We study the question of the existence, uniqueness, and continuous dependence on parameters of the Carathéodory solutions to the Cauchy problem for linear partial functional-differential equations of hyperbolic type. A theorem on the Fredholm alternative is also proved. The results obtained are new even in the case of equations without argument deviations, because we do not suppose absolute continuity of the function the Cauchy problem is prescribed on, which is rather usual assumption in the existing literature.
The Cauchy problem for the system of linear generalized ordinary differential equations in the J. Kurzweil sense dx(t) = dA0(t) · x(t) + df0(t), x(t0) = c0 (t ∈ I) with a unique solution x0 is considered. Necessary and sufficient conditions are obtained for a sequence of the Cauchy problems dx(t) = dAk(t) · x(t) + dfk(t), x(tk) = ck (k = 1, 2, . . .) to have a unique solution xk for any sufficiently large k such that xk(t) → x0(t) uniformly on I. Presented results are analogous to the sufficient conditions due to Z. Opial for linear ordinary differential systems. Moreover, efficient sufficient conditions for the problem of well-posedness are given.
We study systems of reaction-diffusion equations with discontinuous spatially distributed hysteresis on the right-hand side. The input of the hysteresis is given by a vector-valued function of space and time. Such systems describe hysteretic interaction of non-diffusive (bacteria, cells, etc.) and diffusive (nutrient, proteins, etc.) substances leading to formation of spatial patterns. We provide sufficient conditions under which the problem is well posed in spite of the assumed discontinuity of hysteresis. These conditions are formulated in terms of geometry of the manifolds defining the hysteresis thresholds and the spatial profile of the initial data.
The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed., Sen Ming, Han Yang, Zili Chen, Ls Yong., and Obsahuje bibliografii