**Giovanni Cimatti**

*On the functional solutions of a system of Partial Differential Equations relevant in mathematical physics*

**Pages:** 423-439

**Received:** 13 July 2010

**Accepted in revised form:** 4 November 2010

**Mathematics Subject Classification (2000):** 35B15, 35J66.

**Keywords:**
Functional solutions, system of PDE, Thermistor problem, Volterra-Fredholm integral equation.

**Abstract:**
We study the system of P.D.E.
\begin{equation*}
\nabla\cdot\biggl[\sum_{j=1}^na_{ij}({\bf u},w)\nabla u_j+b_i({\bf u},w)\nabla
w\biggl]=0,\ i=1,..,n
\end{equation*}
\begin{equation*}
\nabla\cdot(K({\bf u},w)\nabla w)=0\ {\hbox{in}}\ {\Omega}
\end{equation*}
and the class of its solutions
\( ({\bf u}({\bf x}),w({\bf x}))=(u_1({\bf x}),..,u_n({\bf x}),w({\bf x})) \)
which occurs when a
functional relation between \({\bf u}({\bf x}) \) and \(w({\bf x})\) of the form
\({\bf u}({\bf x})={{\bf U}}(w({\bf x}))\) exists.
If the solution satisfies constant
boundary conditions \({{\bf U}}(w)\) is shown to exist and to satisfy a
Volterra-Fredholm integral equation.
An application to the
thermistor problem and to a bifurcation problem of filtration in
porous media is also given.

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