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A nonlocal parabolic problem arising in linear friction welding

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Guo, Y.-J.L. Osaka J. Math. 47 (2010), 33–40 A NONLOCAL PARABOLIC PROBLEM ARISING IN LINEAR FRICTION WELDING YUNG-JEN LIN GUO (Received September 5, 2008) Abstract We study a nonlocal parabolic problem modeling the temperature in a thin region during linear friction welding for a hard material. We derive the structures of steady states of this nonlocal problem and its associated approximated problems. Moreover, some remarks on the parabolic problem are given. 1. Introduction In this paper, we study the following initial boundary value problem: ut D uxx � u�p �R 1 0 u �p(x , t) dx�1C1=p , 0 < x <1, t > 0,(1.1) ux (0, t) D 0, ux (1, t) D 1, t > 0,(1.2) u(x , 0) D u0(x), x � 0,(1.3) where the parameter p > 1 and u0 is a positive smooth function defined on [0, 1). This model arises in the study of linear friction welding for a hard material (cf. [8] and its references). In particular, in the real model the parameter p is close to 4. To study this problem, in [8] they proposed to study the following approximated problem in bounded intervals: ut D uxx � � Z R 0 u�p(x , t) dx � �1�1=p u�p, 0 < x < R, t > 0,(1.4) ux (0, t) D 0, u(R, t) D R, t > 0,(1.5) u(x , 0) D u0(x), 0 � x � R,(1.6) 2000 Mathematics Subject Classification. Primary 34K05, 34A34; Secondary 34K60, 34E05. This work was partially supported by the National Science Council of the Republic of China under the grant NSC 96-2115-M-003-004. 34 Y.-J.L. GUO where R is any positive constant. It is convenient to introduce the following transfor- mation: (1.7) x D Ry, t D R2s, u(x , t) D Rv(y, s). Then (1.4)–(1.6) is reduced to the following problem: vs D vyy � � � Z 1 0 v �p(y, s) dy � �1�1=p v �p , 0 < y < 1, s > 0,(1.8) vy(0, s) D 0, v(1, s) D 1, s > 0,(1.9) v(y, 0) D v0(y) WD u0(Ry)R , 0 � y � 1,(1.10) where � D �(R) WD R1�1=p. It is well-known that the structure of steady states plays an important role in the study of parabolic problem. Therefore, the aim of this pa

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