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ROTHE TIME-DISCRETIZATION METHOD APPLIED TO A QUASILINEAR WAVE EQUATION SUBJECT TO INTEGRAL CONDITIONS ABDELFATAH BOUZIANI AND NABIL MERAZGA Received 27 January 2004 and in revised form 12 February 2004 This paper presents a well-posedness result for an initial-boundary value problem with only integral conditions over the spatial domain for a one-dimensional quasilinear wave equation. The solution and some of its properties are obtained by means of a suitable application of the Rothe time-discretization method. 1. Introduction Recently, the study of initial-boundary value problems for hyperbolic equations with boundary integral conditions has received considerable attention. This kind of conditions has many important applications | ROTHE TIME-DISCRETIZATION METHOD APPLIED TO A QUASILINEAR WAVE EQUATION SUBJECT TO INTEGRAL CONDITIONS ABDELFATAH BOUZIANI AND NABIL MERAZGA Received 27 January 2004 and in revised form 12 February 2004 This paper presents a well-posedness result for an initial-boundary value problem with only integral conditions over the spatial domain for a one-dimensional quasilinear wave equation. The solution and some of its properties are obtained by means of a suitable application of the Rothe time-discretization method. 1. Introduction Recently the study of initial-boundary value problems for hyperbolic equations with boundary integral conditions has received considerable attention. This kind of conditions has many important applications. For instance they appear in the case where a direct measurement quantity is impossible however their mean values are known. In this paper we deal with a class of quasilinear hyperbolic equations T is a positive constant 2v - 2v f x t v 5 x t e 0 1 X 0 T 1.1 dt2 dx2 dt subject to the initial conditions dv v x 0 V0 x d x 0 V1 x 0 x 1 1.2 and the boundary integral conditions 1 v x t dx E t 0 t T fi 1.3 xv x t dx G t 0 t T 0 where f v0 v1 E and G are sufficiently regular given functions. Problems of this type were first introduced in 3 in which the first author proved the well-posedness of certain linear hyperbolic equations with integral condition s . Later Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 3 2004 211-235 2000 Mathematics Subject Classification 35L05 35D05 35B45 35B30 URL http dx.doi.org 10.1155 S1687183904401071 212 On a quasilinear wave equation with integral conditions similar problems have been studied in 1 4 5 6 7 8 16 24 25 by using the energetic method the Schauder fixed point theorem Galerkin method and the theory of characteristics. We refer the reader to 2 9 10 11 12 13 14 15 17 21 22 23 26 for other types of equations with integral conditions. Differently to these works in the .
