Skip to main content
×
Home
    • Aa
    • Aa

The form of blow-up for nonlinear parabolic equations

  • A. A. Lacey (a1)
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.

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

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

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

4 D. 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.

5 D. 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.

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

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

8 H. 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.

10 C. 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? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 7 *
Loading metrics...

Abstract views

Total abstract views: 110 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 27th May 2017. This data will be updated every 24 hours.