Mixed Problem with an Integral Two-Space-Variables Condition for a Class of Hyperbolic Equations

This paper is devoted to the proof of the existence and uniqueness of the classical solution of mixed problems which combine Neumann condition and integral two-space-variables condition for a class of hyperbolic equations. The proof is based on a priori estimate “energy inequality” and the density of the range of the operator generated by the problem considered. 1. Introduction The integral boundary conditions for evolution problems have various applications in chemical engineering, thermoelasticity, underground water flow, and population dynamics. Cannon was the first who drew attention to these problems with an integral one-space-variable condition [1], and their importance has been pointed out by Samarskii [2]. The existence and uniqueness of the classical solution of mixed problem combining a Dirichlet and integral condition for the equation of heat demonstrated by cannon [1] using the potential method. Always using the potential method, Kamynin established in [3] the existence and uniqueness of the classical solution of a similar problem with a more general representation. Subsequently, more works related to these problems with an integral one-space-variable have been published, among them, we cite the work of Benouar and Yurchuk [4], Cannon and Van Der Hoek [5, 6], Cannon-Esteva-Van Der Hoek [7], Ionkin [8], Jumarhon and McKee [9], Kartynnik [10], Lin [11], Shi [12] and Yurchuk [13]. In these works, mixed problems related to one-dimensional parabolic equations of second order combining a local condition and an integral condition was discussed. Also, by referring to the articles of Bouziani [14–16] and Bouziani and Benouar [17–19], the authors have studied mixed problems with integral conditions for some partial differential equations, specially hyperbolic equation with integral condition which has been investigated in Bouziani [20]. The present paper is devoted to the study of problems with a boundary integral two-space-variables condition for second-order hyperbolic equation. 2. Setting of the Problem In the rectangle , with , we consider the hyperbolic equation: where the coefficient is a real-valued function belonging to such that in the rest of the paper, , , , denote strictly positive constants. we adjoin to (1) the initial conditions the Neumann condition and the integral condition where and are known functions. We will assume that the function and satisfy a compatibility conditions with (5), that is, The presence of integral terms in boundary conditions can, in general, greatly complicate the application of standard functional or numerical