δυ +g(u) in Ω × (0,T) withu=v=0 on δΩ × (0,T) andu t =Δu, v t =Δv inB r × (0,T) withδe/δv=f (v), δe/δv=g(u) onδB R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    H. Amann,Parabolic equations and nonlinear boundary conditions, J. Diff. Eqs. 72, 201–269 (1988).Google Scholar
  2. [2]
    G. Caristi and E. Mitidieri,Blow-up estimates of positive solutions of a parabolic system, J. Diff. Eqs.113, 265–271 (1994).Google Scholar
  3. [3]
    K. Deng,The blow-up behavior of the heat equation with Neumann boundary conditions, J. Math. Anal. Appl.188, 641–650 (1994).Google Scholar
  4. [4]
    K. Deng,Global existence and blow-up for a system of heat equations with nonlinear boundary conditions, Math. Methods Appl. Sci.18, 307–315 (1995).Google Scholar
  5. [5]
    J. Escher,Global existence and nonexistence for semilinear parabolic systems with nonlinear boundary conditions, Math. Ann.284, 285–306 (1989).Google Scholar
  6. [6]
    M. Fila,Boundedness of global solutions for the heat equation with nonlinear boundary conditions, Comment. Math. Univ. Carolinae30, 479–484 (1989).Google Scholar
  7. [7]
    A. Friedman and Y. Giga,A single point blow-up for solutions of semilinear parabolic systems, J. Fac. Sci. Univ. Tokyo Sec. IA Math.34, 65–79 (1987).Google Scholar
  8. [8]
    A. Friedman and B. Mcleod,Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J.34, 425–447 (1985).Google Scholar
  9. [9]
    M. G. Garroni and J. L. Menaldi,Green Functions For Second Order Parabolic Integro-differential Problems, Longman Scientific and Technical, New York 1992.Google Scholar
  10. [10]
    B. Gidas, W.-M. Ni and L. Nirenberg,Symmetry and related properties via the maximum principle, Comm. Math. Phys.68, 209–243 (1979).Google Scholar
  11. [11]
    B. Hu and H. M. Yin.The profile near blow-up time for solution of the heat equation with a nonlinear boundary condition, Trans. Amer. Math. Soc.346, 117–136 (1994).Google Scholar
  12. [12]
    H. A. Levine and G. M. Lieberman,Quenching of solutions of parabolic equations with nonlinear boundary conditions in several dimensions, J. Reine Ang. Math. 345, 23–38 (1983).Google Scholar
  13. [13]
    H. A. Levine and L. Payne,Nonexistence theorems for the heat equations with nonlinear boundary conditions and for the porous medium equation backward in time, J. Diff. Eqs.16, 319–334 (1974).Google Scholar
  14. [14]
    C. V. Pao,Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York 1992.Google Scholar

Copyright information

© Birkhäuser Verlag 1996

Authors and Affiliations

  • Keng Deng
    • 1
  1. 1.Department of MathematicsUniversity of Southwestern LouisianaLafayetteUSA

Personalised recommendations