Abstract
ByHiroshi MATANO*§ I. IntroductionThis paper is divided into two chapters. In the first chapter (§ 2^§ 4) we study the asymptotic behavior of solutions of semilinear diffu-sion equations — in particular their behavior around unstable equilibriumsolutions, so as to get a clearer understanding of the situation wherethose unstable equilibrium solutions are placed. The results will thenbe applied to making some stability criteria as well as to establishingtheorems on the existence of stable equilibrium solutions and on thestructure of multiple equilibrium solutions. In the second chapter (§ 5,§ 6) we are confined to Neumann problems and discuss the possibility ofthe existence of non-constant stable equilibrium solutions.Let D denote a bounded domain in R
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