On the blow-up of the non-local thermistor problem

Abstract

The conditions under which the solution of the non-local thermistor problem ut = Δu + λf(u)/(∫Ωf(u)dx) 2, x ε Ω ⊂ ℝN, N≥ 2, t > 0, ∂u(x, t)/∂ν + β(x)u(x, t) = 0, x ε ∂Ω, t > 0, u(x, 0) = u0(x), x ε Ω, blows up are investigated. We assume that f(s) is a decreasing function and that it is integrable in (0, ∞). Considering a suitable functional we prove that for all λ > 0 the solution of the Neumann problem blows up in finite time. The same result is obtained for the Robin problem under the assumption that λ is sufficiently large (λ ≫ 1). In the proof of existence of blow-up for the Dirichlet problem we use the subsolution technique. We are able to construct a blowing-up lower solution under the assumption that either λ > λ* or 0 < λ < λ*, for some critical value λ*, and that the initial condition is sufficiently large provided also that f(s) satisfies the decay condition ∫0∞[sf(s) - s2f′(s)] ds < ∞.