Abstract
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.
Highlights
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. They appear in the case where a direct measurement quantity is impossible; their mean values are known
We deal with a class of quasilinear hyperbolic equations (T is a positive constant):
We look for a weak solution in the following sense
Summary
The study of initial-boundary value problems for hyperbolic equations with boundary integral conditions has received considerable attention. 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 To these works, in the present paper, we employ the Rothe time-discretization method to construct the solution. This method is a convenient tool for both the theoretical and numerical analyses of the stated problem. Under assumptions (H1), (H2), and (H3), problem (1.5)–(1.8) admits a unique weak solution u, in the sense of Definition 2.2, that depends continuously upon the data f , U0, and U1. As n→∞, where the sequences {un}n and {δun}n are defined in (3.18) and (3.19), respectively
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