Admissibility and mean hyperbolicity for evolution equations (Q6553375)

In this article are considered the nonlinear evolution equation \N\[ \N\begin{cases} \partial _t v = A(t) v + f(t,v), \\\Nv(0)= x \in X,\ t \in \mathbb{R}, \end{cases} \tag{1}\N\] \Nand the associated linear evolution equation \N\[ \N\begin{cases} \partial _t u = A(t) u, \\\Nu(0)= x \in X,\ t \in \mathbb{R}, \end{cases} \tag{2}\N\] \Nwhere the continuous operator \(A(t)\) acting on a Banach space \(X\) can be unbounded.\N\NThe authors study the relationship between mean hyperbolicity of the linear system $(2)$, described by mean exponential dichotomy of its evolution family \(\{T(t,s)\}\) and the existence of a bounded mild solution of the perturbed equation $(1)$.\N\NMean hyperbolicity can be seen as a generalization of the uniform (with fixed compression and expansion rate) and the nonuniform (compression and expansion rate varies depending on a constant and initial time) hyperbolicity. By mean hyperbolicity the average compression and expansion rate is still fixed for sufficient long evolution time, but there is a non-hyperbolic behaviour (i.e. coexistence of expansion and contraction behaviour in generalized stable and unstable subspaces) at certain moments for finite evolution time length during the evolution process.\N\NFor their study the authors construct properly admissible function classes (a pair of function classes \((B,C)\) is properly admissible with respect to the evolution family \(\{T(t,s)\}\), if for each perturbation term \(f \in B\) there exist a unique mild solution \(v \in C\) of the perturbed system $(1)$) and invariant decomposition with generalized stable and unstable subspaces.\N\NAs main result the authors obtain first that mean exponential dichotomy for the evolution family \(\{T(t,s)\}\) of the linear system $(2)$ on \(\mathbb{R}\) implies proper admissibility with respect to \(\{T(t,s)\}\) of the constructed pair function classes (i.e. there exists a unique mild solution of the perturbed system $(1)$). And second -- vice versa -- that from proper admissibility of the constructed pair spaces in respect to \(\{T(t,s)\}\) it follows mean exponential dichotomy of the evolution family \(\{T(t,s)\}\) of the linear system $(2)$.\N\NThree examples to illustrate systems with mean hyperbolicity including damped wave equations with variable coefficients are provided and sufficient conditions for roughness of the mean hyperbolicity are presented.