Structure of nonnegative sloutions of diffusive equetious and its applications

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We study the uniqueness of nonnegative solutions of the diffusive equations in unbounded domains. We first study the problem with Minoru Murata, and obtain an optimal sufficient condition for the uniqueness of the nonnegative solutions of the whole Euclidean space to hold. Furthermore Ishige study the uniqueness of nonnegative solutions of the Cauchy-Dirichlet problem to the diffusive equations in unbounded domains, and obtain an optimal sufficient condition for the uniiqueness tohold.Next we study the blow-up problem for a semilinear parabolic equation with larqe diffusion. We prove that the solution blows up only near some points determined by the projection of the initial data to the second Neumann eigenspace. In order to study this problem, we need to study the long-time behavior of the nonnegative solution of the heat equation, that is, the Neumann eigenfunctions.