Sharp estimates for fully bubbling solutions of a SU(3) Toda system

In this paper, we obtain sharp estimates of fully bubbling solutions of SU(3) Toda system in a compact Riemann surface. In geometry, the SU(n?+?1) Toda system is related to holomorphic curves, harmonic maps or harmonic sequences of the Riemann surface to ${\mathbb{CP}^n}$ . In order to compute the Leray?CSchcuder degree for the Toda system, we have to obtain accurate approximations of the bubbling solutions. Our main goals in this paper are (i) to obtain a sharp convergence rate, (ii) to completely determine the locations, and (iii) to derive the ${\partial _z^2}$ condition, a unexpected and important geometric constraint.     

Some existence results for solutions to SU(3) Toda system

We study SU(3) Toda system in non-abelian relativistic self-dual gauge theory. In the range of parameters where the corresponding Trudinger-Moser inequality fails, we show the existence of the solution by a different variational formulation from Lucia-Nolasco's [15]. This work was supported by a grant of the Japan-Korea Scientific Cooperation Program - Joint Research “Mathematical analysis and mathematical science for self-interacting particles.” The second author was partially supported by Grant-in-Aid for Scientific Research (No. 16740103), Japan Society for the Promotion of Science. Mathematics Subject Classification (2000) 35B40 - 35J50 - 35J60 - 49Q99 - 58E15 - 58J05 - 70S15     

Asymptotic behavior of bounded solutions for a system of neutral functional differential equations

Consider the system of neutral functional differential equations

     

A semilinear parabolic system in a bounded domain

Consider the system

Asymptotic behavior of a reaction-diffusion system in bacteriology

This paper is concerned with the asymptotic behavior of the solution for a coupled system of reaction-diffusion equations which describes the bacteria growth and the diffusion of histidine and buffer concentrations. Under the basic boundary condition of Neumann type or mixed type the coupled system can have infinitely many steady-state solutions. The present paper gives some explicit information on the asymptotic limit of the time-dependent solution in relation to these steady states. This information exhibits some rather distinct properties of the solutions between the Neumann boundary problem and the Dirichlet or mixed boundary problem.     

Asymptotic behavior of a sixth-order Cahn-Hilliard system

Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, of a sixth-order Cahn-Hilliard system. This system is based on a modification of the Ginzburg-Landau free energy proposed in [Torabi S., Lowengrub J., Voigt A., Wise S., A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2105), 1337–1359], assuming isotropy.     

Finite time blow-up for a reaction-diffusion system in bounded domain

This paper mainly considers the coupled parabolic system in a bounded domain: u _t = Δu + u ^α v ^p , v _t = Δv + u ^q v ^β in Ω × (0, T) with null Dirichlet boundary value condition which had been discussed by Wang in (Z Angew Math Phys 51:160–167, 2000). The aim of this paper is to solve the open problem mentioned in the Remark of Wang (Z Angew Math Phys 51:160–167, 2000).     

Asymptotic behavior of bounded solutions to a nonhomogeneous second-order evolution equation of monotone type

By using previous results of Djafari Rouhani [B. Djafari Rouhani, Ergodic theorems for nonexpansive sequences in Hilbert spaces and related problems, Ph.D. Thesis, Yale university, 1981, part I, pp. 1-76; B. Djafari Rouhani, Asymptotic behaviour of quasi-autonomous dissipative systems in Hilbert spaces, J. Math. Anal. Appl. 147 (1990) 465-476; B. Djafari Rouhani, Asymptotic behaviour of almost nonexpansive sequences in a Hilbert space, J. Math. Anal. Appl. 151 (1990) 226-235] for dissipative systems, we study the asymptotic behavior of solutions to the following system of second-order nonhomogeneous evolution equations:

     

On small motions of a “liquid-gas” system in a bounded domain

We study small motions and free oscillations of a compressible stratified liquid, the structure of the spectrum, and the basis property of a system of eigenvectors and obtain asymptotic relations for eigenvalues. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1326–1334, October, 2006.     

Canonical solution of the SU(3) ↓ SO(3) reduction problem from the SU(3) pattern calculus

Because of its numerous applications to physics, there have been many solutions published on the problem of reducing a given irreducible representation (irrep) of the unitary unimodular group SU(3) into irreps of the proper orthogonal subgroup SO(3). Such solutions are generally based on an arbitrary construction of a nonorthogonal basis of the highest weight space for an irrep of SO(3), followed by an equally arbitrary orthonormalization procedure. This paper presents a unique solution of this problem based on the intrinsic structure of the multiplicity function, which is a function M _L(p, q) giving the number of times irrep[L] of SO(3) is contained in irrep[pq0] of SU(3). This structure is implemented uniquely into the reduction problem through the use of the SU(3) pattern calculus.Work performed under the auspices of the U.S. Department of Energy.     

Existence of solutions for a degenerate auasilinear elliptic system in bounded domain

Using variational methods, we study the existence of weak solutions forthe degenerate quasilinear elliptic system$$\left\{\begin{array}{ll}- \mathrm{div}\Big(h_1(x)|\nabla u|^{p-2}\nabla u\Big) = F_{u}(x,u,v) &\text{ in } \Omega,\\-\mathrm{div}\Big(h_2(x)|\nabla v|^{q-2}\nabla v\Big) = F_{v}(x,u,v) &\text{ in } \Omega,\\u=v=0 & \textrm{ on } \partial\Omega,\end{array}\right.$$where $\Omega\subset \mathbb R^N$ is a smooth bounded domain, $\nabla F= (F_u,F_v)$ stands for the gradient of $C^1$-function $F:\Omega\times\mathbb R^2 \to \mathbb R$, the weights $h_i$, $i=1,2$ are allowed to vanish somewhere,the primitive $F(x,u,v)$ is intimately related to the first eigenvalue of acorresponding quasilinear system.