We consider a free boundary problem for a system of partial differential equations, which arises in a model of tumor growth with a necrotic core. For any positive number R there exists a radially symmetric stationary solution with free boundary r = R and interior boundary r = ρ.The system depends on a positive parameter μ, and for a sequence of values μ2 < μ3 < · · · there also exist branches o...

I construct a simple model of the convective burning layer during a type I X-ray burst to investigate the effects convection has on the stability of the layer to nonradial oscillations. A linear perturbation analysis demonstrates that the region is stable to nonradial oscillations when energy transport is convection-dominated, but it is unstable when energy transport is radiation-dominated. Thu...

We consider solutions u(t) to the 3d focusing NLS equation i∂tu+∆u+ |u|2u = 0 such that ‖xu(t)‖L2 = ∞ and u(t) is nonradial. Denoting by M [u] and E[u], the mass and energy, respectively, of a solution u, and byQ(x) the ground state solution to −Q+∆Q+ |Q|2Q = 0, we prove the following: if M [u]E[u] < M [Q]E[Q] and ‖u0‖L2‖∇u0‖L2 > ‖Q‖L2‖∇Q‖L2 , then either u(t) blows-up in finite positive time o...

Self-similar solutions provide good descriptions for the gravitational collapse of spherical clouds or stars when the gas obeys a polytropic equation of state, p = Kργ (with γ ≤ 4/3, and γ = 1 corresponds to isothermal gas). We study the behaviors of nonradial (nonspherical) perturbations in the similarity solutions of Larson, Penston and Yahil, which describe the evolution of the collapsing cl...

A large class of multidimensional nonlinear Schrödinger equations admit localized nonradial standing wave solutions that carry nonzero intrinsic angular momentum. Here we provide evidence that certain of these spinning excitations are spectrally stable. We find such waves for equations in two space dimensions with focusing-defocusing nonlinearities, such as cubic-quintic. Spectrally stable wave...

Recent analyses of photometric data on globular clusters and galaxies enabled us to study more closely the long-periodic amplitude/phase modulation (Blazhko effect) in classical variables. In the frequency spectra of these stars we see either a doublet or an equallyspaced triplet with a very small frequency separation close to the main component. None of the available theoretical models are abl...

Abstract In this work we address the question of existence nonradial domains inside a nonconvex cone for which mixed boundary overdetermined problem admits solution. Our approach is variational, and consists in proving minimizers, under volume constraint, associated torsional energy functional. particular give condition on domain D sphere spanning ensures that spherical sector not minimizer. Si...

In this note we present the solution of some isoperimetric problems in open convex cones of R in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition i...

By a combination of variational and topological techniques in the presence invariant cones, we detect new type positive axially symmetric solutions Dirichlet problem for elliptic equation $$ -\Delta u + = a(x)|u|^{p-2}u an annulus $A \subset \mathbb{R}^N$ ($N\ge3$). Here $p>2$ is allowed to be supercritical $a(x)$ but possibly nonradial function with additional symmetry monotonicity properties,...