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Thermal runaway in a nonlocal problem modelling Ohmic heating. II: General proof of blow-up and asymptotics of runaway. (English) Zbl 0849.35058

Summary: We consider the nonlocal problem (with homogeneous Dirichlet boundary conditions) \[ u_t= u_{xx}+ \lambda f(u)\Biggl/ \Biggl( \int^1_{-1} f(u) dx\Biggr)^2,\quad -1< x< 1. \] It is found that for the case of decreasing \(f\) then: (i) for \(\int^\infty_0 f(s) ds= \infty\) there is a unique steady state which is globally asymptotically stable; (ii) for \(\int^\infty_0 f(s)ds< \infty\) then the problem can be scaled so that \(\int^\infty_0 f(s) ds= 1\) in which case: (a) for \(\lambda< 8\) there is a unique steady state which is globally asymptotically stable; (b) for \(\lambda= 8\) there is no steady state and \(u\) is unbounded; (c) for \(\lambda> 8\) there is no steady state and \(u\) blows up for all \(x\), \(- 1< x< 1\). Some formal asymptotic estimates for the local behaviour of \(u\) as it blows up are obtained.
For Part I, see [ibid., No. 2, 127–144 (1994; Zbl 0843.35008)].

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
45K05 Integro-partial differential equations
78A35 Motion of charged particles

Citations:

Zbl 0843.35008
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References:

[1] Lacey, Euro. J. Appl. Math. 6 pp 127– (1994)
[2] Cimatti, Quart. J. Appl. Math. 47 pp 117– (1989)
[3] DOI: 10.1137/0143090 · Zbl 0543.35047
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