On the initial boundary problem for hyperbolic equations with exponential degeneration t^ 12/7
DOI:
https://doi.org/10.26577/JMMCS202412113Keywords:
Degenerate equations, degree of degeneracy, hyperbolic equations, a priori estimateAbstract
Degenerate equations have been and are the object of numerous studies. They have not only theoretical but also practical significance. Let us only point out the fact that they arise when modeling subsonic and supersonic processes flows in a gaseous environment, filtration processes and movement of groundwater, in climate forecasts, etc.
Mathematically, the degeneracy of a differential equation can be different. In this paper we consider a degenerate equation of the form ∂t(tβ∂tu(x, t))− ∆u(x, t) = f(x, t). In a bounded cylindrical domain, when the degree of degeneracy β = 12/7, we have established the unique solvability of the Cauchy-Dirichlet problem for the considered degenerate hyperbolic equation. Based on the solution of the spectral problem for the Laplace operator with Dirichlet conditions are introduced spectral decompositions of the right side of the differential equation and the desired solution to the Cauchy-Dirichlet problem.
For the Fourier coefficients we obtain a family of Cauchy problems for a degenerate second-order ordinary differential equation, moreover, the second initial the condition must be met with weight. The latter is determined by the degree of degeneracy of the equation. The solutions to each of the Cauchy problems are represented by using Bessel functions. A priori estimates are established, on the basis of which is established the solvability of the initial-boundary problems for a degenerate hyperbolic equation.
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