Propagation of solutions of a nonlinear functional differential equation
The authors extend their previous results in Rend. Semin. Mat. Univ. Padova 80, 55-63 (1988; Zbl 0671.34066), on the propagation of solutions of seminlinear delay equations to the nonlinear functional differential equation \(\dot x(t)=f(x(t))+g(x_ t),x(0)=h\in X\), \(x_ 0\in L^ p=L^ p(-r,0;X)\), where X is a Hilbert space, \(-f+\alpha I: D(f)\subseteq X\to X\) is m-accretive for some \(\alpha >0\) and g: \(L^ p\to X\) satisfies a Lipschitz condition. Let \(T_ f(t)\) and \(T_ A(t)\) denote the semigroups generated by f and by \(A\{\phi,h\}=\{\phi ',f(h)+g(\phi)\},\) \(D(A)=\{\{\phi,h\},\quad \phi \in W^{1,p}(- r,0;X),\quad \phi (0)=h\in D(f)\}.\) The authors prove that if \(\{X_ t\}_{t\geq 0}\) is a family of closed subspaces of X and \(T_ f(t)\) propagates \(\{X_ t\}\), then \(T_ A(t)\) propagates \(\{Y_ t\}\), where \(Y_ t\subseteq L^ p\times X\) is constructed from \(X_ t\).
nonlinear functional differential equation
scientific article; zbMATH DE number 4198518