State-dependent delay differential equations on \(H^1\) (Q6616073)

The paper provides a novel approach to initial value problems for solutions of differential equations with state-dependent delay of the form \N\[ \dot x(t) = G(t, x_t), \quad x_0= \phi \in (C^0[-h,0], \mathbb{R}^n). \tag{1} \] \NThe novelty is the use of the Sobolev space \(H^1([-h,0], \mathbb{R}^n)\) (briefly: \(H^1\)) instead of \(C^k\) spaces. The main condition of Theorem 1.1, which gives local (in time) solutions of (1), is that \(G\) is Lipschitz continuous in the second argument, if that argument is restricted to \(H^1\) functions with derivative in \(L^{\infty}\), and the initial state is also from this class. These functions are identified with the Lipschitz continuous functions in \(H^1\) in Remark 4.2(d), and this condition is obviously similar to the well-known `almost Lipschitz continuous' property from reference [\textit{J. Mallet-Paret} et al., Topol. Methods Nonlinear Anal. 3, No. 1, 101--162 (1994; Zbl 0808.34080)].\N\NA special case of Theorem 1.1 for equations with state-dependent point-delay like \N\[\dot x(t) = g[t, x(t), x(t- r(x_t))]\] \Nis proved in Theorem 4.1 of Section 4, and in this case solutions on a uniform existence interval of the form \([0,T]\) can be constructed. Section 5 then proves a more detailed version of Theorem 1.1, and here the existence time depends on the initial state.\N\NWhile the general technical approach is necessarily of Picard-Lindelöf type, important specific techniques of this work are:\N\N1) the use of exponentially weighted norms on \(H^1\) (which allows to construct a solution on all of \([0, T]\) in Theorem 4.9, without `gluing together' local solutions);\N\N2) the use of an auxiliary equation involving orthogonal projection within \(H^1\) onto a space of Lipschitz functions (equation (7));\N\N3) the fact that this projection is `superfluous' for short enough time (Theorem 4.10).\N\NUnlike the solution manifold approach from reference [\textit{H.-O. Walther}, J. Differ. Equations 195, No. 1, 46--65 (2003; Zbl 1045.34048)], initial values do not have to satisfy compatibility conditions in this paper. On the other hand, the approach presented here does (so far) not give differentiability w.r. to initial values. Further, in situations where the conditions of both approaches are satisfied, the more general solutions constructed in Theorem 4.1 of the present work will enter the solution manifold of [loc. cit., Zbl 1045.34048] after time \(h\), as pointed out in Remark 4.2(b).\N\NThe paper concludes with two example equations and considerations for future research, in particular, involving the retraction (projection) technique.