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

Zbl 0623.00009