We follow the approach of [13] to study the N-dimensional generalized MHD system with fractional Laplacians as dissipative and diffusive terms in various anisotropic spaces. In particular, we obtain small initial data results with anisotropic Sobolev space type norms for which, depending on the power of the fractional Laplacians, we may decrease the regularity index in many directions to zero or even negative, in the expense of increasing the rest. Similar results in anisotropic Besov type spaces are also obtained.

We describe the asymptotic profile of sign changing solutions with small energy to the singularly perturbed nonlinear elliptic equation $-\varepsilon ^{2}\Delta_{g}u+u=|u|^{p-2}u$ on an $n$-dimensional compact Riemannian manifold $M$ without boundary, $p\in(2,2^{\ast}),$ as $\varepsilon \rightarrow0.$

The renormalization group method of Chen, Goldenfeld, and Oono offers a comprehensive approach to formally computing asymptotic expansions of the solutions to singular perturbation problems and multi-scale problems. In particular, the RG method applies to a broad array of problems customarily treated with disparate methods, such as the method of multiple scales, boundary layer theory, matched asymptotic expansions, Poincare-Lindstedt theory, the WKBJ method with and without turning points, the method of averaging, and others. For problems in which the expansions are in powers of the small parameter, it has been shown that the RG method leads to uniformly valid asymptotic expansions of the solutions. In addition, it has been shown that the RG condition constitutes an invariance condition, that the RG method is effectively a resummation technique, and that it is equivalent to normal form theory for certain broad classes of perturbed ordinary differential equations. However, there has not yet been an analysis of the validity of the RG method, i.e.,i> a demonstration that it leads to uniformly valid asymptotic expansions, for problems in which the expansions also involve logarithms of the small parameter as gauge functions. The aim of this article is to provide this justification of the RG method. In particular, we extend the approach developed in DeVille, et ali>. Physica D 237 no. 8, 1029 {-}1052] from the classes of autonomous and non-autonomous perturbations considered there to include the non-autonomous systems subject to singular perturbations for which the solutions involve logarithmic gauge functions. This framework is built upon the relationship between the RG method and normal form theory. We apply the RG method to three successively-more complex examples and also elucidate the common general features.

In this paper, we study one-dimensional $p$-Laplace equation, whose weight function has a critical power at the origin. We show the existence of infinitely many bifurcation branches emanating from the same single point and also study the existence and multiplicity of sign-changing solutions. Moreover, we investigate the global shape of bifurcation branches.

We study the asymptotic behavior of solutions to fully nonlinear second order parabolic equations including a generalized curvature flow equation which was introduced by Mullins in 1957 as a model of evaporation-condensation. We prove that, in the multi-dimensional half space, solutions of the problem with prescribed contact angle asymptotically converge to a self-similar solution of the associated problem under a suitable rescaling. Several properties of the profile function of the self-similar solution are also investigated. We show that the profile function has a corner and that the angles are determined by points at which the equation is degenerate. We also study the depth of the groove, which is represented by the value of the profile function at the boundary. Among other results it turns out that, as the contact angle tends to zero, the depth of the groove is well approximated by the linearized problem.

The Cauchy problem for the Maxwell-Klein-Gordon equations in Lorenz gauge in two and three space dimensions is locally well-posed for low regularity data without finite energy. The result relies on the null structure for the main bilinear terms, which was shown to be not only present in Coulomb gauge but also in Lorenz gauge by Selberg and Tesfahun, who proved global well-posedness for finite energy data in three space dimensions. This null structure is combined with product estimates for wave-Sobolev spaces given systematically by d'Ancona, Foschi and Selberg.

We study the well-posedness of renormalized entropy solutions for quasilinear anisotropic degenerate parabolic-hyperbolic equation of the type $\partial_{t}u+\text{div}f(u)=\nabla\cdot(a(u)\nabla u)$ in a bounded domain with general $L^{1}$ initial data and homogeneous Dirichlet boundary condition. We use the device of doubling variables to prove the uniqueness and use the vanishing viscosity method to prove the existence.