On nonlinear perturbations of the Schrödinger equation with discontinuous coefficients

The results of [2] by W. J&;#228;ger and Y. Saito on the Schrödinger equation with discontinuous coefficients are extended to nonlinear perturbations of the equation.     

Smoothness of solutions of second-order hyperbolic equations with discontinuous coefficients

Translated from Matematicheskie Zametki, Vol. 49, No. 6, pp. 3–11, June, 1991.     

Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. II

We use energy methods to prove the existence and uniqueness of solutions of the Dirichlet problem for an elliptic nonlinear second-order equation of divergence form with a superlinear tem [i.e., g(x, u)=v(x) a(x)⋎u⋎ ^p−1u,p>1] in unbounded domains. Degeneracy in the ellipticity condition is allowed. Coefficients a _i,j(x,r) may be discontinuous with respect to the variable r. University of Catania, Italy. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 12, pp. 1601–1609, December, 1997.     

A finite differences method for a two-dimensional nonlinear hyperbolic equation in a class of discontinuous functions

A numerical scheme is proposed for a scalar two-dimensional nonlinear first-order wave equation with both continuous and piecewise continuous initial conditions. It is typical of such problems to assume formal solutions with discontinuities at unknown locations, which justifies the search for a scheme that does not rely on the regularity of the solution. To this end, an auxiliary problem which is equivalent to, but has more advantages then, the original system is formulated and shown that regularity of the solution of the auxiliary problem is higher than that of the original system. An efficient numerical algorithm based on the auxiliary problem is derived. Furthermore, some results of numerical experiments of physical interest are presented.     

Convergence of FEM with interpolated coefficients for semilinear hyperbolic equation

To solve spatially semidiscrete approximative solution of a class of semilinear hyperbolic equations, the finite element method (FEM) with interpolated coefficients is discussed. By use of semidiscrete finite element for linear problem as comparative function, the error estimate in L^∞$L^{\infty }$-norm is derived by the nonlinear argument in Chen [Structure theory of superconvergence of finite elements, Hunan Press of Science and Technology, Changsha, 2001 (in Chinese)]. This indicates that convergence of FEMs with interpolated coefficients for a semilinear equation is similar to that of classical FEMs.     

Strongly nonlinear degenerate elliptic equations with discontinuous coefficients. I

This paper is concerned with the existence and uniqueness of variational solutions of the strongly nonlinear equation $$ - \sum\limits_1^m {_i \frac{\partial }{{\partial x_i }}\left( {\sum\limits_1^m {_j a_{i,j} (x, u(x))\frac{{\partial u(x)}}{{\partial x_j }}} } \right) + g(x, u(x)) = f(x)} $$ with the coefficients a _i,j (x, s) satisfying an eHipticity degenerate condition and hypotheses weaker than the continuity with respect to the variable s. Furthermore, we establish a condition on f under which the solution is bounded in a bounded open subset Ω of R^m.     

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.     

Existence theorem for a stochastic wave equation with discontinuous unbounded coefficients

We prove an existence theorem for stochastic hyperbolic equations with measurable locally bounded coefficients. A solution of a stochastic hyperbolic equation is understood as a martingale solution of the stochastic inclusion corresponding to the equation.     

The finite element method for nonlinear elliptic equations with discontinuous coefficients

Summary The study of the finite element approximation to nonlinear second order elliptic boundary value problems with discontinuous coefficients is presented in the case of mixed Dirichlet-Neumann boundary conditions. The change in domain and numerical integration are taken into account. With the assumptions which guarantee that the corresponding operator is strongly monotone and Lipschitz-continuous the following convergence results are proved: 1. the rate of convergenceO(h ) if the exact solutionuH ¹ () is piecewise of classH ¹⁺ (0<1);2. the convergence without any rate of convergence ifuH ¹ () only.     

Higher integrability for nonlinear elliptic equations with variable growth and discontinuous coefficients

In this paper, based on the theory of variable exponent spaces, we study the higher integrability for a class of nonlinear elliptic equations with variable growth and discontinuous coefficients. Under suitable assumptions, we obtain a local gradient estimate in Orlicz space for weak solution.     

Homogenization and corrector for the wave equation with discontinuous coefficients in time

In this paper we analyze the homogenization of the wave equation with bounded variation coefficients in time, generalizing the classical result, which assumes Lipschitz-continuity. We start showing a general existence and uniqueness result for a general sort of hyperbolic equations. Then, we obtain our homogenization result comparing the solution of a sequence of wave equations to the solution of a sequence of elliptic ones. We conclude the paper making an analysis of the corrector. Firstly, we obtain a corrector result assuming that the derivative of the coefficients in the time variable is equicontinuous. This result was known for non-time dependent coefficients. After, we show, with a counterexample, that the regularity hypothesis for the corrector theorem is optimal in the sense that it does not hold if the time derivative of the coefficients is just bounded.     

A uniqueness result for a nonlinear hyperbolic equation

Summary We prove the uniqueness of the solution of a Cauchy problem for a nonlinear and non-local hyperbolic equation, which arises in a model of the dynamic of cardiac muscle.The authors were supported by G.N.A.F.A., G.N.I.M. and I.A.N. of C.N.R. and by M.P.I.     

Global attractor for hyperbolic equation with nonlinear damping and linear memory

In this paper, we study the existence of global attractors for the hyperbolic equation with nonlinear damping and linear memory.     

Existence and blow up for a nonlinear hyperbolic equation with anisotropy

In this paper, we study a nonlinear wave equation with anisotropy and a source term. Under some appropriate assumptions on the parameters, and with certain initial data, we obtain several results on the existence of local solutions and the blow up of solutions in finite time.     

A method for finding coefficients of a quasilinear hyperbolic equation

The inverse problem of finding the coefficients q(s) and p(s) in the equation u _tt = a ² u _xx + q(u)u _t ? p(u)u _x is investigated. As overdetermination required in the inverse setting, two additional conditions are set: a boundary condition and a condition with a fixed value of the timelike variable. An iteration method for solving the inverse problem is proposed based on an equivalent system of integral equations of the second kind. A uniqueness theorem and an existence theorem in a small domain are proved for the inverse problem to substantiate the convergence of the algorithm.