IJPAM: Volume 55, No. 3 (2009)

THE GENERALIZED NONLINEAR HEAT EQUATION
AND ITS SPECTRUM

Wanchak Satsanit$^1$, Amnuay Kananthai$^2$
$^{1,2}$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

\begin{displaymath}\frac{\partial}{\partial t}u(x,t)+c^2(-\circledast)^{k}u(x,t)=f(x,t,u(x,t))\,,\end{displaymath}

where $\circledast^{k}$ is the operator iterated $k$-times, defined by

\begin{displaymath}\circledast^{k}=\left[\left(\sum^{p}_{i=1}\frac{\partial^2}{\...
...}_{j=p+1}\frac{\partial^2}{\partial x_j^2}\right)^3\right]^k\,,\end{displaymath}

where $p+q=n$ is the dimension of the Euclidean space $\mathbb{R}^n$, $u(x,t)$ is an unknown for $(x,t)=(x_1,x_2,\dots,x_n,t)\in\mathbb{R}^n\times(0,\infty)$, $k$ 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.

Received: July 28, 2009

AMS Subject Classification: 35K05, 42A38

Key Words and Phrases: diamond operator, ultra-hyperbolic, tempered distribution, Fourier transform

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 3