In this paper, we study the nonlinear equation of the form <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02ent02.gif"> where is <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02tex03.gif" align=absmiddle>the ultra-hyperbolic operator iterated k-times, defined by <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02ent03.gif" align=absmiddle>, p + q = n is the dimension of the Euclidean space <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02tex02.gif" align=absmiddle>n, (x, t) = (x1, x2,..., xn, t) <img border=0 src="../../../../../../img/revistas/cam/v28n2/a01ent09.gif" align=absmiddle><img border=0 src="../../../../../../img/revistas/cam/v28n2/a02tex02.gif" align=absmiddle>n× (0, <img border=0 src="../../../../../../img/revistas/cam/v28n2/a06tex01.gif">), 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 <img border=0 src="../../../../../../img/revistas/cam/v28n2/a02tex02.gif" align=absmiddle>n × (0, <img border=0 src="../../../../../../img/revistas/cam/v28n2/a06tex01.gif">). 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.
ultra-hyperbolic heat equation; the Dirac delta distribution; the spectrum; Fourier transform
