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Generic properties of equilibrium solutions by perturbation of the boundary. (English) Zbl 0656.35069

Dynamics of infinite dimensional systems, Proc. NATO Adv. Study Inst., Lisbon/Port. 1986, NATO ASI Ser., Ser. F 37, 129-139 (1987).
[For the entire collection see Zbl 0623.00009.]
We study equilibrium solutions of \[ (1)\quad u_ t=\Delta u+f(x,u,\nabla u)\quad in\quad \Omega \subset R^ n \] with boundary conditions. \[ (2)\quad u=0\quad or\quad \partial u/\partial N=g(x,u)\quad on\quad \partial \Omega \] and of the corresponding damped wave equation with \(``u_{tt}+r(x)u_ t''\) in place of \(``u_ t''\). The equilibrium problem \[ (3)\quad \Delta u\cdot +f(x,u,\nabla u)=0\quad in\quad \Omega,\quad with\quad boundary\quad conditions, \] are the same in each case, but the eigenvalue problem for the linearization \[ (4)\quad \Delta v+\sum^{n}_{j=1}b_ j(x)\partial v/\partial x_ j+(c(x)-g(x,\lambda))v=0\quad in\quad \Omega \] has \(g(x,\lambda)=\lambda\) in the parabolic case, \(g(x,\lambda)=\lambda^ 2+r(x)\lambda\) for the wave equation. Here \[ (b_ j,c)(x)=(\partial f/\partial \beta_ j,\partial f/\partial \gamma)(x,u(x),\nabla u(x)) \] where u solves (3) for f \((x,\gamma,\beta_ 1,...,\beta_ n)\). If v(x) is a nontrivial solution of (4) for some \(\lambda\in C\), then \(e^{\lambda t}v(x)\) is a nontrivial solution of the linearization of (1), or of the corresponding wave equation, about the equilibrium.
Under various hypotheses about f and g, we can prove that - for most choices of the bounded smooth region \(\Omega\)- all equilibrium solutions u are simple and (with more restrictive hypotheses), all equilibria are hyperbolic, i.e. (4) has no non-trivial solutions when Re \(\lambda\) \(=0\).

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B20 Perturbations in context of PDEs
35J65 Nonlinear boundary value problems for linear elliptic equations

Citations:

Zbl 0623.00009