On the Well-Posedness of the Cauchy Problem for the Generalized Telegraph Equations

This paper establishes the uniform well-posedness of the Cauchy problem for generalized telegraph equations with variable coefficients, of which the classical telegraph equation is a particular case. The well-posedness of a mathematical problem is one of the main requirements for its numerical solution.
For the classical telegraph equation, Riemann's method enables us to solve the Cauchy problem in the class of twice continuously differentiable functions explicitly. The question of stability of the solution in dependence on the initial data, which requires us to work in suitable metric spaces, usually is not discussed; however, it appears to be one of the most important questions once the existence and uniqueness of the solution are known. In this note we use the theory of continuous semigroups of linear operators to establish the uniform well-posedness of the Cauchy problem in the spaces of integrable functions with exponential weight for several classes of differential equations with variable coefficients. We obtain the exact solution to the Cauchy problem and indicate conditions on the coefficients ensuring that the problem is uniformly well-posed in certain functional spaces. These results imply the uniform well-posedness of the Cauchy problem for the classical telegraph equation with constant coefficients.

Keywords

telegraph equation; well-posedness; semigroups; cosine function; Cauchy problem; fractional powers of operators.

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