Abstract

KANANTHAI, Amnuay  and  NONLAOPON, Kamsing. On the generalized nonlinear ultra-hyperbolic heatequation related to the spectrum. Comput. Appl. Math. [online]. 2009, vol.28, n.2, pp. 157-166. ISSN 1807-0302.  http://dx.doi.org/10.1590/S1807-03022009000200002.

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.

Keywords : ultra-hyperbolic heat equation; the Dirac delta distribution; the spectrum; Fourier transform.