On the nonlinear Diamond operator related to the wave equation

摘要

In this paper, we study the solution of nonlinear equation lozenge(k)u(x) = f(x,Delta(k-1)square(k)u(x)) where lozenge(k) is the Diamond operator iterated k times and is defined by lozenge(k) = ((Sigma(i=1)(p) partial derivative(2)/partial derivativex(i)(2))(2) - (Sigma(j=p+1)(p+q) partial derivative(2)/partial derivativex(j)(2))(2))(k) or the operator lozenge(k) can be expressed by lozenge(k) = square(k)Delta(k) = Delta(k)square(k), The operators Delta(k) and square(k) are Laplacian and the ultrahyperbolic operator iterated k times, defined by square(k) = (partial derivative(2)/partial derivativex(1)(2) + partial derivative(2)/partial derivativex(2)(2) +...+ partial derivative(2)/partial derivativex(p)(2) - partial derivative(2)/partial derivativex(p+1)(2) -...-partial derivative(2)/partial derivativex(p+q)(2))(k) and Delta(k) = (partial derivative(2)/partial derivativex(1)(2) + partial derivative(2)/partial derivativex(2)(2) +...+ partial derivative(2)/partial derivativex(n)(2))(k), p + q = n is the dimension of the n-dimensional Euclidean space R-n, x = (x(1),x(2),...,x(n)) is an element of R-n, k is a nonnegative integer, u(x) is an unknown and f is a given function. It is found that, the existence of the solution u(x) of such equation depending on the conditions of f and Delta(k-1)square(k)u(x) and moreover such solution u(x) related to the wave equation depending on the conditions of p, q and k. (C) 2002 Elsevier Science Ltd. All rights reserved. [References: 6]