Blow-up of Radially Symmetric Solutions of a Non-local Problem Modelling Ohmic Heating




Tzanetis, Dimitrios E.

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Southwest Texas State University, Department of Mathematics


We consider a non-local initial boundary-value problem for the equation ut = ∆u + λƒ(u) / (∫Ω ƒ(u) dx)2, x ∈ Ω ⊂ ℝ2, t > 0, where u represents a temperature and ƒ is a positive and decreasing function. It is shown that for the radically symmetric case, if ∫∞0 ƒ(s) ds < ∞ then there exists a critical value λ* > 0 such that for λ < λ* there is no stationary solution and u blows up, whereas for λ < λ* there exists at least one stationary solution. Moreover, for the Dirichlet problem with -s ƒ'(s) < ƒ(s) there exists a unique stationary solution which is asymptotically stable. For the Robin problem, if λ < λ* then there are at least two solutions, which if λ = λ* at least one solution. Stability and blow-up of these solutions are examined in this article.



Nonlocal parabolic equations, Blow-up, Global existence, Steady states


Tzanetis, D. E. (2002). Blow-up of radially symmetric solutions of a non-local problem modelling Ohmic heating. <i>Electronic Journal of Differential Equations, 2002</i>(11), pp. 1-26.


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