Abstract:
This article discusses the problem with nonlocal boundary and integral conditions for a loaded fourth-order hyperbolic equation with impulsive effects. The investigated equation can be interpreted as a generalized Bussinesque-Lev equation, which arises when modeling transverse vibrations of short thick rods, as well as when studying wave processes in periodically layered media. The considered equation generally has nonsmooth coefficients belonging to the $L_p$ space. The solution to the posed problem is sought in a functional space of the Sobolev type. The representation of elements of this space is used significantly. The elements of this functional space allow discontinuities of the first kind along lines parallel to the characteristic curves. Using this representation, the problem is reduced to an equivalent integral equation. It is proven that for establishing a homeomorphism of the operator corresponding to the posed problem between certain pairs of Banach spaces, it is necessary and sufficient for the corresponding integral equation to have a unique solution. Furthermore, the existence and uniqueness of the solution to the posed problem are established. Next, the corresponding conjugate integral equation is constructed, and using an a priori estimate, the existence and uniqueness of its solution are proven. The fundamental solution to the posed problem is defined as a particular case of the conjugate integral equation. Based on the fundamental solution, an integral representation of the solution to the posed problem is obtained
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