Problems for an ultrahyperbolic equation in the half-space with the boundedness condition for the solution.

*(English. Russian original)*Zbl 1131.35370
Differ. Equ. 42, No. 2, 261-268 (2006); translation from Differ. Uravn. 42, No. 2, 245-251 (2006).

From the introduction: Consider the ultrahyperbolic equation

\[ \frac{\partial^2u}{\partial x_1^2}+ \frac{\partial^2 u}{\partial x_2^2}+ \frac{\partial^2 u}{\partial x_3^2}= \frac{\partial^2u}{\partial y^2}+ \frac{\partial^2u}{\partial z^2}\tag{1} \]

in the half-space \(z\geq 0\) with additional conditions on the boundary hyperplane \(z =0\) and with the condition that the solution exists and remains bounded in the entire domain,

\[ |u({\mathbf x},y,z)|\leq m, \qquad z\geq 0.\tag{2} \]

Although the boundedness condition is quite natural, it complicates the situation. If we pose two conditions at \(z=0\), i.e., consider the Cauchy problem

\[ (3)\quad u({\mathbf x},y,0)= f({\mathbf x},y), \qquad (4)\quad \frac{\partial u}{\partial z}({\mathbf x},y,0)= g({\mathbf x},y), \]

then the solution either does not exist in the entire half-space \(z\geq 0\) or is unbounded as \(z\to\infty\) [D. P. Kostomarov, Dokl. Math. 67, No. 3, 377–381 (2003); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 390, No. 4, 443–337 (2003; Zbl 1247.35070)]. The boundedness condition (2) makes the Cauchy problem overdetermined. The contradiction can be eliminated in two ways: one either rejects the boundedness condition and returns to the Cauchy problem in the traditional statement (1), (3), (4) or modifies the initial conditions (3) and (4). The modified Cauchy problem is considered in Section 2. Section 3 deals with the boundary value problem (1)–(3) with a single condition on the boundary hyperplane \(z=0\). This problem proves to be underdetermined. We discuss some cases of additional conditions that refine the statement of problem (1)–(3) and provide the uniqueness of the solution. In Section 4, we discuss the asymptotics of solutions for large \(z\). In Section 5, we consider the case in which the space \(X\) is one-dimensional rather than three-dimensional and the ultrahyperbolic equation (1) becomes the ordinary wave equation.

\[ \frac{\partial^2u}{\partial x_1^2}+ \frac{\partial^2 u}{\partial x_2^2}+ \frac{\partial^2 u}{\partial x_3^2}= \frac{\partial^2u}{\partial y^2}+ \frac{\partial^2u}{\partial z^2}\tag{1} \]

in the half-space \(z\geq 0\) with additional conditions on the boundary hyperplane \(z =0\) and with the condition that the solution exists and remains bounded in the entire domain,

\[ |u({\mathbf x},y,z)|\leq m, \qquad z\geq 0.\tag{2} \]

Although the boundedness condition is quite natural, it complicates the situation. If we pose two conditions at \(z=0\), i.e., consider the Cauchy problem

\[ (3)\quad u({\mathbf x},y,0)= f({\mathbf x},y), \qquad (4)\quad \frac{\partial u}{\partial z}({\mathbf x},y,0)= g({\mathbf x},y), \]

then the solution either does not exist in the entire half-space \(z\geq 0\) or is unbounded as \(z\to\infty\) [D. P. Kostomarov, Dokl. Math. 67, No. 3, 377–381 (2003); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 390, No. 4, 443–337 (2003; Zbl 1247.35070)]. The boundedness condition (2) makes the Cauchy problem overdetermined. The contradiction can be eliminated in two ways: one either rejects the boundedness condition and returns to the Cauchy problem in the traditional statement (1), (3), (4) or modifies the initial conditions (3) and (4). The modified Cauchy problem is considered in Section 2. Section 3 deals with the boundary value problem (1)–(3) with a single condition on the boundary hyperplane \(z=0\). This problem proves to be underdetermined. We discuss some cases of additional conditions that refine the statement of problem (1)–(3) and provide the uniqueness of the solution. In Section 4, we discuss the asymptotics of solutions for large \(z\). In Section 5, we consider the case in which the space \(X\) is one-dimensional rather than three-dimensional and the ultrahyperbolic equation (1) becomes the ordinary wave equation.

##### MSC:

35L82 | Pseudohyperbolic equations |

35N05 | Overdetermined systems of PDEs with constant coefficients |

35B40 | Asymptotic behavior of solutions to PDEs |

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\textit{D. P. Kostomarov}, Differ. Equ. 42, No. 2, 261--268 (2006; Zbl 1131.35370); translation from Differ. Uravn. 42, No. 2, 245--251 (2006)

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##### References:

[1] | Kostomarov, D.P., Zadacha Koshi dlya ul’tragiperbolicheskikh uravnenii (The Cauchy Problem for Ultrahyperbolic Equations), Moscow, 2003. · Zbl 1065.35003 |

[2] | Kostomarov, D.P., Dokl. RAN, 2003, vol. 390, no. 4, pp. 443–447. |

[3] | Kostomarov, D.P., Dokl. RAN, 2004, vol. 396, no. 5, pp. 597–600. |

[4] | Kostomarov, D.P., Dokl. RAN, 2005, vol. 400, no. 4, pp. 449–453. |

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