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Volume 1, 1995

Discrete & Continuous Dynamical Systems - A

1995 , Volume 1 , Issue 4

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Subharmonic solutions in the restricted three-body problem
Hildeberto E. Cabral and  Zhihong Xia
1995, 1(4): 463-474 doi: 10.3934/dcds.1995.1.463 +[Abstract](41) +[PDF](219.7KB)
In this paper, we study the subharmonic bifurcations in the restricted three-body problem. By study the Melnikov integrals for the subharmonic solutions, we obtain the precise bifurcation scenario nearby the circular solutions when one of the two primaries is small.
Schrödinger equations with nonlinearity of integral type
Nakao Hayashi and  Tohru Ozawa
1995, 1(4): 475-484 doi: 10.3934/dcds.1995.1.475 +[Abstract](30) +[PDF](186.4KB)
We consider the Cauchy problem for the nonlinear Schrödinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. Concerning the problem with finite initial time, we prove the global well-posedness in the largest space $L^2(\mathbb R^2)$. Concerning the problem with infinite initial time, we prove the existence of modified wave operators on a dense set of small and sufficiently regular asymptotic states.
Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence
Robert E. Miller
1995, 1(4): 485-502 doi: 10.3934/dcds.1995.1.485 +[Abstract](27) +[PDF](262.1KB)
Allaire's results for elliptic problems on non-homogeneous media and on periodically perforated domains are extended to time-dependent problems. The main emphasis of the paper is to apply the method of two-scale convergence to problems in which the damping term has the same order spatial derivative as the stiffness term.
Breakdown of solutions to $\square u+u_t=|u|^{1+\alpha}$
Tatsien Li and  Yi Zhou
1995, 1(4): 503-520 doi: 10.3934/dcds.1995.1.503 +[Abstract](43) +[PDF](209.9KB)
In the paper we give an upper bound for the life-span of the mild solution to the Cauchy problem for semilinear equations $\square u+u_t=|u|^{1+\alpha}$ ($\alpha >0,$ constant) with certain small initial data. This shows the sharpness of the lower bound obtained in [2] on the life-span of classical solutions to the Cauchy problem for fully nonlinear wave equations with linear dissipation with small initial data.
On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data
Kazuhiro Ishige
1995, 1(4): 521-546 doi: 10.3934/dcds.1995.1.521 +[Abstract](30) +[PDF](278.0KB)
For any nonnegative Radon measure $\mu$, we prove the existence of solutions for the Cauchy problem:

$ u_t =\Delta\phi(u)\qquad\text{in}\quad R^N\times(0,T);\qquad u(\cdot,0) =\mu(\cdot)\ge 0\quad \text{in}\quad R^N, $

where $\phi'(s)$ ~ $\log^m s$, $m<-1$, as $s\to\infty$. On the other hand, for the case $m\ge -1$, we give a sufficient condition for the solvability of the Cauchy problem.

Extended wellposedness of optimal control problems
T. Zolezzi
1995, 1(4): 547-553 doi: 10.3934/dcds.1995.1.547 +[Abstract](26) +[PDF](134.8KB)
A concept of wellposedness is applied to control problems monitored by ordinary differential equations. This concept does not impose uniqueness of the optimal control, and requires strong convergence of every asymptotically minimizing sequence corresponding to small perturbations of the initial state. Sufficient conditions for such a form of wellposedness are obtained via Tikhonov wellposedness of the pointwise maximization of the Hamiltonian function.
On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves
Tung Chang , Gui-Qiang Chen and  Shuli Yang
1995, 1(4): 555-584 doi: 10.3934/dcds.1995.1.555 +[Abstract](55) +[PDF](2154.4KB)
We are concerned with the Riemann problem for the two-dimensional compressible Euler equations in gas dynamics. The central point at this issue is the dynamical interaction of shock waves, centered rarefaction waves, and contact discontinuities that connect two neighboring constant initial states in the quadrants. The Riemann problem is classified into eighteen genuinely different cases. For each configuration, the structure of the Riemann solution is analyzed using the method of characteristics, and corresponding numerical solution is illustrated by contour plots using an upwind averaging scheme that is second order in the smooth region of the solution. In the first paper we mainly focus on the interaction of shocks and rarefaction waves. The theory is developed from an analysis of the structure of the Euler equations and their Riemann solutions in [CC, ZZ] and the MmB scheme [WY].
Invariant regions under Lax-Friedrichs scheme for multidimensional systems of conservation laws
Hermano Frid
1995, 1(4): 585-593 doi: 10.3934/dcds.1995.1.585 +[Abstract](37) +[PDF](195.8KB)
We establish necessary and sufficient conditions for the invariance of a region in the state space, under the Lax-Friedrichs scheme applied to a multidimensional system of conservation laws. We also give some examples of application of the invariance principle proved here.

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