Generalized uniformly continuous semigroups and semilinear hyperbolic systems with regularized derivatives

We adopt the theory of uniformly continuous operator semigroups for use in Colombeau generalized function spaces. The main objective is to find a unique solution to a class of semilinear hyperbolic systems with singularities. The idea of regularized derivatives is to transform unbounded differential operators into bounded, integral ones. This idea is used here to permit working with uniformly continuous operators.     

Generalized uniformly continuous semigroups and semilinear hyperbolic systems with regularized derivatives

We adopt the theory of uniformly continuous operator semigroups for use in Colombeau generalized function spaces. The main objective is to find a unique solution to a class of semilinear hyperbolic systems with singularities. The idea of regularized derivatives is to transform unbounded differential operators into bounded, integral ones. This idea is used here to permit working with uniformly continuous operators.     

Generalized solution to a semilinear hyperbolic system with a non-Lipshitz nonlinearity

Let

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

Global analytic solutions to hyperbolic systems

The aim of this paper is to extend to some classes of systems the global existence of analytic solutions to scalar equations of Kirchhoff type. Received March 17, 1997.     

Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems

In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form:

We analyze the singular convergence, as , in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps:

(i)

We single out algebraic ``structure conditions' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories.

(ii)

We deduce ``energy estimates ', uniformly in , by assuming the existence of a symmetrizer having the so-called block structure and by assuming ``dissipativity conditions' on .

(iii)

We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard.

Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.

     

Large solutions to non-monotone semilinear elliptic systems

We present the existence of entire large positive radial solutions for the non-monotonic system Δu=p(|x|)g(v), Δv=q(|x|)f(u) on Rn where n?3. The functions f and g satisfy a Keller-Osserman type condition while nonnegative functions p and q are required to satisfy the decay conditions and . Further, p and q are such that min(p,q) does not have compact support.     

Entire large solutions to semilinear elliptic systems

We consider the problem of existence of positive solutions to the elliptic system Δu=p(|x|)vα, Δv=q(|x|)uβ on Rn (n?3) which satisfies . The parameters α and β are positive, and the nonnegative functions p and q are continuous and min{p(r),q(r)} does not have compact support. We show that if αβ?1, then such a solution exists if and only if the functions p and q satisfy

     

Generalized solutions to free boundary problems for hyperbolic systems of functional partial differential equations

Summary Local a.e. solutions to a free boundary (Stefan) problem for a quasilinear hyperbolic system of functional PDE's of first order in two independent variables and diagonal form are investigated. The formulation includes retarded arguments and hereditary Volterra terms.     

Homogenization for semilinear hyperbolic systems with oscillatory data

The behavior of multi-dimensional discrete Boltzmann systems with highly oscillatory data is studied. Homogenized equations for the mean solutions are obtained. Uniform convergence of the oscillatory solutions of the discrete Boltzmann equations to the solutions of the corresponding homogenized equations is established. Moreover, we find that the weak limits of the oscillatory solutions for a model of Broadwell type are not continuous functions of the discrete velocities. Generalization of the above results to problems with multiple-scale initial data is also established.     

非线性自治振动系统同宿解的广义双曲函数摄动法

提出广义的双曲函数摄动法,用于求解强非线性自治振子的同宿解,克服一般摄动步骤中派生方程须存在显式精确同宿解的限制．以广义双曲函数作为摄动步骤的基本函数,拓展了基于双曲函数的摄动法的适用范围．对同时含2,3次和含4次强非线性项的系统进行求解分析,验证了方法的有效性和精度．     

Global behavior of solutions to an inverse problem for semilinear hyperbolic equations

This paper is concerned with global in time behavior of solutions for a semilinear, hyperbolic, inverse source problem. We prove two types of results. The first one is a global nonexistence result for smooth solutions when the data is chosen appropriately. The second type of results is the asymptotic stability of solutions when the integral constraint vanishes as t goes to infinity. Bibliography: 22 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 120–134.     

Stability of travelling wave solutions to a semilinear hyperbolic system with relaxation

We study a semilinear hyperbolic system with relaxation and investigate the asymptotic stability of travelling wave solutions with shock profile. It is shown that the travelling wave solution is asymptotically stable, provided the initial disturbance is suitably small. Moreover, we show that the time convergence rate is polynomially (resp. exponentially) fast as t→∞ if the initial disturbance decays polynomially (resp. exponentially) for x→∞. Our proofs are based on the space–time weighted energy method. Copyright © 2008 John Wiley & Sons, Ltd.     

Estimates on the weak solution of semilinear hyperbolic systems

The Cauchy problem for a semilinear hyperbolic system of the type