| Summary: | 博士 === 國立臺灣師範大學 === 數學研究所 === 92 === In this thesis, we first study an initial value problem for a semilinear ordinary differential equation. This problem is closely related to the blow-up behaviour of a semilinear parabolic equation. Under some restriction, we characterize the structure of solutions and derive the uniqueness of positive global solution of this initial value problem.
Next, we stusy a nonlinear parabolic equation with a superlinear reaction term. By studying the backward self-similar
solutions for this equation, we construct a finite number of
self-similar single-point blow-up patterns with different
oscillations.
Finally, we study a boundary value problem for a third order differential equation which arises in the study of self-similar solutions of the steady free convection problem for a vertical heated impermeable flat plate embedded in a porous medium.
We consider the structure of solutions of the initial value problem for this third order differential equation. First, we classify the solutions into 6 different types. Then, by transforming the third order equation into a second order equation, with the help of some comparison principle we are able to derive the structure of solutions. This answers some of the open questions proposed by Belhachmi, Brighi, and Taous in 2001. To obtain a further distinctions of the solution structure, we introduce a new change of variables to
transform the third order equation into a system of two first order equations. Then by the phase plane analysis we can obtain more information on the structure of solutions.
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