On the solutions and conservation laws of the model for tumor growth in the brain

We study the problem of tumor growth and its monitoring ranging from the simple model for the radially symmetric to the more complex case being the radially non-symmetric one. In each case, we take killing rate of the cancer cells dependent on the concentration of the cells. A number of invariant reductions whose further analysis leads to exact solutions are obtained. Conservation laws for the model are also studied.     

Symmetry Breaking of Extremals for the Caffarelli-Kohn-Nirenberg Inequalities in a Non-Hilbertian Setting

We provide an explicit necessary condition to have that no extremal for the best constant in the Caffarelli-Kohn-Nirenberg inequality is radially symmetric.     

Radially Symmetric Solutions of the <Emphasis Type="Italic">p</Emphasis>-Laplace Equation with Gradient Terms

We consider the Dirichlet problem for the p-Laplace equation with nonlinear gradient terms. In particular, these gradient terms cannot satisfy the Bernstein—Nagumo conditions. We obtain some sufficient conditions that guarantee the existence of a global bounded radially symmetric solution without any restrictions on the growth of the gradient term. Also we present some conditions on the function simulating the mass forces, which allow us to obtain a bounded radially symmetric solution under presence of an arbitrary nonlinear source.     

Global existence of the radially symmetric solutions of the Navier–Stokes equations for the isentropic compressible fluids

We study the isentropic compressible Navier–Stokes equations with radially symmetric data in an annular domain. We first prove the global existence and regularity results on the radially symmetric weak solutions with non‐negative bounded densities. Then we prove the global existence of radially symmetric strong solutions when the initial data ρ₀, u ₀ satisfy the compatibility condition for some radially symmetric g ∈ L². The initial density ρ₀ needs not be positive. We also prove some uniqueness results on the strong solutions. Copyright © 2004 John Wiley & Sons, Ltd.     

Radially symmetric wave maps from (1 + 2)-dimensional Minkowski space to the sphere

By a blow-up analysis as in [8] for a related problem we rule out concentration of energy for radially symmetric wave maps from the (1+ 2)-dimensional Minkowski space to the sphere. When combined with the local existence and regularity results of Christodoulou and Tahvildar-Zadeh for this problem, our result implies global existence of smooth solutions to the Cauchy problem for radially symmetric wave maps for smooth radially symmetric data. Received: 1 November 2000; in final form: 12 April 2001 / Published online: 1 February 2002     

On the shape of solutions to an integral system related to the weighted Hardy–Littlewood–Sobolev inequality

We study the Euler–Lagrange system for a variational problem associated with the weighted Hardy–Littlewood–Sobolev inequality. We show that all the nonnegative solutions to the system are radially symmetric and have particular profiles around the origin and the infinity. This paper extends previous results obtained by other authors to the general case.     

Stability of the vortex defect in the Landau–de Gennes theory for nematic liquid crystals

We analyze the radially symmetric solution corresponding to the vortex defect (the so-called melting hedgehog) in the Landau–de Gennes theory for nematic liquid crystals. We prove the existence, uniqueness and stability results of the melting hedgehog.     

<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-concentration of blow-up solutions for two-coupled nonlinear Schrödinger equations with harmonic potential

In this paper, we consider the blow-up solutions of Cauchy problem for twocoupled nonlinear Schrödinger equations with harmonic potential. We establish the lower bound of blow-up rate. Furthermore, the L ² concentration for radially symmetric blow-up solutions is obtained.     

Absence of Critical Mass Densities for a Vibrating Membrane

We study the dependence of the eigenvalues of a N-dimensional vibrating membrane upon variation of the mass density. We prove that the elementary symmetric functions of the eigenvalues depend real-analytically on the mass density and that such functions have no critical points with constant mass constraint. In particular, the elementary symmetric functions of the eigenvalues, hence all simple eigenvalues, have no local maxima or minima on the set of those mass densities with a prescribed total mass.     

Stabilization of a thermoelastic Mindlin–Timoshenko plate model revisited

This paper is a continuation of our work in Grobbelaar-Van Dalsen (Appl Anal 90:1419–1449, 2011) where we showed the strong stability of models involving the thermoelastic Mindlin–Timoshenko plate equations with second sound. For the case of a plate configuration consisting of a single plate, this was accomplished in radially symmetric domains without applying any mechanical damping mechanism. Further to this result, we establish in this paper the non-exponential stability of the model for a particular configuration under mixed boundary conditions on the shear angle variables and Dirichlet boundary conditions on the displacement and thermal variables when the heat flux is described by Fourier’s law of heat conduction. We also determine the rate of polynomial decay of weak solutions of the model in a radially symmetric region under Dirichlet boundary conditions on the displacement and thermal variables and free boundary conditions on the shear angle variables.     

