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Phase transition dynamics with memory. (English) Zbl 1048.45004

Summary: We consider the following partial integral-differential equations which generalize the classical phase field equations with a non-conserved order parameter and describe the process of phase transitions where memory effects are present, \[ \begin{cases} u_t+ \varepsilon^2\varphi_t= \int^t_0 a_1(t-\tau)\Delta u(\tau)d\tau\\ \varepsilon^2\varphi_t= \int^t_0 a_2(t-\tau)\bigl[ \varepsilon^2\Delta \varphi+f(\varphi) +\varepsilon u\bigr] (\tau)d\tau, \end{cases} \] where \(\varepsilon\) is a small parameter. The functions \(u\) and \(\varphi\) represent the temperature field and order parameter respectively. The kernels \(a_1\) and \(a_2\) are assumed to be piecewise continuous, differentiable at the origin, scalar-valued functions on \((0,\infty)\) with \(a_1(\infty)= a_2 (\infty)=0\), independent of \(\varepsilon\) and such that they satisfy the following conditions \(\int^\infty_0 a_i(t)dt <\infty\), \(\int_0^\infty \overline \alpha_i(s) ds<\infty\) and \(\int_0^\infty\overline \alpha_i(s)s\, ds<\infty\). By means of a formal asymptotic analysis we show that to the leading order and under suitable assumption on the kernels, the integro-differential system behave like a system of partial differential equations obtained by considering appropriate exponentially decreasing kernels.

MSC:

45K05 Integro-partial differential equations
45G15 Systems of nonlinear integral equations
45M05 Asymptotics of solutions to integral equations
80A22 Stefan problems, phase changes, etc.
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