# American Institute of Mathematical Sciences

ISSN:
1937-1632

eISSN:
1937-1179

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## Discrete & Continuous Dynamical Systems - S

June 2012 , Volume 5 , Issue 3

Issue dedicated to Michel Chipot
on the occasion of his 60th birthday

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2012, 5(3): i-i doi: 10.3934/dcdss.2012.5.3i +[Abstract](820) +[PDF](90.4KB)
Abstract:
Michel Chipot was born in 1949 in Gérardmer (88), France.
Michel started his mathematical research activities in the 70's, and defended first a "Thèse de 3ème cycle" at the university of Nancy I in 1975, where he was assistant since the end of 1971.

2012, 5(3): 369-397 doi: 10.3934/dcdss.2012.5.369 +[Abstract](1073) +[PDF](554.9KB)
Abstract:
In this paper, we study the div-curl-grad operators and some elliptic problems in the half-space $\mathbb{R}^n_+$, with $n\geq 2$. We consider data in weighted Sobolev spaces and in $L^1$.
2012, 5(3): 399-417 doi: 10.3934/dcdss.2012.5.399 +[Abstract](1271) +[PDF](420.5KB)
Abstract:
We introduce a notion of variational convergence for bifunctionals in an abstract setting. Then we apply this convergence to the asymptotic analysis of a junction problem in order to capture the gradient oscillations in the joint by considering the energy functional as a bifunctional of Sobolev-function/Young measure arguments. The well known asymptotic model described in terms of Sobolev-functions is obtained by eliminating the Young-measure argument considered as an internal variable through a marginal map. Furthermore, the surface energy of the classical model can be considered as a relaxation of a Dirichlet condition.
2012, 5(3): 419-426 doi: 10.3934/dcdss.2012.5.419 +[Abstract](1130) +[PDF](197.3KB)
Abstract:
We are interested in a nonlocal conservation law which describes the morphodynamics of sand dunes sheared by a fluid flow, recently proposed by Andrew C. Fowler and studied by [1,2]. We prove that constant solutions of Fowler's equation are non-linearly unstable. We also illustrate this fact using a finite difference scheme.
2012, 5(3): 427-434 doi: 10.3934/dcdss.2012.5.427 +[Abstract](1129) +[PDF](337.5KB)
Abstract:
In this paper we compare some families of weigthted Sobolev spaces which are commonly used for solving partial differential equations in unbounded domains. The first result is an identity between two particular spaces. The second result is another identity which generalises partially the first one.
2012, 5(3): 435-447 doi: 10.3934/dcdss.2012.5.435 +[Abstract](1138) +[PDF](352.9KB)
Abstract:
The dynamics of magneto-viscoelastic materials is described by a nonlinear system which couples the equation of the magnetization, given in Gibert form, and the viscoelastic integro-differential equation for the displacements. We study the general three-dimensional case and establish a theorem for the existence of weak solutions. The existence is proved by compactness of the approximated penalty problem.
2012, 5(3): 449-472 doi: 10.3934/dcdss.2012.5.449 +[Abstract](1584) +[PDF](544.1KB)
Abstract:
This paper explores certain concepts which extend the notions of (forward) self-similar and asymptotically self-similar solutions. A self-similar solution of an evolution equation has the property of being invariant with respect to a certain group of space-time dilations. An asymptotically self-similar solution approaches (in an appropriate sense) a self-similar solution to first order approximation for large time. Such solutions have a definite long-time asymptotic behavior, with respect to a specific time dependent spatial rescaling. After reviewing these fundamental concepts and the basic known results for heat equations on $\mathbb{R}^N$, we examine the possibility that a global solution might not be asymptotically self-similar. More precisely, we show that the asymptotic form of a solution can evolve differently along different time sequences going to infinity. Indeed, there exist solutions which are asymptotic to infinitely many different self-similar solutions, along different time sequences, all with respect to the same time dependent rescaling. We exhibit an explicit relationship between this phenomenon and the spatial asymptotic behavior of the initial value under a related group of dilations. In addition, we show that a given solution can exhibit nontrivial asymptotic behavior along different time sequences going to infinity, and with respect to different time dependent rescalings.
2012, 5(3): 473-483 doi: 10.3934/dcdss.2012.5.473 +[Abstract](993) +[PDF](383.5KB)
Abstract:
