Differentiability of parabolic semi-flows in \(L^p\)-spaces and inertial manifolds (Q1592762)

In the main body of the paper the author considers in a bounded domain \(\Omega\) a scalar parabolic equation \(u_t-\Delta u=f(x,u)\) with the homogeneous Dirichlet boundary condition and with an inital data \(u_0\) from \(L^p (\Omega)\). The nonlinear term \(f(x,u)\) is measurable in \(x\) for fixed \(u\), \(f (\cdot,0) \in L^\infty (\Omega)\), and it is of class \(C^k\) \((k\geq 1)\) in \(u\) for fixed \(x\) with uniformly bounded derivatives of order less or equal \(k\). It is shown that such a problem defines a global compact differentiable semiflow \(\Phi\in C(L^p\times [0,\infty), L^p)\) and that its derivative \(d\Phi (\cdot, t)\) belongs to \(C^{s-1}(L^p,L (L^p,L^q))\) where \(1<p\), \(q<\infty\), \(s\leq {p(n+2) \over n}\). Also the estimate of a difference between the derivative and the heat semigroup \(U(t)\) is given; namely, \[ \bigl\|d_u\Phi(u,t) [v]-U(t) [v]\bigr \|_p\leq Kte^{Kt} \|v\|_p, \] where \(K\) denotes the supremum of \(|f_u |\). In the last few pages of the paper the author discusses properties of the spectrum of the Laplace-Beltrami operator \(\Delta_M\) and shows -- considering equation \(u_t-\Delta_M u=f(x,u)\) with \(f\in C^1\) such that \(|f_u|\leq K\), \(|f_{uu} |\leq K\), \(f(x,0)=0\) -- the existence of an inertial manifold. These latter considerations come back to the author's previous article in [Nonlinear Anal., Theory Methods Appl. 28, No. 7, 1227-1248 (1997; Zbl 0874.34047)].