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Grow-up of critical solutions for a non-local porous medium problem with Ohmic heating source

  • Evangelos A. Latos
  • Dimitrios E. TzanetisEmail author
Article

Abstract

We investigate the behaviour of solution uu(x, t; λ) at λ =  λ* for the non-local porous medium equation \({u_t = (u^n)_{xx} + {\lambda}f(u)/({\int_{-1}^1} f(u){\rm d}x)^2}\) with Dirichlet boundary conditions and positive initial data. The function f satisfies: f(s),−f ′ (s) > 0 for s ≥ 0 and s n-1 f(s) is integrable at infinity. Due to the conditions on f, there exists a critical value of parameter λ, say λ*, such that for λ > λ* the solution u = u(x, t; λ) blows up globally in finite time, while for λ ≥ λ* the corresponding steady-state problem does not have any solution. For 0 < λ < λ* there exists a unique steady-state solution w = w(x; λ) while u = u(x, t; λ) is global in time and converges to w as t → ∞. Here we show the global grow-up of critical solution u* =  u(x, t; λ*) (u* (x, t) → ∞, as t → ∞ for all \({x\in(-1,1)}\).

Mathematics Subject Classification (2000)

Primary 35K55 Secondary 35B05 

Keywords

Non-local parabolic problems Porous medium Grow-up of solutions 

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of Mathematics, School of Applied Mathematical and Physical SciencesNational Technical University of AthensAthensGreece

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