Nonuniqueness of the traveling wave speed for harmonic heat flow

Given any wave speed c∈R, we construct a traveling wave solution of ut=Δu+²|∇u|u in an infinitely long cylinder, which connects two locally stable and axially symmetric steady states at x₃=±∞. Here u is a director field with values in S²⊂R³: |u|=1. The traveling wave has a singular point on the cylinder axis. In view of the bistable character of the potential, the result is surprising, and it is intimately related to the nonuniqueness of the harmonic map flow itself. We show that for only one wave speed the traveling wave behaves locally, near its singular point, as a symmetric harmonic map.     

Nonuniqueness in a Free Boundary Problem from Combustion

We study a parabolic free boundary problem with a fixed gradient condition which serves as a simplified model for the propagation of premixed equidiffusional flames. We give a rigorous justification of an example due to J.L. Vázquez that the initial data in the form of two circular humps leads to the nonuniqueness of limit solutions if the supports of the humps touch at the time of their maximal expansion. A. Petrosyan was supported in part by NSF grant DMS-0701015. N.K. Yip was supported in part by NSF grant DMS-0406033.     

A one-dimensional nonlinear heat equation with a singular term

In this paper we are concerned with the Dirichlet problem for the one-dimensional nonlinear heat equation with a singular term:

     

Nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws

We consider the Riemann problem for a system of two decoupled, nonstrictly hyperbolic, Burgers-like conservation equations with added artificial viscosity. We analytically establish two different vanishing viscosity limits for the solution of this system, which correspond to the two cases where one of the viscosities vanishes much faster than the other. This is done without altering the initial condition as is necessary with travelling wave methods. Numerical evidence is then provided to show that when the two viscosities vanish at the same rate, the solution converges to a limit that lies strictly between the two previously established limits. Finally, we use control theory to explain the mechanism behind this nonuniqueness behavior, which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur.     

Nonuniqueness of solutions for a singular diffusion problem

In this paper, we consider a singular diffusion problem and show, by constructing a counterexample, that the weak solution to the problem is not unique. The proof consists of several steps. First, we prove that there exists a maximal weak solution to the problem. We show that the support of the continuous maximal weak solution cannot decrease in time. Then we cite an example of a nonnegative continuous function with shrinking support that also solves the problem, and therefore the problem possesses at least two weak solutions for some continuous nonnegative initial data.     

On nonuniqueness of solutions to elliptic equations for probability measures

We consider the set of all probability measures μ on Rd satisfying an elliptic equation L^∗μ=0 in the weak worm. We give sufficient conditions in order that this set contains at least two different elements. We also construct new examples of nonuniqueness for such equations.     

非唯一的有理L_p逼近

设f∈L_p\R_m~1,p≥1。如果f在R_m~1中有无穷多个最佳逼近,则这些最佳逼近必有一致收敛的子序列,并且其极限函数也是f的最佳逼近。如果f在R_m~1中的最佳逼近都是非退化的临界点,则这些最佳逼近仅有有限个。f在R_m~1中的形如P_1(x)/(1-λx)~n的最佳逼近仅有有限个。     

Nonuniqueness and selections in spline approximation

This paper studies problems of nonuniqueness for the metric projection ofC(T),T a compact Hausdorff space, onto a finite-dimensional subspaceG, and discusses the results for polynomial spline approximation. Among others, we prove that the metric projection ofC[a, b] ontoS _k,n , the space of polynomial splines of degree less than or equal ton withk simple knots in (a, b), is lower semicontinuous on an open, dense subset ofC[a, b] and, consequently, any standard selection of the projection is continuous on this subset. We further show that continuous selections are not so easy to construct.Communicated by Ronald A. DeVore.