We study a nonlocal parabolic problem airing in the modeling of linear friction welding. Using so... more 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.
u{x, 0) = u0(x), x € [-1,1], (1.3) where 0<a<l,/?>0, Z>0, and uq(x) >0, Vx G [-1, Z]. Without los... more u{x, 0) = u0(x), x € [-1,1], (1.3) where 0<a<l,/?>0, Z>0, and uq(x) >0, Vx G [-1, Z]. Without loss of generality, we may assume that uq(x) is smooth and bounded above by 1 such that uo(±Z) = 1. Since uo(x) is positive, the local (in time) existence and uniqueness of a classical solution of the problem (1.1)-(1.3) are trivial (see [8]). Many results in quenching, such as single point quenching and profiles, are similar to those blow-up results ([3], [5] and the references therein). The system (1.1)-(1.3) in the case a < 1 models fast diffusion and absorption (see [8]). The case a-1 was studied in [1], [2], [6], [7]. To study the asymptotic behavior of quenching, a simple Lyapunov function can be constructed in the case a = 1. When a < 1, however, such an explicit Lyapunov function is not available. In this paper, we use the idea of [4] to construct a Lyapunov function, which is rather technical. We say that the solution u quenches if the minimum of u(-,t) reaches zero at some finite time. We shall always assume that u quenches at a quenching time T < oo. In this case the right-hand side of Eq. (1.1) becomes singular and we no longer have a classical solution after this quenching time. A point (c, T) is said to be a quenching point if there is a sequence {(xn,tn)} such that xn-> c, tn-> T, and u(xn,tn)-> 0 as n-> oo. It is shown in [8] that there can only be finitely many quenching points that stay a positive distance away from the boundary |x| = I for any positive initial data. The purpose of this paper is to study how the solution tends to zero. For simplicity we shall only consider the symmetric case, i.e., the case that uq is symmetric with respect to x = 0 and is monotone increasing in \x\. It has been shown that (0, T) is the only possible
When a crack s propagates in an elastic medium the stress intensity factors evolve with the tip x... more When a crack s propagates in an elastic medium the stress intensity factors evolve with the tip x(s) of s. In this paper we derive formulae which describe the evolution of these stress intensity factors for a homogeneous isotropic elastic medium under plane strain conditions. Denoting by ψ = ψ(x, s) the stress potential (ψ is biharmonic and has zero traction along the crack s) and by κ(s) the curvature of the crack at the tip x(s), we prove that the stress intensity factors A 1 (s), A 2 (s), as functions of s, satisfy: dA 1 ds = 3 2 κ(s)A 2 (s) +β 1 (s), dA 2 ds = − 3 2 κ(s)A 1 (s) +β 2 (s), whereβ 2 (s),β 1 (s) are stress intensity factors of the tangential derivative of ψ − 4κ(s)A 2 (s)r 3/2 cos 1 2 θ in the polar coordinate system at x(s) with θ = 0 in the direction of the crack at x(s). The case of antiplane shearing is also briefly considered; in this case ψ is harmonic.
In this paper, we study a nonlocal parabolic problem arising in the study of microelectro mechani... more In this paper, we study a nonlocal parabolic problem arising in the study of microelectro 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 global (in time) solution to the maximal solution as the time tends to infinity. Finally, we provide some quenching criteria.
Discrete and Continuous Dynamical Systems - Series S, 2010
The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo ... more The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D.
We study a two-point free boundary problem for a quasilinear parabolic equation. This problem ari... more We study a two-point free boundary problem for a quasilinear parabolic equation. This problem arises in the model of flame propagation in combustion theory. It also arises in the study of the motion of interface moving with curvature in which the studied problem is confined in the conical region bounded by two straight lines and the interface has prescribed touching angles with these two straight lines. Depending on these two touching angles, there are three different cases, namely, area expanding, preserving, and shrinking cases. We first give a proof of the global existence in the expanding and preserving cases. Then the convergence to a line in the preserving case is derived. Finally, in the shrinking case, we show the finite time vanishing and the convergence of the solution to self-similar solution.
