A study on the qualitative theory of solutions for some parabolic equations with nonlinear boundary conditions

This thesis is devoted to a study on the qualitative theory of solutions for some parabolic equations with nonlinear boundary conditions of the radiation type. In part I, we consider Fujita-type equations with nonlinear boundary conditions of the radiation type, which are prototypes of nonlinear heat equations. We show not only the local well-posedness but also the uniform boundedness of global solutions, the comparison principle, the critical phenomena associated with the non-existence of global solutions, and the structural stability. Due to the nonlinearity of boundary conditions, since the argument using integral equations for semi-linear problems does not work well, we here rely on the theory of nonlinear evolution equations associated with subdifferential operators. In part II, we are concerned with a reaction-diffusion system arising from a nuclear reactor. We here study the qualitative properties of the solution as in Part I. In particular, as an application of the comparison theorem in Part I, we solve an open problem of the existence of blow-up solutions.