Journal of Inequalities and Applications 
Volume 7 (2002), Issue 4, Pages 517-553
doi:10.1155/S1025583402000267

On vanishing at infinity solutions of higher order linear hyperbolic equations

T. Kiguradze1 and I. P. Stavroulakis2

 1l. Javakhishvili Tbilisi State University, Faculty of Physics, 3 Chavchavadze Ave., Tbilisi 380028, Georgia
2Department of Mathematics, University of Ioannina, Ioannina 45110, Greece
Received 10 April 2001; Revised 5 July 2001

Abstract

In the half strip Db=[0,+∞)×[0,b] the linear hyperbolic equation ∂m+nu∂xm∂yn=p0(x,y)u+p1(y)∂mu∂xm+p2(x)∂nu∂yn with coefficients p0∈Lloc2(Db),p1∈L2([0,b]) and p2∈Lloc2(ℝ) is considered. Sufficient conditions of existence of solutions to this equation satisfying the conditions ∂ku(x,y)∂yk|y=0=0  (k=0,…,n0−1),∂ku(x,y)∂yk|y=b=0  (k=0,…,n−n0−1),∂ju(x,y)∂xj|x=0=φj(y),  limx→+∞∂ju(x,y)∂xj=0  (j=0,…,m0−1) are established, where m0 and n0 are the integral parts of m/2 and n/2.