Quenching for a degenerate parabolic problem due to a concentrated nonlinear source

Abstract

Abstract.Let q, a, T, and b be any real numbers such that q > 0, a > 0, T > 0, and 0 < b < 1. This article studies the following degenerate semilinear parabolic first initial-boundary value problem with a concentrated nonlinear source situated at b:where <5 (x) is the Dirac delta function, / is a given function such that limu^c-f(u) = oo for some positive constant c, and f{u) and f'(u) are positive for 0 < u < c. It is shown that the problem has a unique continuous solution u before m&x{u(x,t) : 0 < x < 1} reaches c~, u is a strictly increasing function of t for 0 < x < 1, and if max{u(x,i) : 0 < x < 1} reaches c~, then u attains the value c only at the point b. The problem is shown to have a unique a* such that a unique global solution u exists for a < a*, and max{u(x,t) : 0 < x < 1} reaches c~ in a finite time for a > a*; this a* is the same as that for q = 0. A formula for computing a* is given, and no quenching in infinite time is deduced.

A quenching criterion for a multi-dimensional parabolic problem due to a concentrated nonlinear source

“…As an illustration, an explosion described by a quenching model occurs at a finite temperature while when described by a blow-up model, it happens at an infinitely high temperature. Chan [2] gave a criterion for the solution u of a one-dimensional parabolic first initial-boundary value problem to quench in a finite time. It turns out that the forcing term f (u) need not be superlinear in u in order for a quenching to occur.…”

Quenching criteria for a parabolic problem due to a concentrated nonlinear source in an infinite strip

“…This proves the lemma. The proof of the following result is similar to that of Theorem 4 of Chan and Jiang [1] for a first initial-boundary value problem (cf. Chan and Kaper [2], and Chan and Tragoonsirisak [4]).…”

A multi-dimensional blow-up problem due to a concentrated nonlinear source in ℝ^{ℕ}

“…We note that the critical value α * is determined as the supremum of all positive values α for which a solution U of (4.2) exists. The proof of the next result (showing that the solution u exists globally when α = α * ) for the case f (0) > 0 is a modification of that for Theorem 7 of Chan and Jiang [1] for a degenerate one-dimensional problem in a bounded domain. 3) where for N = 3,…”

“…License or copyright restrictions may apply to redistribution; see http://www.ams.org/license/jour-dist-license.pdf Lemma 3.1 of Chan and Tragoonsirisak [4] states that for t ≥ 1 and any x ∈B, 1) where ω N denotes the surface area of an N -dimensional unit sphere.…”