Opuscula Math. 28, no. 1 (2008), 47-62

Opuscula Mathematica

# Classical and weak solutions for semilinear parabolic equations with Preisach hysteresis

Mathias Jais

Abstract. We consider the solvability of the semilinear parabolic differential equation $\frac{\partial u}{\partial t}(x,t)- \Delta u(x,t) + c(x,t)u(x,t) = \mathcal{P}(u) + \gamma (x,t)$ in a cylinder $$D=\Omega \times (0,T)$$, where $$\mathcal{P}$$ is a hysteresis operator of Preisach type. We show that the corresponding initial boundary value problems have unique classical solutions. We further show that using this existence and uniqueness result, one can determine the properties of the Preisach operator $$\mathcal{P}$$ from overdetermined boundary data.

Keywords: hysteresis, parabolic, inverse problem, uniqueness.

Mathematics Subject Classification: 35K55, 35R30.

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• Mathias Jais
• Cardiff University, School of Computer Science, Cardiff, CF24 3AA, U.K.
• Revised: 2007-04-30.
• Accepted: 2007-05-10.