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The Cauchy problem for a second-order nonlinear hyperbolic equation with initial data on a line of parabolicity

Published online by Cambridge University Press:  17 April 2009

John M.S. Rassias
John M.S. Rassias
Affiliation:
II Dervenakion Str, Daphne, Athens, Greece.
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Abstract

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In this paper we study the Cauchy problem for the second order nonlinear hyperbolic partial differential equation

with initial conditions

where

and |u|, |ux|, |uy| < ∞, y ≥ 0, r = r(x) ∈ C4(·), ν = ν(x) ∈ C4(·).

These conditions on k, H, f, r, and ν are assumed to be satisfied in some sufficiently small neighborhood of the segment I, y = 0, in the upper half-plane y > 0

This paper generalizes the results obtained by N.A. Lar'kin (Differencial'nye Uravnenija 8 (1972), 76–84), who has treated the special case H = H(x, y, u); that is, the quasi-linear hyperbolic equation (*).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 20 , Issue 3 , June 1979 , pp. 345 - 365
Copyright
Copyright © Australian Mathematical Society 1979

References

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