Nonlinear hyperbolic systems with nonlocal boundary conditions

A system of nonlinear hyperbolic equations with boundary conditions of renewal type is studied as a general mathematical model for structured biological populations.     

Kirchhoff systems with dynamic boundary conditions

We are interested in the study of the global non-existence of solutions of hyperbolic nonlinear problems, governed by the p-Kirchhoff operator, under dynamic boundary conditions, when p>p_n with p_n<2. The systems involve nonlinear external forces and may be affected by a perturbation of the type |u|^p−2u. Several models already treated in the literature are covered in special subcases, and concrete examples are provided for the source term f and the external nonlinear boundary damping Q.     

Phase-field systems with nonlinear coupling and dynamic boundary conditions

We consider phase-field systems of Caginalp type on a three-dimensional bounded domain. The order parameter fulfills a dynamic boundary condition, while the (relative) temperature is subject to a homogeneous boundary condition of Dirichlet, Neumann or Robin type. Moreover, the two equations are nonlinearly coupled through a quadratic growth function. Here we extend several results which have been proven by some of the authors for the linear coupling. More precisely, we demonstrate the existence and uniqueness of global solutions. Then we analyze the associated dynamical system and we establish the existence of global as well as exponential attractors. We also discuss the convergence of given solutions to a single equilibrium.     

Existence of chaotic oscillations in second-order linear hyperbolic PDEs with implicit boundary conditions

This paper establishes rigorously mathematical theorems that guarantee the existence of chaotic oscillations in the systems of second-order linear hyperbolic PDEs. It separately considers the systems with nonlinear explicit boundary conditions (EBCs) and nonlinear implicit boundary conditions (IBCs) as well as those with such IBCs subjected to small perturbations, where IBCs include EBCs as special cases but the latter cannot in general be expressed by the former. Numerical examples are demonstrated to illustrate the effectiveness of theoretical results.     

Some sufficient conditions for oscillation of impulsive delay hyperbolic systems with Robin boundary conditions

This paper deals with the oscillation problems of delay hyperbolic systems with impulses. Some sufficient conditions for oscillations of impulsive delay hyperbolic systems with Robin boundary conditions are obtained and the criteria of oscillation of the systems are established.     

Linear symmetric hyperbolic systems with characteristic boundary

The author studies the mixed problem for the linear symmetric hyperbolic systems with maximally non-negative and characteristic boundary condition. Existence of a unique solution is proved inside a suitable class of functions of weighted Sobolev type which takes account of the loss of regularity in the normal direction to the characteristic boundary.     

Quasilinear hyperbolic stefan problem with nonlocal boundary conditions

Using the method of contracting mappings, we prove, for small values of time, the existence and uniqueness of a generalized Lipschitz solution of a mixed problem with unknown boundaries for a hyperbolic quasilinear system of first-order equations represented in terms of Riemann invariants with nonlocal (nonseparated and integral) boundary conditions.     

Nontrivial solutions for asymptotically linear hamiltonian systems with Lagrangian boundary conditions

In this article, we study the existence of nontrivial solutions for a class of asymptotically linear Hamiltonian systems with Lagrangian boundary conditions by the Galerkin approximation methods and the L-index theory developed by the first author.     

On minimalization of time varying linear systems with well-posed boundary conditions

The notion of minimal realization of a pair of finite rank operators between Hilbert spaces is introduced. Constructions of such minimal realizations are made. A minimal realization of an integral operator with semi-separable kernel is constructed for the case that one of the realizations has analytic coefficients. A consequence for the characterization of minimal stability of an analytic minimal system is given.     

Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions

We consider a linear system of PDEs of the form 1 $$\begin{aligned} & \begin{aligned} u_{tt} - c\Delta u_t - \Delta u &= 0 \quad\text{in } \varOmega\times (0,T)\\ u_{tt} + \partial_n (u+cu_t) - \Delta_\varGamma(c \alpha u_t + u)& = 0 \quad\text{on } \varGamma_1 \times(0,T)\\ u &= 0 \quad\text{on } \varGamma_0 \times(0,T) \end{aligned} \\ &\quad (u(0),u_t(0),u|_{\varGamma_1}(0),u_t|_{\varGamma_1}(0)) \in {\mathcal{H}} \end{aligned}$$ on a bounded domain Ω with boundary Γ=Γ ₁∪Γ ₀. We show that the system generates a strongly continuous semigroup T(t) which is analytic for α>0 and of Gevrey class for α=0. In both cases the flow exhibits a regularizing effect on the data. In particular, we prove quantitative time-smoothing estimates of the form ∥(d/dt)T(t)∥?|t|^?1 for α>0, ∥(d/dt)T(t)∥?|t|^?2 for α=0. Moreover, when α=0 we prove a novel result which shows that these estimates hold under relatively bounded perturbations up to 1/2 power of the generator.     

Open boundary conditions for hyperbolic equations

A previously developed general procedure for deriving accurate difference equations to describe conditions at open boundaries for hyperbolic equations is extended and further illustrated by means of several examples of practical importance. Problems include those with both incoming and outgoing waves at the boundary, the use of locally cylindrical and spherical wave approximations at each point of the boundary, and nonlinear wave propagation. Reflected waves in all cases are minimal and less than 10^?2 of the incident wave.     

Problems with nonlinear boundary conditions for a hyperbolic equation

Two problems with nonlinear boundary conditions are studied. Existence and uniqueness theorems are proved for generalized solutions to each problem.     

Necessary conditions for hyperbolic systems

We consider the Cauchy problem for first order hyperbolic systems that have characteristic points of higher multiplicity. This means that the determinant of the principal symbol has multiple characteristic points. In the case where, on a multiple characteristic point, the principal symbol has corank 2, we give necessary conditions for the well posedness of the Cauchy problem. These conditions involve a suitably defined noncommutative determinant of the full symbol of the system.     

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

This paper is devoted to initial boundary value problems for quasi-linear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow. __________ Translated from Chinese Annals of Mathematics, Ser. A, 1982, 3(2): 223–232