Uniqueness of solutions to certain hyperbolic boundary value problems in asemi-infinite strip

It is well known that boundary value problems for hyperbolic equations are in general "not well posed" problems. This paper is concerned with the uniq ueness of solutions to boundary value problems for the hyperbolic equation u(xx) - Qu =u(u). Here Q is a function of the variable x alone, and satisfi es the following conditions: (a) Q:[0, infinity)-->R; (b) Q is Lebesgue integrable on any compact subinterval of [0, infinity); (c) Q(x)-->infinity as x-->infinity. The boundary value problems are considered in an infinite strip D = {(x, t) : 0 < x < infinity, 0 < t < T}, and the classical boundary conditions are r eplaced by a limiting form of boundary conditions taken from within the inf inite strip. In doing so, we avoid the difficulties associated with the par tial derivatives of the solution on the boundary, including the particular difficulties associated with the corners. Boundary value problems for the m ore general hyperbolic equation {(Pu-x)(x) - Qu}w = (pu(t))(t) - qu are als o considered in D. Finally, as a special case we give necessary and suffici ent conditions for uniqueness of solutions to boundary value problems for t he hyperbolic equation u(xx) - x(2)u = u(u) in D. In this case, we note tha t the conditions on the solution It and its derivatives are much more relax ed.