On a Generalized Euler–Poisson–Darboux Equation

Solutions of the following Cauchy problem are obtained by means of Fourier transform methods: \[\begin{gathered} \sum_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_i^2 }} - \frac{{\partial ^2 u}}{{\partial t^2 }} - \frac{\lambda }{t}\frac{{\partial u}}{{\partial t}}} = c^2 u, \hfill \\ \left. \begin{gathered} u(x,t) = f(x) \hfill \\ u_t (x,t) = 0 \hfill \\ \end{gathered} \right\}\quad {\text{at }} t = 0. \hfill \\ \end{gathered} \] Special cases for the parameters $\lambda $ and c are considered and the regularity of the solutions is studied in some detail.

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