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Compactness properties for families of quasistationary solutions of some evolution equations. (English) Zbl 1008.47065

In passing to the limit in certain phase field models, one encounters sequences of space-time dependent functions \(\{ \theta_n \}\), \(\{ \chi_n \}\) for which the sum \(\theta_n + \chi_n\) converges in some \(L^p\)-space as \(n \uparrow \infty\) while the time integrals of some functional evaluated on \(\theta_n, \chi_n\) are uniformly bounded. P. I. Plotnikov and V. N. Starovoitov [Differ. Equations 29, No. 3, 395-404 (1993; Zbl 0802.35165)] used compactness and reflexivity arguments to prove an abstract result concerning the question when \(\theta_n\) and \(\chi_n\) converge separately. In the present paper, a general setting for this and related problems is developed which provides necessary and sufficient conditions for their solvability. These conditions rely on general topological and coercivity properties and do not require reflexivity.

MSC:

47J25 Iterative procedures involving nonlinear operators
80A22 Stefan problems, phase changes, etc.
37L05 General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
47J35 Nonlinear evolution equations
49J45 Methods involving semicontinuity and convergence; relaxation

Citations:

Zbl 0802.35165
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References:

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