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A Cauchy problem for an ultrahyperbolic equation. (English. Russian original) Zbl 1029.35069
Differ. Equ. 38, No. 8, 1155-1161 (2002); translation from Differ. Uravn. 38, No. 8, 1085-1090 (2002).
From the text: Consider the ultrahyperbolic equation \[ {\partial^2 u\over \partial x^2_1}+ {\partial^2 u\over\partial x^2_2}+ {\partial^2 u\over\partial x^2_3}= {\partial^2 u\over\partial t^2_2}+ {\partial^2 u\over\partial t^2_3}.\tag{1} \] We refer to the spaces \(X_3\) and \(T_3\) as the geometric space and the time space, respectively, and equip them with spherical coordinate \(r,\alpha,\beta\) and \(t,\theta,\varphi\). For Eq. (1), we pose the Cauchy problem with initial conditions on the sphere \(t=t_0\) in the space \(T_3\): \[ u(r,\alpha, \beta,t_0, \theta,\varphi)= f(r, \alpha, \beta,\theta, \varphi),\quad {\partial u\over\partial t}(r,\alpha, \beta, t_0,\theta, \varphi)= g(r,\alpha,\beta, \theta,\varphi). \tag{2} \] The solution will be sought outside the sphere, i.e., for \(t\geq t_0\). A characteristic feature of problem (1), (2) is the fact that the variables \(t_1, t_2\), and \(t_3\) play equal roles, which is not the case in the classical statement of initial conditions on the hyperplane \(t_3=0\).
Let us consider two special cases of problem (1), (2), which reveal two opposite tendencies in it.
1. Let the initial functions \(f\) and \(g\) be independent of the angles \(\theta\) and \(\varphi\). Then problem (1), (2) has spherical symmetry in the time space \(T_3:u=u (r,\alpha, \beta,t)\). In this case the ultrahyperbolic equation (1) can be reduced by the change \(u=\nu/t\) of the unknown function to the usual wave equation \(\Delta\nu= \partial^2\nu/ \partial t^2\), and the solution of the reduced problem is given by the Poisson formula.
2. Let the initial functions \(f\) and \(g\) be independent of the spatial coordinates \(r,\alpha\), and \(\beta\). Then we obtain the Cauchy problem for the Laplace equation in the time space \(T_3\) with initial conditions on the sphere.
Depending on the specific form of the initial conditions (2), the solution of the general problem combines the properties of both the solution of the hyperbolic problem 1 and the solution of the elliptic problem 2 to some extent. In the present paper, we analyze this phenomenon.

MSC:
35G10 Initial value problems for linear higher-order PDEs
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