Patterns in Parabolic Problems with Nonlinear Boundary Conditions
نویسنده
چکیده
In this paper we show the existence of stable nonconstant equilibrium (patterns) for reaction-diffusion equations with nonlinear boundary conditions on small domains connected by thin channels. We prove the convergence of eigenvalues and eigenfunctions of the Laplace operator is such domains. This information is used to show that the asymptotic dynamic of the heat equations in this domain is equivalent to the asymptotic dynamics of a two ordinary differential equations diffusively (weakly) coupled. The main tools employed are the invariant manifold theory and uniform trace theorem.
منابع مشابه
A MIXED PARABOLIC WITH A NON-LOCAL AND GLOBAL LINEAR CONDITIONS
Krein [1] mentioned that for each PD equation we have two extreme operators, one is the minimal in which solution and its derivatives on the boundary are zero, the other one is the maximal operator in which there is no prescribed boundary conditions. They claim it is not possible to have a related boundary value problem for an arbitrarily chosen operator in between. They have only considered lo...
متن کاملStable and Unstable Manifolds for Quasilinear Parabolic Problems with Fully Nonlinear Dynamical Boundary Conditions
We develop a wellposedness and regularity theory for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we construct and investigate stable and unstable local invariant manifolds near a given equilibrium. In a companion paper we treat center, center–stable and center–unstable manifolds for such problems and investigate their stability p...
متن کاملCenter Manifolds and Attractivity for Quasilinear Parabolic Problems with Fully Nonlinear Dynamical Boundary Conditions
We construct and investigate local invariant manifolds for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions and study their attractivity properties. In a companion paper we have developed the corresponding solution theory. Examples for the class of systems considered are reaction–diffusion systems or phase field models with dynamical boundary co...
متن کاملHigher order multi-point fractional boundary value problems with integral boundary conditions
In this paper, we concerned with positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions. We establish the criteria for the existence of at least one, two and three positive solutions for higher order m-point nonlinear fractional boundary value problems with integral boundary conditions by using a result from the theory of fixed...
متن کاملBlow-up in the Parabolic Problems under Nonlinear Boundary Conditions
In this paper, I consider nonlinear parabolic problems under nonlinear boundary conditions. I establish respectively the conditions on nonlinearities to guarantee that ( , ) u x t exists globally or blows up at some finite time. If blow-up occurs, an upper bound for the blow-up time is derived, under somewhat more restrictive conditions, lower bounds for the blow-up time are also derived.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2002