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The Cauchy problem for semilinear hyperbolic equation with characteristic degeneration on the initial hyperplane
Author(s) -
Zhang Kangqun
Publication year - 2018
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.4750
Subject(s) - mathematics , hyperplane , bessel function , mathematical analysis , initial value problem , cauchy problem , hyperbolic partial differential equation , heat equation , cauchy distribution , partial differential equation , geometry
In this paper, we study the well posed‐ness of Cauchy problem for a class of hyperbolic equation with characteristic degeneration on the initial hyperplane. By a delicate analysis of two integral operators in terms of Bessel functions, we give the uniform weighted estimates of solutions to the linear problem with a parameter m ∈(0,1) and establish local and global existences of solution to the semilinear equation. Meanwhile, we derive the existence of solutions to semilinear generalized Euler‐Poisson‐Darboux equation with a negative parameter α ∈(−1,0).