Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 149, pp. 1-20.
Title: Exact behavior of singular solutions to Protter's problem
with lower order terms
Authors: Aleksey Nikolov (Univ. of Sofia, Bulgaria)
Nedyu Popivanov (Univ. of Sofia, Bulgaria)
Abstract:
For the (2+1)-D wave equation Protter formulated (1952) some boundary
value problems which are three-dimensional analogues of the Darboux
problems on the plane. Protter studied these problems in a 3-D domain,
bounded by two characteristic cones and by a planar region. Now it is
well known that, for an infinite number of smooth functions in the
right-hand side, these problems do not have classical solutions, because of
the strong power-type singularity which appears in the generalized
solution. In the present paper we consider the wave equation involving
lower order terms and obtain new a priori estimates describing the exact
behavior of singular solutions of the third boundary value problem.
According to the new estimates their singularity is of the same order as in
case of the wave equation without lower order terms.
Submitted May 8, 2012. Published August 29, 2012.
Math Subject Classifications: 35L05, 35L20, 35D05, 35A20.
Key Words: Wave equation; boundary value problems; generalized solutions;
singular solutions; propagation of singularities.