A blow-up criterion for a degenerate parabolic problem due to a concentrated nonlinear source
by
R. Boonklurb
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010 USA
Coauthors: C. Y. Chan, Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010 USA

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,

xq ut (x, t) - uxx (x, t) = ad(x-b) f(u(x, t)) for 0 < x < 1, 0 < t ≤ T,

u(x, 0) = y(x) for 0 ≤ x ≤ 1,

u(0, t) = u(1, t) = 0 for 0 < t ≤ T,

where d(x) is the Dirac delta function, and f and y are given functions. It is shown that there exists a unique number b^* ∈ ( 0, 1/2) such that u never blows up for b ∈ ( 0, b^*] ∪[ 1-b^*, 1) , and u always blows up in a finite time for b ∈ (b^*, 1-b^*). To illustrate our main results, two examples are given.

Date received: March 26, 2007