Asymptotic Behavior of a Nonlocal Parabolic Problem in Ohmic Heating Process
نویسندگان
چکیده
In this paper, we consider the asymptotic behavior of the nonlocal parabolic problem ut = ∆u+ λf(u) ( ∫ Ω f(u)dx )p , x ∈ Ω, t > 0, with homogeneous Dirichlet boundary condition, where λ > 0, p > 0, f is nonincreasing. It is found that: (a) For 0 < p ≤ 1, u(x, t) is globally bounded and the unique stationary solution is globally asymptotically stable for any λ > 0; (b) For 1 < p < 2, u(x, t) is globally bounded for any λ > 0; (c) For p = 2, if 0 < λ < 2|∂Ω|2, then u(x, t) is globally bounded, if λ = 2|∂Ω|2, there is no stationary solution and u(x, t) is a global solution and u(x, t) → ∞ as t → ∞ for all x ∈ Ω, if λ > 2|∂Ω|2, there is no stationary solution and u(x, t) blows up in finite time for all x ∈ Ω; (d) For p > 2, there exists a λ∗ > 0 such that for λ > λ∗, or for 0 < λ ≤ λ∗ and u0(x) sufficiently large, u(x, t) blows up in finite time. Moreover, some formal asymptotic estimates for the behavior of u(x, t) as it blows up are obtained for p ≥ 2.
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