We study a nonlocal parabolic problem airing in the modeling of linear friction welding. Using so... more 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.
u{x, 0) = u0(x), x € [-1,1], (1.3) where 0<a<l,/?>0, Z>0, and uq(x) >0, Vx G [-1, Z]. Without los... more u{x, 0) = u0(x), x € [-1,1], (1.3) where 0<a<l,/?>0, Z>0, and uq(x) >0, Vx G [-1, Z]. Without loss of generality, we may assume that uq(x) is smooth and bounded above by 1 such that uo(±Z) = 1. Since uo(x) is positive, the local (in time) existence and uniqueness of a classical solution of the problem (1.1)-(1.3) are trivial (see [8]). Many results in quenching, such as single point quenching and profiles, are similar to those blow-up results ([3], [5] and the references therein). The system (1.1)-(1.3) in the case a < 1 models fast diffusion and absorption (see [8]). The case a-1 was studied in [1], [2], [6], [7]. To study the asymptotic behavior of quenching, a simple Lyapunov function can be constructed in the case a = 1. When a < 1, however, such an explicit Lyapunov function is not available. In this paper, we use the idea of [4] to construct a Lyapunov function, which is rather technical. We say that the solution u quenches if the minimum of u(-,t) reaches zero at some finite time. We shall always assume that u quenches at a quenching time T < oo. In this case the right-hand side of Eq. (1.1) becomes singular and we no longer have a classical solution after this quenching time. A point (c, T) is said to be a quenching point if there is a sequence {(xn,tn)} such that xn-> c, tn-> T, and u(xn,tn)-> 0 as n-> oo. It is shown in [8] that there can only be finitely many quenching points that stay a positive distance away from the boundary |x| = I for any positive initial data. The purpose of this paper is to study how the solution tends to zero. For simplicity we shall only consider the symmetric case, i.e., the case that uq is symmetric with respect to x = 0 and is monotone increasing in \x\. It has been shown that (0, T) is the only possible
When a crack s propagates in an elastic medium the stress intensity factors evolve with the tip x... more When a crack s propagates in an elastic medium the stress intensity factors evolve with the tip x(s) of s. In this paper we derive formulae which describe the evolution of these stress intensity factors for a homogeneous isotropic elastic medium under plane strain conditions. Denoting by ψ = ψ(x, s) the stress potential (ψ is biharmonic and has zero traction along the crack s) and by κ(s) the curvature of the crack at the tip x(s), we prove that the stress intensity factors A 1 (s), A 2 (s), as functions of s, satisfy: dA 1 ds = 3 2 κ(s)A 2 (s) +β 1 (s), dA 2 ds = − 3 2 κ(s)A 1 (s) +β 2 (s), whereβ 2 (s),β 1 (s) are stress intensity factors of the tangential derivative of ψ − 4κ(s)A 2 (s)r 3/2 cos 1 2 θ in the polar coordinate system at x(s) with θ = 0 in the direction of the crack at x(s). The case of antiplane shearing is also briefly considered; in this case ψ is harmonic.
In this paper, we study a nonlocal parabolic problem arising in the study of microelectro mechani... more In this paper, we study a nonlocal parabolic problem arising in the study of microelectro 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 global (in time) solution to the maximal solution as the time tends to infinity. Finally, we provide some quenching criteria.
Discrete and Continuous Dynamical Systems - Series S, 2010
The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo ... more The reaction-diffusion system modeling the dorsal-ventral patterning during the zebrafish embryo development, developed in [Y.-T. Zhang, A.D.
We study a two-point free boundary problem for a quasilinear parabolic equation. This problem ari... more We study a two-point free boundary problem for a quasilinear parabolic equation. This problem arises in the model of flame propagation in combustion theory. It also arises in the study of the motion of interface moving with curvature in which the studied problem is confined in the conical region bounded by two straight lines and the interface has prescribed touching angles with these two straight lines. Depending on these two touching angles, there are three different cases, namely, area expanding, preserving, and shrinking cases. We first give a proof of the global existence in the expanding and preserving cases. Then the convergence to a line in the preserving case is derived. Finally, in the shrinking case, we show the finite time vanishing and the convergence of the solution to self-similar solution.
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