## 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

### Keywords:

thermal runaway; nonlocal problem; blow-up; Ohmic heating

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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