This paper considers asymptotic approximations to the solutions of the semilinear parabolic equation: u_t=triangle open u+f(u), (0.1) where the function f(u) is such that the solution to (0.1) blows up in a finite time T_b. In order to control the explosive behavior of this problem, we consider a "perturbation" to (0.1) defined by: u~#epsilon#_t=triangle open u~#epsilon#+f(u~#epsilon#)v~(#epsilon#), v~#epsilon#_t=triangle open v~#epsilon#-#epsilon#f(u~#epsilon#)v~#epsilon#, (0.2) where #epsilon# is a small positive number. The boundary and initial conditions on u~#epsilon# are those of u. For v~#epsilon#, the initial and boundary conditions are chosen to be 1. Note that system (0.2) belongs to a class of coupled semilinear parabolic equations, with positive solutions and "mass control" property, (see Ref. 10). The solution {u~#epsilon#, v~#epsilon#} of such systems is known to be global. As such, (0.2) appears to be a regular perturbation to a singular problem (0.1). In this work, our basic theorem is a convergence proof for u~#epsilon# and u~#epsilon#_t to u and u_t, respectively, in the L~(infinity) norm. These results constitute a framework for designing in subsequent work, numerical algorithms for the computation of blow-up times (see Ref. 6).
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