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

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: x(q)mu(t)(x, t) - u(xx) (x, t) = a(2)delta(x - b)f (u(x, t)) for 0 < x < 1, 0 < t less than or equal to T, u(x, 0) = psi(x) for 0 < x < 1, mu(0,t) = mu(1,t)=0 for 0 < t < T, where 6(x) is the Dirac delta function, and f and V) 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.