BLOW-UP FOR SEMIDISCRETIZATION OF A LOCALIZED SEMILINEAR HEAT EQUATION

This paper concerns the study of the numerical approximation for the following initial-boundary value problem: { u(t)(x, t) - u(xx)(x, t) + epsilon f(u(0, t)), (x, t) is an element of (-l, l) x (0, T), u(- l, t) = 0, u(l, t) = 0, t is an element of (0, T), u(x, 0) = u(0)(x) >= 0, x is an element of (-l, l), where f : [0, infinity) -> [0, infinity) is a C-2 convex, nondecreasing function, integral(infinity) d sigma/f(sigma) < infinity, l = 1/2 and epsilon is a positive parameter. Under some assumptions, we prove that the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also show that the semidiscrete blow-up time in certain cases converges to the real one when the mesh size tends to zero. Finally, we give some numerical experiments to illustrate our analysis.