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Nonsmooth kernels in a phase relaxation problem with memory. (English) Zbl 0958.35158

The authors study the foflowing system of partial differential equations with convolution integrals modelling linear memory effects: \[ \begin{aligned} \partial_t(\varphi_0 \theta+\psi_0 \chi+\varphi* \theta+\psi* \chi)-k_0 \Delta \theta -k*\Delta \theta=f,\\ \alpha\partial_t \chi\in\beta (\theta, \chi)-H^{-1} (\chi)\end{aligned} \tag{1} \] for \((x,t)\in\Omega \times(0,T) \subseteq \mathbb{R}^3 \times (0,T)\). Here \(\theta(x,t)\) denotes the temperature, \(\chi(x,t)\) the fraction of the liquid component in a mixture, \(\varphi_0,\psi_0,k_0\) and \(\alpha\) are positive constants, \(f\) represents a known function which depends both on the heat supply and on the given past history of \(\theta\) and \(\chi\). The nonlinear function \(\beta\) is uniformly Lipschitz continuous, and \[ H^{-1}(r)=\begin{cases} \{0\}\times(+\infty,0], \quad &\text{if }r=0,\\ \bigl\{(r,0) \bigr\}, \quad &\text{if }0<r<1,\\ \{1\}\times [0,\infty), \quad &\text{if }r=1,\\ \varnothing,\quad & \text{otherwise}. \end{cases} \] The evolution equation (1) for the fraction \(\chi\) can model undercooling or superheating.
In a previous article [P. Colli and M. Grasseli, J. Integral Equations Appl. 5, No. 1, 1-22 (1993; Zbl 0781.45009)] an existence and uniqueness theorem for the initial-boundary value problem with homogeneous Dirichlet data was proved under the assumption that the memory kernels \(\varphi\) and \(k\) are absolutely continuous. In the article under consideration it is shown that the same result holds if \(\varphi\) and \(k\) are just square integrable and if \(k\) satisfies an integral inequality assuring the second law of thermodynamics to hold. The last section of the article contains a few remarks on the regularity of the solution.

MSC:

35R70 PDEs with multivalued right-hand sides
35Q72 Other PDE from mechanics (MSC2000)
74N30 Problems involving hysteresis in solids
74F05 Thermal effects in solid mechanics

Citations:

Zbl 0781.45009
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References:

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