Skip to main content
×
Home
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 20
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Li, Lin-Lin Sun, Hong-Rui and Zhang, Quan-Guo 2016. Existence and Nonexistence of Global Solutions for a Semilinear Reaction-Diffusion System. Journal of Mathematical Analysis and Applications,


    Beck, Mélanie Gander, Martin J. and Kwok, Felix 2015. B-Methods for the Numerical Solution of Evolution Problems with Blow-Up Solutions Part I: Variation of the Constant. SIAM Journal on Scientific Computing, Vol. 37, Issue. 6, p. A2998.


    Nie, Yuanyuan Zhou, Qian Zhou, Mingjun and Xu, Xiaoli 2015. Quenching Phenomenon of a Singular Semilinear Parabolic Problem. Journal of Dynamical and Control Systems, Vol. 21, Issue. 1, p. 81.


    Li, Zhongping and Du, Wanjuan 2014. Life span and secondary critical exponent for degenerate and singular parabolic equations. Annali di Matematica Pura ed Applicata, Vol. 193, Issue. 2, p. 501.


    Zhu, Liping and Zhang, Zhengce 2014. Non-Self-Similar Dead-Core Rate for the Fast Diffusion Equation with Dependent Coefficient. Abstract and Applied Analysis, Vol. 2014, p. 1.


    Lindsay, Alan E. 2013. An asymptotic study of blow up multiplicity in fourth order parabolic partial differential equations. Discrete and Continuous Dynamical Systems - Series B, Vol. 19, Issue. 1, p. 189.


    Ru, Qiang and Duan, Zhi-wen 2010. Existence and nonexistence of the global solution for the semilinear parabolic system on Riemannian manifold. Nonlinear Analysis: Theory, Methods & Applications, Vol. 72, Issue. 2, p. 856.


    Giga, Yoshikazu Seki, Yukihiro and Umeda, Noriaki 2009. Mean Curvature Flow Closes Open Ends of Noncompact Surfaces of Rotation. Communications in Partial Differential Equations, Vol. 34, Issue. 11, p. 1508.


    Seki, Yukihiro 2008. On directional blow-up for quasilinear parabolic equations with fast diffusion. Journal of Mathematical Analysis and Applications, Vol. 338, Issue. 1, p. 572.


    Ma, Zhongtai and Wen, Guochun 2007. Iterative approximation of solutions for semilinear parabolic equations system. Journal of Computational and Applied Mathematics, Vol. 209, Issue. 2, p. 167.


    Giga, Yoshikazu and Umeda, Noriaki 2006. On blow-up at space infinity for semilinear heat equations. Journal of Mathematical Analysis and Applications, Vol. 316, Issue. 2, p. 538.


    Budd, C. J. Galaktionov, V. A. and Chen, Jianping 1998. Focusing blow-up for quasilinear parabolic equations. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 128, Issue. 05, p. 965.


    King, J.R. 1994. Asymptotic results for nonlinear outdiffusion. European Journal of Applied Mathematics, Vol. 5, Issue. 03,


    Hu, Bei 1992. A nonlinear nonlocal parabolic equation for channel flow. Nonlinear Analysis: Theory, Methods & Applications, Vol. 18, Issue. 10, p. 973.


    Stuart, A. M. and Floater, M. S. 1990. On the computation of blow-up. European Journal of Applied Mathematics, Vol. 1, Issue. 01, p. 47.


    Chipot, M. and Weissler, F. B. 1989. Some Blowup Results for a Nonlinear Parabolic Equation with a Gradient Term. SIAM Journal on Mathematical Analysis, Vol. 20, Issue. 4, p. 886.


    Bebernes, J. and Troy, W. 1987. Nonexistence for the Kassoy Problem. SIAM Journal on Mathematical Analysis, Vol. 18, Issue. 4, p. 1157.


    Bebernes, J. and Troy, W. 1987. Existence for the modified Kassoy problem. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 106, Issue. 1-2, p. 131.


    Friedman, Avner and Lacey, Andrew A. 1987. The Blow-Up Time for Solutions of Nonlinear Heat Equations with Small Diffusion. SIAM Journal on Mathematical Analysis, Vol. 18, Issue. 3, p. 711.


    Lacey, A. A. 1986. Global blow-up of a nonlinear heat equation. Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Vol. 104, Issue. 1-2, p. 161.


    ×
  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 98, Issue 1-2
  • January 1984, pp. 183-202

The form of blow-up for nonlinear parabolic equations

  • A. A. Lacey (a1)
  • DOI: http://dx.doi.org/10.1017/S0308210500025609
  • Published online: 14 November 2011
Abstract
Synopsis

Semilinear parabolic equations of the form u1 = ∇2u + δf(u), where f is positive and is finite, are known to exhibit the phenomenon of blow-up, i.e. for sufficiently large S, u becomes infinite after a finite time t*. We consider one-dimensional problems in the semi-infinite region x>0 and find the time to blow-up (t*). Also, the limiting behaviour of u as t→t*- and x→∞ is determined; in particular, it is seen that u blows up at infinity, i.e. for any given finite x, u is bounded as t→t*. The results are extended to problems with convection.

The modified equation xu, = uxx +f(u) is discussed. This shows the possibility of blow-up at x =0 even if u(0, f) = 0. The manner of blow-up is estimated.

Finally, bounds on the time to blow-up for problems in finite regions are obtained by comparing u with upper and lower solutions.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

1R. O. Ayeni . On the thermal runaway of variable viscosity flows between concentric cylinders. Z. Angew. Math. Phys. 33 (1982), 408413.

2H. Fujita . On the nonlinear equations Δu+eu = 0 and ut =Δu +eu. Bull. Amer. Math. Soc. 75 (1969), 132135.

3A. K. Kapila . Reactive-diffusive system with Arrhenius kinetics: dynamics of ignition. SIAM J. Appl. Math. 39 (1980), 2136.

4D. R. Kassoy and J. Poland . The thermal explosion confined by a constant temperature boundary: I The induction period solution. SIAM J. Appl. Math. 39 (1980). 412430.

5D. R. Kassoy and J. Poland . The thermal explosion confined by a constant temperature boundary: II The extremely rapid transient. SIAM J. Appl. Math. 41 (1981), 231246.

6A. A. Lacey . The spatial dependence of supercritical reacting systems. IMA J. Appl. Math. 27 (1981). 7184.

7A. A. Lacey . Mathematical analysis of thermal runaway for spatially inhomogeneous reactions.SIAMJ. Appl. Math. 43 (1983), 13501366.

8H. A. Levine . Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: the method of unbounded Fourier coefficients. Math. Ann. 214 (1975), 205220.

10C. V. Pao . Nonexistence of global solutions and bifurcation analysis of a boundary-value problemof parabolic type. Proc. Amer. Math. Soc. 65 (1977), 245251.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Proceedings of the Royal Society of Edinburgh Section A: Mathematics
  • ISSN: 0308-2105
  • EISSN: 1473-7124
  • URL: /core/journals/proceedings-of-the-royal-society-of-edinburgh-section-a-mathematics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×