In this paper, we are concerned with a system of equations governing the evolution of non-isothermal compressible fluids of Korteweg type, that can be used to the phase transition. It is found that the linear third-order capillarity behaves like damping of density fluctuation, which inspires us to investigate the dissipative structure for generally hyperbolic–parabolic systems with symmetric damping and Korteweg-type dispersion. Consequently, we establish the global well-posedness of solutions in the framework of hybrid Besov spaces by Friedrichs’ approximation procedure. In order to overcome the difficulty arising from the loss of derivative in nonlinear a priori estimates, the “gauge” function technique is mainly employed, which may be new in the study of hyperbolic–parabolic dissipative systems. Finally, we deduce optimal decay rates of solutions by using the Lyapunov-type energy method, provided that the low-frequency part of initial data belongs to [Formula: see text] with [Formula: see text].
In this paper, we study the Hyperbolic Nonlinear Schrödinger (HNLS) equation on general tori of dimensions [Formula: see text]. In particular, we establish sharp well-posedness results up to the endpoint cases for all odd nonlinearities. For higher dimensions and nonlinearities, we further extend the well-posedness theory to the critical regularity. For this purpose, we prove the scale-invariant Strichartz estimates and multi-linear estimates for hyperbolic Schrödinger equations on general tori.
In this paper, we study the existence, uniqueness and homogenization of a nonlinear hyperbolic problem with nonlinear boundary damping term [Formula: see text] and a boundary source term [Formula: see text] where [Formula: see text] At first, we prove the existence and uniqueness of a solution related to this problem through using the Faedo–Galerkin method. Besides, under certain assumptions on the problem’s coefficients and according to the oscillating test function method, we pass to the limit (when [Formula: see text]) and we get a nonlinear hyperbolic problem with zero Dirichlet boundary condition.
Our objective is to establish a generalized Oleinik-type inequality for a convex scalar balance law with nonlinear source term g(u), in the spirit of the classical results obtained independently by Lax and Oleinik for scalar conservation laws. From this inequality, and under a generic condition linking the flux function and the source term, we deduce a BV regularity result for the unique entropy solution. If this condition is not satisfied, we obtain instead a fractional regularity result, showing that the solution lies in BVs for some s is an element of (0, 1), as in the homogeneous case g = 0.
This paper investigates the large time optimal growth estimates of solutions for an elastic plate system under the Green-Naghdi hyperbolic thermal law in the inviscid case. By employing WKB asymptotic analysis and Fourier analysis, we establish sharp large time estimates in the L2 norm for both the vertical deflection of plate and the thermal displacement, which grow in lower dimensions. A key technical tool is the Fourier splitting method equipped with suitably chosen time-dependent splitting functions, which effectively handles non-summable singularities for small frequencies as the viscous dissipation vanishes. Our results show the crucial influence of viscous dissipation and the growth mechanism in the viscoelastic plate system studied by Chen et al. [Large time asymptotic behavior for the elastic plate system with type II heat conduction. Part I: The viscous case, Acta Math. Sin. (Chin. Ser.) (2026)].
