# IJPAM: Volume 55, No. 3 (2009)

THE GENERALIZED NONLINEAR HEAT EQUATION
AND ITS SPECTRUM

Wanchak Satsanit, Amnuay Kananthai
Department of Mathematics
Chiang Mai University
Chiang Mai, 50200, THAILAND
e-mail: aunphue@live.com

Abstract.In this paper, we study the nonlinear equation of the form

where is the operator iterated -times, defined by

where is the dimension of the Euclidean space , is an unknown for , is a positive integer and is a positive constant, is the given function in nonlinear form depending on and . On suitable conditions for , , , and the spectrum, we obtain the unique solution of such equation. Moreover, if we put , we obtain the solution of non-linear heat equation.