Quenching Behavior of Parabolic Problems with Localized Reaction Term

Let p, q, T be positive real numbers, B = {x ∈ R n : ∥x∥ < 1}, ∂B = {x ∈ R n : ∥x∥ = 1}, x ∗ ∈ B, △ be the Laplace operator in R n . In this paper, the following the initial boundary value problem with localized reaction term is studied: ut(x, t) = ∆u(x, t) + 1 (1 − u(x, t)) p + 1 (1 − u(x ∗ , t)) q , (x, t) ∈ B × (0, T ), u(x, 0) = u0(x), x ∈ B, where u0 ≥ 0. The existence of the unique classical solution is established. When x ∗ = 0, quenching criteria is given. Moreover, the rate of change of the solution at the quenching point near the quenching time is studied.