Study of singularities arising in nonlinear partial differential differential equations and asymptotic methods

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(1) Dynamics of blow-up solutions Some blow-up solutions of a nonlinear heat equation can be continued beyond the blow-up time in a certain weak sense. Matano studied the dynamics of such solutions from the point of view of dynamical systems.(2) Motion of interfaces with random deviation In a class of diffusion equations involving a small parameter, say ε, solutions develop sharp transition layers, or interfaces, as ε→0. Funaki considered the case where the equation involves a random deviation term.(3) Estimate of blow-up time in a nonlinear heat equation Yanagida sutdied blow-up phenomena in a nonlinear heat equation and extended the classical results of Fujita and others.(4) Motion of interface in competition systems Mimura studied the behavior of interfaces that arise in the singular limit of a three-species competition-diffusion system.(5) Order-preserving systems in the presence of symmetry Matano extended the existing theory on order-preserving dynamical systems in the presence of symmetry. He then applied the general theory to the stability analysis of traveling waves and other problems.