On Some Dynamical Aspects of Parabolic Equations with Variable Domain

Abstract

We consider the scalar parabolic equation

$$ \left( 1 \right)_\varepsilon \left\{ {\begin{array}{*{20}c} {u_t = \Delta u - ku + b\left( {x,u} \right)\quad in\quad D_\varepsilon } \\ {\frac{{\partial u}} {{\partial n}} = 0\quad on\quad \partial D_\varepsilon } \\ \end{array} } \right. $$

where k > 0 is fixed, b: Rⁿ × R → R is smooth, for some constant M we have |b(x,u)| < M, |b_u(x,u)| < M for all (x,u)∈ Rⁿ × R, and {D_ε} is a family of open smooth domains in Rⁿ such that for 0 ≤ ε ≤ ε' ≤ 1, D_ε is contained in D_ε', and |D_ε−D_ε'| → 0 as ε → ε'⁺, where |.| denotes the Lebesgue measure in Rⁿ. We assume also that each D_ε is connected and there is a ball B in Rⁿ which contains every D_ε.