Do global attractors depend on boundary conditions?
The paper deals with the scalar partial differential equations \[ u_t= u_{xx}+ f(x, u, u_x) \tag{1} \] on the unit interval \(0\leq x\leq 1\) with boundary conditions \[ (1- a_i) u+ a_i u_x= 0, \quad i=0, 1, \qquad \text{at } \quad x=0,\;x=1 \tag{2} \] and where \(0\leq a_i\leq 1\), \(i=0, 1\). For the nonlinearities \(f\in C^2\), such dissipation conditions are imposed that guarantee the local semiflow of \(x\)-profiles of solutions \(u(t, \cdot)\in X^a\), \(t\geq 0\) to be globally defined and dissipative. The state space \(X^a\) is the Sobolev space \(H^2\) intersected with boundary conditions (2), \(a= (a_0, a_1)\). The system \((1)+ (2)\) possesses a global attractor \(A^a_f \subset X^a\) which is the principal object of the author's investigation. Global attractors are called orbit equivalent, if there exists a homeomorphism between them which maps orbits to orbits. For given boundary conditions \(a\), the author defines the attractor class \(A(a)\) as the set of all orbit equivalence classes of global attractors \(A^a_f\). Here \(f\in C^2\) are assumed to be dissipative with only hyperbolic equilibria. The author proves: The global attractor class \(A (a)\) does not depend on the boundary conditions (2) given by \(a= (a_0, a_1)\). The proof is based on the Morse-Smale property and a geometric criterion for hyperbolicity of the set of equilibrium solutions \(u_t= 0\) of \((1)+(2)\). This interesting paper is well written and organized.
scientific article; zbMATH DE number 913529