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Discrete & Continuous Dynamical Systems - A

1997 , Volume 3 , Issue 2

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Minimizing movements of the Mumford and Shah energy
Antonin Chambolle and Francesco Doveri
1997, 3(2): 153-174 doi: 10.3934/dcds.1997.3.153 +[Abstract](68) +[PDF](286.4KB)
Abstract:
We study a simplified model of fracture propagation introduced by L. Ambrosio and A. Braides, based on the evolution by minimizing movements of the Mumford-Shah energy. In the two-dimensional case, we show that under a few additional assumptions on the "fracture" the movement solves the heat equation, with (weak) Neumann boundary conditions, and we are able to give some estimate on the decrease of the Mumford-Shah energy.
Existence and blow up of small amplitude nonlinear waves with a negative potential
Walter A. Strauss and Kimitoshi Tsutaya
1997, 3(2): 175-188 doi: 10.3934/dcds.1997.3.175 +[Abstract](67) +[PDF](208.9KB)
Abstract:
Consider a nonlinear wave equation in three space dimensions with zero mass together with a negative potential. If the potential is sufficiently short-range, then it does not alter the global existence of small-amplitude solutions. On the other hand, if the potential is sufficiently large, it will force some solutions to blow up in a finite time.
Linearization near a locally nonunique invariant manifold
George Osipenko
1997, 3(2): 189-205 doi: 10.3934/dcds.1997.3.189 +[Abstract](62) +[PDF](531.6KB)
Abstract:
Theorem of C. Pugh and M. Shub states that a flow can be linearized near normally hyperbolic compact invariant manifold. A normally hyperbolic manifold has the property of local uniqueness. This paper gives conditions for linearization of a flow near an invariant manifold without the assumption of its local uniqueness. These conditions are wider than the normally hyperbolicity condition.
Existence of stable and unstable periodic solutions for semilinear parabolic problems
E. N. Dancer and Norimichi Hirano
1997, 3(2): 207-216 doi: 10.3934/dcds.1997.3.207 +[Abstract](64) +[PDF](187.0KB)
Abstract:
In this paper, we show the existence of stable and unstable $T-$periodic solutions for a semilinear parabolic equation

$\frac{\partial u}{\partial t} - \Delta u = g(x,u) + h( t, x ),\quad \text{in} \quad (0,T) \times \Omega$

$u=0 ,\quad \text{on}\quad (0,T) \times \partial \Omega$

$u(0) = u(T),\quad \text{in} \quad \overline \Omega$

where $\Omega \subset R^N$ is a bounded domain with a smooth boundary, $g:\overline{\Omega} \times R \rightarrow R$ is a continuous function such that $g(x,\cdot )$ has a superlinear growth for each $x \in \overline{\Omega} $ and $h:(0,T) \times \Omega \to R$ is a continuous function.

A parabolic integro-differential equation arising from thermoelastic contact
Walter Allegretto, John R. Cannon and Yanping Lin
1997, 3(2): 217-234 doi: 10.3934/dcds.1997.3.217 +[Abstract](60) +[PDF](205.2KB)
Abstract:
In this paper we consider a class of integro-differential equations of parabolic type arising in the study of a quasi-static thermoelastic contact problem involving a critical parameter $\alpha$. For $\alpha <1$, the problem is first transformed into an equivalent standard parabolic equation with non-local and non-linear boundary conditions. Then the existence, uniqueness and continuous dependence of the solution upon the data are demonstrated via solution representation techniques and the maximum principle. Finally the asymptotic behavior of the solution as $ t \rightarrow \infty$ is examined, and we show that the non-local term has no impact on the asymptotic behavior for $ \alpha <1$. The paper concludes with some remarks on the case $\alpha >1$.
Regularization of discontinuous vector fields in dimension three
Jaume Llibre and Marco Antonio Teixeira
1997, 3(2): 235-241 doi: 10.3934/dcds.1997.3.235 +[Abstract](86) +[PDF](154.6KB)
Abstract:
In this paper vector fields around the origin in dimension three which are approximations of discontinuous ones are studied. In a former work of Sotomayor and Teixeira [6] it is shown, via regularization, that Filippov's conditions are the natural ones to extend the orbit solutions through the discontinuity set for vector fields in dimension two. In this paper we show that this is also the case for discontinuous vector fields in dimension three. Moreover, we analyse the qualitative dynamics of the local flow in a neighborhood of the codimension zero regular and singular points of the discontinuity surface.
On instant extinction for very fast diffusion equations
Yun-Gang Chen, Yoshikazu Giga and Koh Sato
1997, 3(2): 243-250 doi: 10.3934/dcds.1997.3.243 +[Abstract](51) +[PDF](158.1KB)
Abstract:
In this paper we prove instant extinction of the solutions to Dirichlet and Neumann boundary value problem for some quasilinear parabolic equations whose diffusion coefficient is singular when the spatial gradient of unknown function is zero.
Multiple periodic solutions of Hamiltonian systems with strong resonance at infinity
Laura Olian Fannio
1997, 3(2): 251-264 doi: 10.3934/dcds.1997.3.251 +[Abstract](71) +[PDF](159.6KB)
Abstract:
An asymptotically linear Hamiltonian system with strong resonance at infinity is considered. The existence of multiple periodic solutions is proved via variational methods in an equivariant setting.
Concerning the well-posedness of a nonlinearly coupled semilinear wave and beam--like equation
George Avalos
1997, 3(2): 265-288 doi: 10.3934/dcds.1997.3.265 +[Abstract](65) +[PDF](295.5KB)
Abstract:
In this work, we show the local existence and uniqueness of a coupled hyperbolic/parabolic system, where the coupling is partially accomplished through a strongly nonlinear term of polynomial growth. We show ultimately that the degree of the nonlinearity allowed depends upon the smoothness of a "piece" of the initial data and the geometry where the equations take place, and under a relatively mild imposition of smoothness, one can solve the system for nonlinearities of arbitrary polynomial bound.
On the bifurcation from critical homoclinic orbits in n-dimensional maps
Flaviano Battelli and Claudio Lazzari
1997, 3(2): 289-303 doi: 10.3934/dcds.1997.3.289 +[Abstract](64) +[PDF](223.2KB)
Abstract:
We consider the problem of existence of homoclinic orbits in systems like $x_{n+1}=f(x_n)+\mu g(x_n,\mu )$, $x\in \mathbb{R} ^N$, $\mu \in \mathbb{R}$, when the unperturbed system $x_{n+1}=f(x_n)$ has an orbit ${ \gamma _n } _{n\in \mathbb{Z}}$ homoclinic to an expanding fixed point (snap-back repeller) and such that $f'(\gamma _n)$ is invertible for any $n \ne 0$ but $f'(\gamma _0)$ is not. We show that, if a certain analytical condition is satisfied, homoclinic orbits of the perturbed equation occur in pair on one side of $\mu =0$ while are not present on the other side.

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