Analysis of an elliptic-parabolic free boundary problem modelling the growth of non-necrotic tumor cord

We study a free boundary problem modelling the growth of a tumor cord in which tumor cells live around and receive nutrient from a central blood vessel. The evolution of the tumor cord surface is governed by Darcy's law together with a surface tension equation. The concentration of nutrient in the tumor cord satisfies a reaction-diffusion equation. In this paper we first establish a well-posedness result for this free boundary problem in some Sobolev-Besov spaces with low regularity by using the analytic semigroup theory. We next study asymptotic stability of the unique radially symmetric stationary solution. By making delicate spectrum analysis for the linearized problem, we prove that this stationary solution is locally asymptotically stable provided that the constant c representing the ratio between the diffusion time of nutrient and the birth time of new cells is sufficiently small.     

The improved decay rate for the heat semigroup with local magnetic field in the plane

We consider the heat equation in the presence of compactly supported magnetic field in the plane. We show that the magnetic field leads to an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta. The proof employs Hardy-type inequalities due to Laptev and Weidl for the two-dimensional magnetic Schrödinger operator and the method of self-similar variables and weighted Sobolev spaces for the heat equation. A careful analysis of the asymptotic behaviour of the heat equation in the similarity variables shows that the magnetic field asymptotically degenerates to an Aharonov–Bohm magnetic field with the same total magnetic flux, which leads asymptotically to the gain on the polynomial decay rate in the original physical variables. Since no assumptions about the symmetry of the magnetic field are made in the present work, it gives a normwise variant of the recent pointwise results of Kova?ík (Calc Var doi:10.1007/s00526-011-0437-4) about large-time asymptotics of the heat kernel of magnetic Schrödinger operators with radially symmetric field in a more general setting.     

On radially symmetric solutions of the compressible isentropic self-gravitating fluid

In this paper, we mainly prove the global existence of weak solutions to the Cauchy problem for the Navier–Stokes system of compressible isentropic self-gravitating fluids in ${\mathbb{R}}^3$ when the Cauchy data are radially symmetric. It extends Feireisl’s existence theorem, Ducomet et al. (2001) [16], to the case $4/3<\gamma \le 3/2$ for radially symmetric initial data, where $\gamma$ is the specific heat ratio in the pressure. If the total mass is less than a certain critical mass, this conclusion also holds for $\gamma =4/3$. Furthermore, for the case of annular domain, we point out the global existence radially symmetric strong solutions when the radially symmetric initial data satisfy the compatibility condition and the initial density need not be positive.     

带调和势的非线性Schrdinger方程爆破解的L~2集中率

本文讨论了带调和势的具有临界幂的非线性Schrodinger方程,得到其爆破解在t→T(爆破时间)的L2集中率．     

Multiple positive radial solutions on annuli for nonlinear Neumann problems with large growth

We consider a classical semilinear elliptic equation with Neumann boundary conditions on an annulus in R ^N . The nonlinear term is the product of a radially symmetric coefficient with a pure power. We prove that if the power is sufficiently large, the problem admits at least three distinct positive and radial solutions. In case the coefficient is constant, we show that none of the three solutions is constant. The methods are variational and are based on the study of a suitable limit problem.     

Solitary waves in Abelian Gauge Theories with strongly nonlinear potentials

We study the existence of radially symmetric solitary waves for a system of a nonlinear Klein–Gordon equation coupled with Maxwell's equation in presence of a positive mass. The nonlinear potential appearing in the system is assumed to be positive and with more than quadratical growth at infinity.     

带调和势的非线性Schrdinger方程爆破解的L~2集中率

本文讨论了带调和势的具有临界幂的非线性Schrodinger方程,得到其爆破解在t→T(爆破时间)的L2集中率．     

Radially symmetric growth of nonnecrotic tumors

The growth of tumors is an important subject in recent research. We present here a mathematical model for the growth of nonnecrotic tumors in all the three regimes of vascularisation. This leads to a free-boundary problem which we treat by means ODE techniques. We prove the existence of a unique radially symmetric stationary solution. It is also shown that, if the initial tumor is radially symmetric, there exists a unique radially symmetric solution of the evolution equation, which exists for all times. The asymptotic behaviour of this solution will be discussed in relation to the parameters characterizing cell proliferation and cell death.     

Chern–Simons limit of the standing wave solutions for the Schrödinger equations coupled with a neutral scalar field

We show the existence of standing wave solutions to the Schrödinger equation coupled with a neutral scalar field. We also verify the Chern–Simons limit for these solutions. More precisely we prove that solutions to Eqs. (1.3)–(1.4) converge to the unique positive radially symmetric solution of the nonlinear Schrödinger equation (1.6) as the coupling constant q goes to infinity.