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A nonlocal parabolic problem arising in linear friction welding
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.
... [6] and references therein). For some related works on nonlocal parabolic problems, we also refer the reader to [1][2][3][4][5][6]. ...
... The steady states of (1.1) has been studied in [5]. The main purpose of this paper is to answer the question raised in [5], namely, whether the solution of (1.1) exists globally (in time). ...
... The steady states of (1.1) has been studied in [5]. The main purpose of this paper is to answer the question raised in [5], namely, whether the solution of (1.1) exists globally (in time). In [6], numerical simulations indicate that the solution of (1.1) exists globally. ...
We study a nonlocal parabolic problem airing in the modeling of linear friction welding. Using some a priori estimates, we derive the global in time existence of solution of this nonlocal problem.
The heat equation with a nonlocal nonlinearity , subject to is studied. Stability-instability is analyzed and finite time quenching results are given. Discussions are also extended to more general problems.
In this paper, we study a nonlocal parabolic problem arising in the study of a micro-electro mechanical system. The nonlocal nonlinearity involved is related to an integral over the spatial domain. We first give the structure of stationary solutions. Then we derive the convergence of a global (in time) solution to the maximal solution as the time tends to infinity. Finally, we provide some quenching criteria.
A non-local parabolic equation modelling linear friction welding is studied. The equation applies on the half line and contains
a non-linearity of the form . For f(u) = eu, global existence and convergence to a steady state are proved. Numerical calculations are also carried out for this case
and for f(u) = (– u)1/a.
We study the long-time behavior of solutions of Burgers' equation with nonlocal nonlinearities u t = u xx + "uu x + 1 2 Gamma aku(Delta; t)k pGamma1 + b Delta u, 0 ! x ! 1, a; " 2 , b ? 0, p ? 1, subject to u(0; t) = u(1; t) = 0. A stability--instability analysis is given in some detail, and some finitetime blow-up results are given. The research of Deng and Levine was supported by NSF Grant DMS--8822788, that of Kwong by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U. S. Department of Energy, under Contract W-31-109-Eng-38. 1 Introduction In this paper we consider the following initial-boundary value problem: 8 ? ? ? ? ! ? ? ? ? : u t = u xx + "uu x + 1 2 i aku(Delta; t)k pGamma1 + b j u; 0 ! x ! 1; t ? 0; u(0; t) = u(1; t) = 0; t ? 0; u(x; 0) = u 0 (x); 0; x 1: (P) Here a, b, ", and p are given constants, with " 0 (without loss of generality), b ? 0, and p ? 1; and u 0 (Delta) is a continuous function with u 0 (0) = ...
The method of subsolution and the comparison principle are applied to the study of the blow-up behavior of the solution of an integrodifferential equation arising from the nuclear reactor dynamics. We give a blow-up criterion and study the blow-up set and the blow-up rate.
We study the solution for the initial boundary value problem of a nonlocal semilinear
heat equation. It is well-known that the solution quenches in finite time for certain
choices of initial data. We first prove that there is only one quenching point for
symmetric initial data with one peak. Then we derive a quenching rate estimate. It turns
out that the constant in the quenching rate estimate depends on the solution itself due to
the nonlocal nonlinearity.
In this paper we examine the first initial boundary value problem for ut=uxx + (1 – u)–, > 0, > 0,on (0, 1) (0, ) from the point of view of dynamical systems. We construct the set of stationary solutions, determine those which are stable, those which are not and show that there are solutions with initial data arbitrarily close to unstable stationary solutions which quench (reach one in finite time). We also examine the related problem ut=uxx, 0 x t > 0;u(0,t)=0, (1 – u(1, t))–. The set of stationary solutions for this problem, and the dynamical behavior of solutions of the time dependent problem are somewhat different.