Rotstein, H. G.; Domoshnitsky, A. I.; Nepomnyashchy, A. A. Phase transition dynamics with memory. (English) Zbl 1048.45004 Funct. Differ. Equ. 5, No. 3-4, 439-451 (1998). 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. Cited in 6 Documents 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. Keywords:partial integral-differential equations; phase field equations; phase transitions; memory effects; asymptotic analysis; system PDFBibTeX XMLCite \textit{H. G. Rotstein} et al., Funct. Differ. Equ. 5, No. 3--4, 439--451 (1998; Zbl 1048.45004)