The parabolic Cauchy problem and quenching

This article treats the Cauchy problem for the equation u(t) = Deltau + f(t, x, u) with initial values u(0, x) = phi (x) in R-N with an eye to the study of qu enching phenomena. Among the results are an existence theorem under the sol e assumption of continuity of f, existence of maximal and minimal solutions without a monotonicity assumption regarding f, an extension of the results on growth of solutions and on uniqueness that were obtained by Aguirre, Es cobedo, and Herrero in [1] and [3] for power functions f(u) = u(p) with 0 < p < 1 to a larger class of nonlinearities, using a new technique, and furt hermore, new results on the behavior at infinity of the solution in just on e fixed direction when the behavior of the initial function in this directi on is known. The results are extended to parabolic systems, and some applic ations to quenching problems are discussed.