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Forward-backward parabolic equations and hysteresis. (English) Zbl 0928.35084

J. Math. Sci., New York 93, No. 5, 747-766 (1999) and Zap. Nauchn. Semin. POMI 233, 183-209 (1996).
Summary: The following initial-boundary value problem for the forward-backward parabolic equation in a bounded region \(\Omega\subseteq \mathbb{R}^d\), \(1\leq d\leq 3\), is considered: \[ \Omega\times (0,T):u_t =\Delta\varphi (u),\;\partial \Omega \times(0,T): \nabla\varphi(u) \cdot n=0, \]
\[ \Omega:u (\cdot,0)= u_0\in L_\infty (\Omega),\;\varphi(u_0) \in H_1(\Omega). \] It is assumed that the function \(\varphi\) decreases monotonically on the interval \((-1,1)\), increases outside it, and that \(| u_0|\geq 1\). It is proved that this problem has entropy solutions which describe a phase transition process with hysteresis.

MSC:

35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
35K65 Degenerate parabolic equations

Citations:

Zbl 0921.00015
Full Text: DOI EuDML

References:

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