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A Principle of Reduced Stability for Reaction-Diffusion Equations

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A semilinear elliptic boundary value problem, Au+f(x, u, lambda)=0 (with f u(x, u, lambda) bounded below)can be shown to be equivalent to a finite-dimensional problem,B(c, lambda)=0∈ℝd (c∈ℝd ), in the sensethat their solution sets, which are not necessarily singletons,are in a one-to-one correspondence (c(u)↔u(c)).The function B(c, lambda) is called the bifurcation function.It is shown that, for any solution u(c), the number of negative(resp. zero) eigenvalues of the matrix B c(c, lambda) is identicalto the number of negative (resp. zero) eigenvalues of the linearizedelliptic operator Av+f u(x, u(c), lambda) v. Thisresults in a version of the principle of reduced stability forthe problem u t+Au+f(x, u, lambda)=0 and its reductionc\'+B(c, lambda)=0. Copyright 1998 Academic Press.

Document Type: Research Article

Affiliations: Department of Mathematics, Iowa State University, Ames, Iowa, 50011-2066

Publication date: January 1, 1998

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