Research Article Received 14 October 2008 Published online 10 June 2009 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/mma.1150 MOS subject classification: 35 L 70; 35 L 20; 35 K; 35 R Mixed nonlocal problem for a nonlinear singular hyperbolic equation Said Mesloub ∗ † ‡ Communicated by A. Belleni-Morante In this paper, we study a nonlocal mixed problem for a nonlinear hyperbolic equation. Based on some a priori estimates and some density arguments, we prove the well posedness of the associated linear problem. The existence and uniqueness of the weak solution of the nonlinear problem are then established by applying an iterative process based on the obtained results for the linear problem. Copyright © 2009 John Wiley & Sons, Ltd. Keywords: nonlinear hyperbolic equation; nonlocal condition; a priori estimate; Bessel operator; iterative method 1. Introduction During the last three decades, many physical phenomena have been modeled by one-dimensional partial differential equations combining an integral condition over the spacial domain of the desired solution and one classical condition (Dirichlet–Neumann). The integral condition has the form:    (x)u(x,t)dx = k(t), where (x) and k(t) are given functions. This kind of condition can represent physically a moment, a total energy, a total mass or a mean. Many methods were used to investigate the existence and uniqueness of the solution of mixed problems with purely integral conditions or which combine classical and integral conditions. Mixed problems with nonlocal conditions such as boundary integral conditions have many important applications in chemical diffusion, thermoelasticity, heat conduction processes, population dynamics, vibration problems, nuclear reactor dynamics, control theory, medical science, biochemistry, underground-water flow, transmission theory and certain biological processes. For example, many processes in porous media can be described by second-order hyperbolic equations with an integral condition [1, 2]. For heat conduction [3, 4], for elasticity and thermoelasticity [5, 6] and plasma physics [7]. It is the reason why such nonlocal mixed problems gained much attention in recent years not only in engineering but also in mathematics community. Note that theoretical study of nonlocal problems is connected with great difficulties since the presence of an integral term in boundary conditions can greatly complicate the application of classical methods of functional analysis, the energetic method, the method of singular integral equations. This is the reason for the existence of only separate results for nonlocal initial boundary value problems. Their investigation requires always a separate study every time. We study a nonlocal mixed problem for a nonlinear hyperbolic equation. The Fourier method has been used to treat a regular case for the linear associated problem to prove the existence and uniqueness of a classical solution [8]. In addition, for the associated linear case, a hyperbolic equation with Neumann weighted integral conditions has been studied in [9], and when the Bessel operator is replaced by the operator (a(x,t)u x ) x with Neumann integral conditions, a hyperbolic equation is treated in Bouziani [10, 11]. For hyperbolic equations with purely integral conditions, the reader should refer to Mesloub and Bouziani [12] and Pulkina [2, 13]. In this paper, we prove the existence and uniqueness of a weak solution of the posed problem. First, we establish for the associated linear problem an a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of deriving the a priori estimate is based on constructing a suitable multiplier. From the resulted energy estimate, it is possible to establish the solvability of the linear problem. Then, by applying an iterative process based on the obtained results for the linear problem, we establish the existence and uniqueness of the weak solution of the nonlinear problem. Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia ∗ Correspondence to: Said Mesloub, Department of Mathematics, College of Sciences, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia. † E-mail: mesloubs@yahoo.com ‡ Permanent address: Departement de Mathematiques, Universite’ de Cheikh Larbi Tebessi, Tebessa 12001, Algeria. Contract/grant sponsor: King Saud University; contract/grant number: No(Math/2008/19) Copyright © 2009 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2010, 33 57–70 57