The Landau-Lifshitz-Bloch equation is an important dynamical model for studying the magnetization offerromagnets above Curie temperature. This article shows global existence and uniqueness of smooth solutions modied equation.

This work is concerned with the construction and analysis of high order product integration methods for a class of Volterra integral equations with logarithmic singular kernel. Sufficient conditions for the methods to be convergent are derived and it is shown that optimal convergence orders are attained if the exact solution is sufficiently smooth. The case of non-smooth solutions is dealt with...

We study the existence and structure of steady-state ngers in two-dimensional solidiication, when the surface energy has a crystalline anisotropy so that the energy-minimizing Wull shape and hence the solid-liquid interface are polygons, and in the one-sided quasi-static limit so that the diiusion eld satisses Laplace's equation in the liquid. In a channel of nite width, this problem is the cry...

True amplitude one-way wave equations provide full waveform solutions that agree with leading order WKBJ solutions of the full, two-way wave equation away from caustics and other anomalies of the ray field. Both a proof of this fact and numerical examples are available in the literature. Near smooth caustics, ray-theoretic solutions fail for both the one-way and two-way wave equations; the WKBJ...

In this paper, we investigate the existence of positive solutions for the ellipticequation $Delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $Omega$ of $R^{n}$, $ngeq2$, with Navier boundary conditions. We show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

We consider the following anisotropic boundary value problem ∇(a(x)∇u) + a(x)u = 0, u > 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ R2 is a bounded smooth domain, p is a large exponent and a(x) is a positive smooth function. We investigate the effect of anisotropic coefficient a(x) on the existence of concentrating solutions. We show that at a given strict local maximum point of a(x), there exist arbitraril...

The implicit function theorem implies that the zero set of a smooth function, the set where the function vanishes, is a smooth hypersurface away from the critical zero set. Hence to study zero sets it is important to understand the structure of the critical zero sets. For solutions of the second order elliptic equations the critical zero sets represent the singular parts of zero sets. They have...

We consider a model for the interaction of a gas with photons. In the article [10], smooth traveling wave solutions called shock profiles have been constructed under a suitable smallness assumption between the asymptotic states. In this work, we construct piecewise smooth traveling wave solutions that connect two asymptotic states with a large jump. In particular, we give a rigorous mathematica...

It is shown for a simple ODE that it has many symmetry groups beyond its usual Lie group symmetries, when its generalized solutions are considered within the nowhere dense differential algebra of generalized functions. 0. Idea and Motivation The standard Lie group theory applied to symmetries of solutions of PDEs, Olver [1-3], deals with classical, and specifically, C-smooth solutions of such e...