The generalized nonlinear heat equation and its spectrum

Abstract

In this paper, we study the nonlinear equation of the form ∂/∂t u(s, t) + c2(-*)ku(x, t) = f(x, t, u(x, t)), where*k is the operator iterated k-times, defined by*k=[(Σi=1p ∂2/ ∂xi2)3 + (Σj=p+1 ∂2/∂xi2)3]k, where p + q = n is the dimension of the Euclidean space ℝn, u(x, t) is an unknown for (x, t) = (x1,x2. . . , X n, t) ∈ ℝn × (0, ∞), k is a positive integer and c is a positive constant, f is the given function in nonlinear form depending on x, t and u(x, t). On suitable conditions for f, p, q, k and the spectrum, we obtain the unique solution u(x, t) of such equation. Moreover, if we put q = 0, k = 1, we obtain the solution of non-linear heat equation. © 2009 Academic Publications.