||In this thesis, we study two different evolutionary problem. The first one is the nodal point position of the Stieltjes string, which is a discrete version of the vibrating strings after separation of variables. We can express the nodal positions of some special solutions in terms of a continued fraction involving the given data (masses and lengths between masses) of the problem. The solutions are associated with the maximal eigenvalues for Dirichlet-Dirichlet problem and Dirichlet-Neumann problem. Our result is new in literature.|
Another problem is about the profile of the blow-up solution of some nonlinear heat
u_t − Δu = |u|^(p−1) u (x, t) ∈ Ω × (0, T),
with initial boundary conditions u(x, 0) = u0(x) and u(x, t) = 0 when (x, t) ∈ ∂Ω × (0, T), where Ω is a bounded convex domain. Using a sophisticated scaling method, Giga-Kohn (1989) gave a uniform estimate of this blow-up solution. We shall follow B. Hu’s book to give a clear exposition to this result, giving more details in the long
proof. We also show in detail that the uniform estimate result still holds when Ω = R^n, which is unbounded. We hope that this account will be useful for beginning graduate students to understand the method. It is believed that the method is important in the development of the subject.