Single-point blow-up for a degenerate parabolic problem due to a concentrated nonlinear source

Abstract

Abstract.Let q be a nonnegative real number, and T be a positive real number. This article studies the following degenerate semilinear parabolic first initial-boundary value problem:where S(x) is the Dirac delta function, and / and ip are given functions. It is shown that the problem has a unique solution before a blow-up occurs, u blows up in a finite time, and the blow-up set consists of the single point b. A lower bound and an upper bound of the blow-up time are also given. To illustrate our main results, an example is given. A computational method is also given to determine the finite blow-up time.

A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source

“…From Theorem 2.4 of Chan and Tian [3], µ is a nondecreasing function of t. Hence, it follows from Theorem 2.6 of Chan and Tian [3] that for 0 ≤ t ≤ θ, µ (x, t) attains its maximum at (b, θ). Thus given any number M > k 0 , it follows from (2.4) and (2.5) that…”

“…When ψ is sufficiently large and f is sufficiently nonlinear, it follows from Theorem 3.3 of Chan and Tian [3] that there is a position b to place the nonlinear concentrated source such that (1.1) blows up in a finite time. To find a position b so that the solution u exists for all t > 0, let us first consider the problem (1.1) with q = 0, namely,…”

“…From Theorem 3.2 of Chan and Tian [3], the blow-up set is the single point x = b. From Chan and Tian [3],…”

“…Chan and Tian [3] proved that under certain conditions, the solution u of the problem (1.1) blows up in a finite time. For q = 0, they and Olmstead and Roberts [4] showed that if the position b of the concentrated source is sufficiently close to x = 0 or x = 1, then u never blows up.…”

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

“…Using Green's second identity, we obtain u(x,t) = a2 I G(x,t;b,T)f(u(b,r))dT. We modify the techniques in proving Lemma 2.3 and Theorems 2.4 and 2.6 of Chan and Tian [6] to show that the integral equation (3) has a unique nonnegative continuous solution.…”