Forward-backward parabolic equations and hysteresis

P. I. Plotnikov

Zapiski Nauchnykh Seminarov POMI, Boundary-value problems of mathematical physics and related problems of function theory. Part 27, Tome 233 (1996), pp. 183-209

Résumé

The following initial-boundary value problem for the forwardbackward parabolic equation in the bounded region $\Omega\in R^d$, $1\le d\le3$, is considered, $$ \begin{gathered} \Omega\times(0,T)\colon\ u_t=\Delta\varphi(u),\qquad\partial\Omega\times(0,T)\colon\ \nabla\varphi(u)\cdot n=0,\\ \Omega\colon\ u(\cdot,0)=u_0\in L_\infty(\Omega),\qquad\varphi(u_0)\in H_1(\Omega). \end{gathered} $$ It is supposed that the function $\varphi$ decreases monotonically on the interval $(-1,1)$ increases outside one and $|u_0|\ge1$. It is proved that this problem has the entropy solutions which describe the phase transition process with hysteresis. Bibl. 11 titles.