\frac{\partial f}{\partial t}={\rm div}\left(\nabla f +\frac{f\nabla V}{T} \right)
\frac{\partial T}{\partial t}=\kappa^\prime\frac{\partial^2T}{\partial x^2}+\kappa\nabla V.\left(\nabla f+\frac{f\nabla V}{T}\right).
with initial data f(x,0) and T(x,0).
We show that there is a unique solution for small times to the coupled nonlinear heat-diffusion equations for f(x,t) and T(x,t) if p is greater than d, nabla V and Delta V being bounded. The method is to show that the integral form of the equation defines a contraction map in a small enough ball around the free solution, in the norm
\begin{math} \|(f(,.,),T(.,.))\|=\int _0^t (\|f(..s)\|_1+\|f( ,s).\|_p)ds +\sup_{x,s\leq t}|T(x,s)| \end{math}
for small t. Our equations have the form of the Fokker-Planck equations of a process X(t). It follows that f(x,t) is the expectation of a process (starting at x) that obeys a stochastic differential equation driven by Brownian motion.
Keywords: Reaction, diffusion, Brownian motion, Smoluchowski, nonlinear.
The paper was reviewed by Lorenza Viola.
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© by Ray Streater, 14/6/00.