It is well known that the linear Korn inequality pervades the theory of three-dimensional linearized elasticity. It is thus conceivable that nonlinear Korn's inequalities could likewise play a role in the theory of three-dimensional nonlinear elasticity. In this paper, we describe the (available to this date) linear and nonlinear Korn's inequalities and we discuss the resemblances, but also the sometimes intriguing differences, that exist between these two kinds of inequalities.
2012, 5(3): 485-505 doi: 10.3934/dcdss.2012.5.485 +[Abstract](1111) +[PDF](430.3KB)
Abstract:
This paper deals with the longtime behavior of the Caginalp phase-field system with coupled dynamic boundary conditions on both state variables. We prove that the system generates a dissipative semigroup in a suitable phase-space and possesses the finite-dimensional smooth global attractor and an exponential attractor.
2012, 5(3): 507-530 doi: 10.3934/dcdss.2012.5.507 +[Abstract](1414) +[PDF](474.1KB)
Abstract:
In this paper we study a Dirichlet problem for an elliptic equation with degenerate coercivity and a singular lower order term with natural growth with respect to the gradient. The model problem is $$$$\left\{\begin{array}{11} -div\left(\frac{\nabla u}{(1+|u|)^p}\right) + \frac{|\nabla u|^{2}}{|u|^{\theta}} = f & \mbox{in \Omega,} \\ u = 0 & \mbox{on \partial\Omega,} \end{array} \right.$$$$ where $\Omega$ is an open bounded set of $\mathbb{R}^N$, $N\geq 3$ and $p, \theta>0$. The source $f$ is a positive function belonging to some Lebesgue space. We will show that, even if the lower order term is singular, it has some regularizing effects on the solutions, when $p>\theta-1$ and $\theta<2$.
2012, 5(3): 531-544 doi: 10.3934/dcdss.2012.5.531 +[Abstract](1079) +[PDF](371.6KB)
Abstract:
Given two $k-$forms $\alpha$ and $\beta$ we derive an identity relating $$%TCIMACRO{\dint _{\Omega}} %BeginExpansion {\displaystyle\int_{\Omega}} %EndExpansion \left( \langle d\alpha;d\beta\rangle+\langle\delta\alpha;\delta\beta \rangle-\langle\nabla\alpha;\nabla\beta\rangle\right)$$ to an integral on the boundary of the domain and involving only the tangential and the normal components of $\alpha$ and $\beta.$ We use this identity to deduce in a very simple way the classical Gaffney inequality and a generalization of it.
2012, 5(3): 545-557 doi: 10.3934/dcdss.2012.5.545 +[Abstract](1032) +[PDF](204.6KB)
Abstract:
Given real Hilbert spaces $\mathcal{X},\mathcal{Y},\mathcal{Z}$, closed convex functions $f : \mathcal{X} \longrightarrow \mathbb{R}\cup\{+\infty\}$, $g : \mathcal{Y} \longrightarrow \mathbb{R}\cup\{+\infty\}$ and linear continuous operators $A : \mathcal{X} \longrightarrow \mathcal{Z}$, $B : \mathcal{Y} \longrightarrow \mathcal{Z}$, we study the following alternating proximal algorithm $$$$\left\{ \begin{array}{ll} x_{n+1}&=argmin\{f(\zeta) + \frac{1}{2\gamma}\|A\zeta - By_n\|^2_z +\frac{\alpha}{2}\|\zeta - x_n\|^2_x; \quad \zeta\in\mathcal{X}\}\\ y_{n+1}&=argmin\{g(\eta) + \frac{1}{2\gamma}\|Ax_{n+1} - B\eta\|^2_z +\frac{\nu}{2}\|\eta - y_n\|^2_y; \quad \eta\in\mathcal{Y}\}, \end{array} \right.$$$$ where $\gamma$, $\alpha$ and $\nu$ are positive parameters. Under suitable conditions, we prove that any sequence $(x_n,y_n)$ generated by $(\mathcal{A})$ weakly converges toward a minimum point of the function $(x,y)\mapsto f(x) + g(y) + \frac{1}{2\gamma}\|Ax - By\|^2_z$ and that the sequence of dual variables $\left(-\frac{1}{\gamma}(Ax_n-By_n)\right)$ strongly converges in $\mathcal{Z}$ toward the unique minimizer of the function $z\mapsto f^*(A^*z)+g^*(-B^*z)+\frac{\gamma}{2}\|z\|^2_z$. An application is given in variational problems and PDE's.
2012, 5(3): 559-566 doi: 10.3934/dcdss.2012.5.559 +[Abstract](1308) +[PDF](349.4KB)
Abstract:
This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyapunov function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in [4].
2012, 5(3): 567-580 doi: 10.3934/dcdss.2012.5.567 +[Abstract](1307) +[PDF](375.8KB)
Abstract:
This paper is concerned with the asymptotic behaviour of the solutions of some semilinear hyperbolic problems. Using the monotonicity hypothesis, convergence results are shown in different spaces depending on the derivative directions of an arbitrary domain $\Omega .$ Some improvements are established when $\Omega$ is a cylinder.
2012, 5(3): 581-590 doi: 10.3934/dcdss.2012.5.581 +[Abstract](1058) +[PDF](371.5KB)
Abstract:
A one parameter family of equations is considered which connects the well-known Perona-Malik equation to standard diffusion. The parameter acts as a regularization parameter which gradually modifies the ill-posed Perona-Malik equation, through a strongly locally well-posed equation, to a strongly globally well-posed one which exhibits a behavior akin to that of standard diffusion, which is itself obtained in the limit. In the locally well-posed regime, the equation is degenerate parabolic and the onset of singularities can not be ruled out. Using a classical regularization approach, a-priori estimates can be derived which allow to go to limit with the regularization parameter globally in time. It is, however, not clear how to obtain a proper definition of weak solution for the limiting equation of interest.
