Parabolic and Elliptic Equations in Unbounded Domains

In this chapter we extend to unbounded domains the method of upper and lower solutions for parabolic and elliptic equations in bounded domains. For the parabolic equation this extension includes the Cauchy problem in ℝ n , a half-space problem in ℝ + n and problems in the exterior of a bounded domain as well as in a general unbounded domain. Similar extension is given to the corresponding elliptic equation, including an exterior problem with nonlinear boundary condition. In the case of the Cauchy problem sufficient conditions for the asymptotic stability and instability of a steady-state solution and the monotone convergence of the time-dependent solution to the maximal and the minimal steady-state solutions are obtained. A characterization of the global existence and the blowing-up behavior of the solution in relation to the spatial dimension n and the growth property of the reaction function are also given. For the elliptic equation in ℝ n an infinite number of radially symmetric positive solutions are constructed and are applied to a special model arising from differential geometry and applied physics. In addition, a Dirichlet boundary-value problem in a general unbounded domain is considered, and a similar iteration process as in the case of a bounded domain is formulated. It is shown that without any prescribed condition at infinity there exist two monotone sequences which converge to a maximal solution and a minimal solution in the same fashion as for the corresponding problem in bounded domains. Sufficient conditions for the positivity and the uniqueness of the solution are given.