The Cauchy–Dirichlet Problem for \(n\)-Dimensional Hyperbolic Equations with a \(p\)-th Degree Trigonometric Source

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This paper investigates the Cauchy–Dirichlet problem for a class of \(n\)-dimensional non-homogeneous hyperbolic equations characterized by a complex source term. The governing equation incorporates first-order derivatives with respect to both time and spatial coordinates, representing systems with internal dissipation (energy loss) and convection. The non-homogeneous part of the equation is given as a \(p\)-th degree power of a multi-harmonic trigonometric sum, which indicates that the temporal dynamics of the system are significantly complex. Since the considered mixed problem is in an \(n\)-dimensional setting, we establish the existence and uniqueness of the generalized solution within the framework of Sobolev spaces, specifically \(H^1_0(\Omega)\). The convergence of the obtained Fourier series is rigorously proved in the energy norm, ensuring the stability of the analytical solution found for \(n\)-dimensional domains. Using the method of eigenfunction expansion (separation of variables) in \(n\)-dimensional rectangular domains, a complete analytical and exact solution has been found. Here, the source term is determined by applying the Binomial Theorem, which allows for an explicit representation of the modal forcing coefficients. The temporal variation of each spatial mode is determined via Duhamel’s principle, and a bounded solution is obtained that accounts for both the initial conditions and external multi-harmonic excitations.

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Related topics: Differential Equations and Boundary Problems · Algebraic and Geometric Analysis · Stability and Controllability of Differential Equations