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ISSN 0012-2661, Differential Equations, 2006, Vol. 42, No. 4, pp. 519–536. c Pleiades Publishing, Inc., 2006. Original Russian Text c R.V. Andrusyak, V.M. Kirilich, A.D. Myshkis, 2006, published in Differentsial’nye Uravneniya, 2006, Vol. 42, No. 4, pp. 489–503. PARTIAL DIFFERENTIAL EQUATIONS Local and Global Solvability of the Quasilinear Hyperbolic Stefan Problem on the Line R. V. Andrusyak, V. M. Kirilich, and A. D. Myshkis Lviv National University, Lviv, Ukraine Moscow State University of Railway Engineering (Railway Communications), Moscow, Russia Received December 16, 2004 DOI: 10.1134/S0012266106040094 1. INTRODUCTION Stefan problems (problems with unknown boundaries) describe a wide variety of processes arising in numerous applied problems modeled by partial differential equations [1–3] (see the bibliography therein). In the present paper, following [3–5 (Sec. 4)], we prove existence and uniqueness theorems for local and global solutions of a problem with unknown boundaries for a quasilinear hyperbolic system of first-order equations of the general form (1) (see below) with two independent variables. The local solvability of hyperbolic Stefan problems of a different form was considered in [3, 6–9]. Global solvability was studied in [10–12] for some specific mixed hyperbolic problems with unknown boundaries. 2. STATEMENT OF THE PROBLEM In the present
Differential Equations – Springer Journals
Published: May 17, 2006
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