To The Memory of My Father ,

This thesis is concerned with the study of the Blow-up phenomena for parabolic problems, which can be defined in a basic way as the inability to continue the solutions up to or after a finite time, the so called blow-up time. Namely, we consider the blow-up location in space and its rate estimates, for special cases of the following types of problems: (i) Dirichlet problems for semilinear equations, (ii) Neumann problems for heat equations, (iii) Neumann problems for semilinear equations, (iv) Dirichlet (Cauchy) problems for semilinear equations with gradient terms. For problems of type (i), (ii), we extend some known blow-up results of parabolic problems with power and exponential type nonlinearities to problems with nonlinear terms, which grow faster than these types of functions for large values of solutions. Moreover, under certain conditions, some blow-up results of the single semilinear heat equation are extended to the coupled systems of two semilinear heat equations. For problems of type (iii), we study how the reaction terms and the nonlinear boundary terms affect the blow-up properties of the blow-up solutions of these problems. The noninfluence of the gradient terms on the blow-up bounds is showed for problems of type (iv). The study of blow-up is considerably more difficult and interesting when the equations involved are PDEs, and indeed, it has become both a kind of industry and an art. Prof. Juan Luis Vazquez, [64]