L Infinity Stability of an Exponentially Decreasing Solution of the Problem Delta u + f(x,u) = 0 or R(n).
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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The equations studied here arise in many fields of mathematical sciences such as population dynamics in mathematical ecology, population genetics, chemical reaction theory, etc. This study concerns the stability of equilibrium solutions of these equations. Among the solutions of nonlinear evolution equations, the practically important ones are those which are stable in a certain sense. However, finding a stable equilibrium solution is in many cases considerably more difficult than just proving the existence of equilibrium solutions. This paper gives a useful sufficient condition for the existence of stable equilibrium solutions. Result presented in this paper is a generalization of the authors former results on equations in bounded domains. However, the equations considered here which are in the whole space R sub n exhibit much more complicated dynamical behavior, and therefore only a few results have been known about the existence of stable equilibrium solutions. The objective of this paper is to make a systematic study of these equations and to give rather a general theorem on the existence of stable equilibrium solutions.
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