On the generalized nonlinear ultra-hyperbolic heatequation related to the spectrum

Abstract

In this paper, we study the nonlinear equation of the form where is the ultra-hyperbolic operator iterated k-times, defined by , p + q = n is the dimension of the Euclidean space n, (x, t) = (x1, x2,..., xn, t) n× (0, ), k is a positive integer and c is a positive constant. On the suitable conditions for f , u and for the spectrum of the heat kernel, we can find the unique solution in the compact subset of n × (0, ). Moreover, if we put k = 1 and q = 0 we obtain the solution of nonlinear equation related to the heat equation. Mathematical subject classification: 35L30, 46F12, 32W30.