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Monotonicity formula for a problem with hysteresis. (English. Russian original) Zbl 1401.35302

Dokl. Math. 97, No. 1, 49-51 (2018); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 478, No. 4, 379-381 (2018).
This short note states without proof a local monotonicity formula for the initial boundary value problem \[ \begin{aligned} \partial_t u - \Delta u + h[u] & = 0 \;\text{ in }\; (0,T)\times \mathcal{U}\;, \\ u(0) & = \varphi \;\text{ in }\; \mathcal{U}\;,\end{aligned} \] supplemented with either Dirichlet or Neumann boundary conditions. Here, \(h[u]\) is a discontinuous nonlinearity built upon a function \(f:\mathbb{R}\to \{-1,1\}\) and is defined as follows: \(h[u](t,x)=f(u(t,x))=-1\) if \(u(t,x)\leq \alpha\), while \(h[u](t,x)=f(u(t,x))=1\) if \(u(t,x)\geq \beta\). When \(u(t,x)\in (\alpha,\beta)\), we set \(h[u](t,x)=f(u(\hat{t}(x),x))\), where \(\hat{t}(x)\leq t\) is the largest time in \([0,t]\) such that \(u(\hat{t}(x),x)\in\mathbb{R}\setminus (\alpha,\beta)\).

MSC:

35R05 PDEs with low regular coefficients and/or low regular data
35R35 Free boundary problems for PDEs
34C55 Hysteresis for ordinary differential equations
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[1] Weiss, G. S., No article title, SIAM J. Math. Anal., 30, 623-644, (1999) · Zbl 0922.35193 · doi:10.1137/S0036141097327409
[2] Apushkinskaya, D. E.; Uraltseva, N. N., No article title, Phil. Trans. R.^Soc. A, 373, 20140271, (2015) · Zbl 1353.35318 · doi:10.1098/rsta.2014.0271
[3] P. Curran, P. Gurevich, and S. Tikhomirov, in Control of Self-Organizing Nonlinear Systems, Ed. by E. Schöll, S. H. L. Klapp, and P. Hövel (Springer, Berlin, 2016), pp. 211-234. · Zbl 1329.93002
[4] Apushkinskaya, D. E.; Uraltseva, N. N., No article title, Interfaces Free Boundaries, 17, 93-115, (2015) · Zbl 1321.35261 · doi:10.4171/IFB/335
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