Riv. Mat. Univ. Parma, Vol. 1, No. 2, 2010

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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