Structural stability for scalar parabolic equations
The author considers the parabolic equation \[ u_ t= u_{xx}+ f(u), \quad 0<x<1, \qquad u(t,0)= u(t,1) =0.\tag{1} \] Here \(f\) is smooth and satisfies \(\limsup f(u) u^{-1}=0\) as \(| u| \to\infty\), and \(f(0)=0\). Equation (1) is rewritten as an evolution equation in the form (2) \(u_ t+ Au= \overline{F} (u)\). The underlying space is \(H_ 0^ 1\) with \(\text{dom} (A)=\{u\mid u\in H^ 2\cap H_ 0^ 1\), \(u_{xx}\in H_ 0^ 1\}\) and \(Au=- u_{xx}\) for \(u\in \text{dom}(A)\). Moreover \(\overline{F} (u)= f(u)\) turns out to be a \(C^ 1\) mapping from \(H_ 0^ 1\) into \(H_ 0^ 1\). Equation (2) admits an absorbing ball \(B_ R\) which gives rise to a truncated function \(F(u)= \theta(| u| (2R)^{-1}) \overline{F} (u)\) (\(|\cdot |\) the norm on \(H_ 0^ 1\)) where \(\theta (s)=1\) for \(s\in [0,1]\), \(\theta (s)=0\) for \(s\in [2,\infty)\) and \(|\theta' (s)|\leq 2\) for \(s\geq 0\). Instead of equation (2) the truncated equation (3) \(u_ t+ Au= F(u)\) is now studied. More precisely one considers a perturbation of (1), i.e. a parabolic equation (4) \(u_ t= u_{xx}+ g(u)\), \(0<x<1\) and \(u(t,0)= u(t,1) =0\) with smooth \(g\). Equation (4) is put into an abstract form similar to (2) and then replaced by a corresponding truncated equation assuming the form (5) \(u_ t+ Au= G(u)\). The main result compares (3) with (5). In somewhat imprecise terms it states: If \(F\), \(G\) are close in a certain specific sense then there is a neighbourhood \(U\) of the attractor of (3) and a neighbourhood \(V\) of the attractor of (5) and a homeomorphism \(\varphi\) from \(U\) onto \(V\) which transforms orbits of (3) into orbits of (5) and preserves the direction of time. The theorem thus expresses a property of structural stability of (3). The proof is based on techniques used in inertial manifold theory and used in similar contexts by some of the authors in the reference list.
scientific article; zbMATH DE number 696890