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Volume 1, 1995

Discrete & Continuous Dynamical Systems - A

1995 , Volume 1 , Issue 3

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Parabolic singular limit of a wave equation with localized boundary damping
Aníbal Rodríguez-Bernal and  Enrique Zuazua
1995, 1(3): 303-346 doi: 10.3934/dcds.1995.1.303 +[Abstract](42) +[PDF](373.8KB)
We will consider the family of wave equations with boundary damping

$\qquad\qquad \qquad\qquad \epsilon u_{t t} -\Delta u + \lambda u =f $ on $\Omega \times (0,T)$

$(P_{\epsilon, \lambda, \Gamma_0})\qquad\qquad u_t + \frac{\partial u}{\partial \vec{n}} =g$ on $\Gamma_1 \times (0,T) $

$\qquad\qquad u=0 $ on $\Gamma_0 \times (0,T)$

where $0< \epsilon \leq \epsilon_0$, $\Omega \subset \mathbb R^N$ is a regular open connected set, $\lambda \geq 0$ and $\Gamma = \Gamma_0\cup \Gamma_1$ is a partition of the boundary of $\Omega$. We will also consider the case where $\Gamma_0$ is empty (see below for more precise assumptions on $\lambda$, $\Omega$ and $\Gamma_0$, $\Gamma_1$).
For this problem the corresponding formal singular perturbation at $\epsilon =0$ is

$\qquad\qquad \qquad\qquad -\Delta u + \lambda u =f$ on $\Omega \times (0,T) $

$(P_{0, \lambda, \Gamma_0}) \qquad\qquad u_t + \frac{\partial u}{\partial \vec{n}} =g$ on $\Gamma_1 \times (0,T) $

$\qquad\qquad u=0 $ on $ \Gamma_0 \times (0,T)$

We are here concerned with the well possedness of both problems for the non--homogeneous case, i.e. $f=f(t,x)$, $g=g(t,x)$, and with the convergence, as $\epsilon$ approaches $0$, of the solutions of $(P_{\epsilon, \lambda, \Gamma_0})$ to solutions of $(P_{0, \lambda, \Gamma_0})$.

A characterization of variational convergence for segmentation problems
Micol Amar and  Andrea Braides
1995, 1(3): 347-369 doi: 10.3934/dcds.1995.1.347 +[Abstract](33) +[PDF](262.7KB)
We characterize the $\Gamma$-convergence of one-dimensional integral functionals with bulk and jump-part energies, by means of a suitable convergence of the integrands.
Optimality conditions for controls of semilinear evolution systems with mixed constraints
Jiongmin Yong
1995, 1(3): 371-388 doi: 10.3934/dcds.1995.1.371 +[Abstract](40) +[PDF](222.1KB)
In this paper, we study an optimal control problem for semilinear evolution equations with a mixed constraint of the state and the control. One of the motivations is the problem with a state dependent control domain. Our main result is the Pontryagin type necessary conditions for the optimal controls. The main tools we use are the Ekeland variational principle and the spike perturbation technique.
Critical values and minimal periods for autonomous Hamiltonian systems
K. Tintarev
1995, 1(3): 389-400 doi: 10.3934/dcds.1995.1.389 +[Abstract](23) +[PDF](190.9KB)
The paper studies periodic solutions of Hamiltonian systems. It states that a range of periods for such solutions can be obtained by diiferentiation of the function of constrained critical values with respect to the variable constraint level. It also shows that when a Hamiltonian is equal to a positive quadratic form plus an oscillatory term, there exist infinitely many solutions with the same period.
Optimal control problems with weakly converging input operators
Giuseppe Buttazzo and  Lorenzo Freddi
1995, 1(3): 401-420 doi: 10.3934/dcds.1995.1.401 +[Abstract](29) +[PDF](235.6KB)
We study the variational convergence, as $\h \rightarrow \infty$, of a sequence of optimal control problems $(\mathcal{P}_h)$ with abstract state equations $A_h(y)=B_h(u)$, where $A_h$ are $G$-converging and the operators $B_h$ acting on the controls are supposed continuously converging, or nonlinear but local, or linear but possibly nonlocal.
Approximate inertial manifolds of exponential order
Ricardo Rosa
1995, 1(3): 421-448 doi: 10.3934/dcds.1995.1.421 +[Abstract](36) +[PDF](258.8KB)
A fairly general class of nonlinear evolution equations with a self-adjoint or non self-adjoint linear operator is considered, and a family of approximate inertial manifolds (AIMs) is constructed. Two cases are considered: when the spectral gap condition (SGC) is not satisfied and an exact inertial manifold is not known to exist the construction is such that the AIMs have exponential order, while when the SGC is satisfied (and hence there exists an exact inertial manifold) the construction is such that the AIMs converge exponentially to the exact inertial manifold.
Interior derivative blow-up for quasilinear parabolic equations
Yoshikazu Giga
1995, 1(3): 449-461 doi: 10.3934/dcds.1995.1.449 +[Abstract](46) +[PDF](194.3KB)
We give examples of a bounded solution whose gradient blows up in a finite time but it stays bounded on the boundary for a class of quasilinear parabolic equations with zero boundary data. The method reflects a geometric argument for curve evolution equations.

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