Singular solutions to Protter's problem for the 3-D wave equation involving lower order terms

Grammatikopoulos, Myron.K. ; Hristov, Tzvetan D. ; Popivanov, Nedyu I.

Electronic Journal of Differential Equations, Tome 2003 (2003).

Résumé

Summary: In 1952, at a conference in New York, Protter formulated some boundary value problems for the wave equation, which are three-dimensional analogues of the Darboux problems (or Cauchy-Goursat problems) on the plane. Protter studied these problems in a 3-D domain $\Omega_0$, bounded by two characteristic cones $\Sigma_1$ and $\Sigma_{2,0}$, and by a plane region $\Sigma_0$. It is well known that, for an infinite number of smooth functions in the right-hand side, these problems do not have classical solutions. Popivanov and Schneider (1995) discovered the reason of this fact for the case of Dirichlet's and Neumann's conditions on $\Sigma_0$: the strong power-type singularity appears in the generalized solution on the characteristic cone $\Sigma_{2,0}$. In the present paper we consider the case of third boundary-value problem on $\Sigma_0$ and obtain the existence of many singular solutions for the wave equation involving lower order terms. Especifically, for Protter's problems in $\mathbb{R}^{3}$ it is shown here that for any $n\in N$ there exists a $C^{n}(\bar{\Omega}_0)$-function, for which the corresponding unique generalized solution belongs to $C^{n}(\bar{\Omega}_0\backslash O)$ and has a strong power type singularity at the point $O$. This singularity is isolated at the vertex $O$ of the characteristic cone $\Sigma_{2,0}$. and does not propagate along the cone. For the wave equation without lower order terms, we presented the exact behavior of the singular solutions at the point $O$.

Détail

Classification : 35L05, 35L20, 35D05, 35A20
Keywords: wave equation, boundary value problems, generalized solutions, singular solutions, propagation of singularities