2012, 5(3): 591-604 doi: 10.3934/dcdss.2012.5.591 +[Abstract](1445) +[PDF](407.2KB)
Abstract:
A rate-independent evolution problem is considered for which the stored energy density depends on the gradient of the displacement. The stored energy density does not have to be quasiconvex and is assumed to exhibit linear growth at infinity; no further assumptions are made on the behaviour at infinity. We analyse an evolutionary process with positively $1$-homogeneous dissipation and time-dependent Dirichlet boundary conditions.
2012, 5(3): 605-629 doi: 10.3934/dcdss.2012.5.605 +[Abstract](1191) +[PDF](492.1KB)
Abstract:
We prove existence of solution of a $p$-curl type evolutionary system arising in electromagnetism with a power nonlinearity of order $p$, $1 < p < \infty$, assuming natural tangential boundary conditions. We consider also the asymptotic behaviour in the power obtaining, when $p$ tends to infinity, a variational inequality with a curl constraint. We also discuss the existence, uniqueness and continuous dependence on the data of the solutions to general variational inequalities with curl constraints dependent on time, as well as the asymptotic stabilization in time towards the stationary solution with and without constraint.
2012, 5(3): 631-639 doi: 10.3934/dcdss.2012.5.631 +[Abstract](964) +[PDF](320.1KB)
Abstract:
We consider the time dependent motion of incompressible viscous fluid with non-homogeneous boundary condition. We suppose that the bounded domain filled by the fluid has at least two boundary components, and the boundary data for the fluid velocity satisfies only the general outflow condition (GOC). The existence of solutions for the stationary problem and time periodic problem is not known in general context. We present results for the Navier-Stokes equations and the Boussinesq equations.
2012, 5(3): 641-656 doi: 10.3934/dcdss.2012.5.641 +[Abstract](1089) +[PDF](712.8KB)
Abstract:
We derive in this paper a numerical scheme in order to calculate solutions of $1D$ transport equations. This $2nd$-order scheme is based on the method of characteristics and consists of two steps: the first step is about the approximation of the foot of the characteristic curve whereas the second one deals with the computation of the solution at this point. The main idea in our scheme is to combine two $2nd$-order interpolation schemes so as to preserve the maximum principle. The resulting method is designed for classical solutions and is unconditionally stable.
2012, 5(3): 657-670 doi: 10.3934/dcdss.2012.5.657 +[Abstract](1111) +[PDF](402.9KB)
Abstract:
We consider the Cauchy problem for a class of nonlinear parabolic equations with variable density in the hyperbolic space, assuming that the initial datum has compact support. We provide simple conditions, involving the behaviour of the density at infinity, so that the support of every nonnegative solution is not compact at some positive time, or it remains compact for any positive time. These results extend to the case of the hyperbolic space those given in [8] for the Cauchy problem in $\mathbb{R}^n$.
2012, 5(3): 671-681 doi: 10.3934/dcdss.2012.5.671 +[Abstract](1331) +[PDF](399.4KB)
Abstract:
We review known and prove new results on blow-up rate of solutions of parabolic problems with nonlinear boundary conditions. We also compare these results and methods of their proofs with corresponding results and methods for the nonlinear heat equation.
2012, 5(3): 683-706 doi: 10.3934/dcdss.2012.5.683 +[Abstract](1323) +[PDF](534.1KB)
Abstract:
We consider the generalized Burgers' equation \begin{eqnarray*} \left\{ \begin{array}{ll} \partial_t u = \partial_x^2u - u \partial_x u + u^p - \lambda u &\textrm{ in } \overline{\Omega} \textrm{ for } t>0, \\ \mathcal{B}(u)=0 & \textrm{ on } \partial \Omega \textrm{ for } t>0, \\ u(\cdot,0) = \varphi \geq 0 & \textrm{ in } \overline{\Omega}, \end{array} \right. \end{eqnarray*} with $p>1$, $\lambda \in \mathbb{R}$, $\Omega$ a subdomain of $\mathbb{R}$, and where $\mathcal{B}(u)=0$ denotes some boundary conditions. First, using some phase plane arguments, we study the existence of stationary solutions under the Dirichlet or the Neumann boundary conditions and prove a bifurcation depending on the parameter $\lambda$. Then, we compare positive solutions of the parabolic equation with appropriate stationary solutions to prove that global existence can occur when $\mathcal{B}(u)=0$ stands for the Dirichlet, the Neumann or the dissipative dynamical boundary conditions $\sigma \partial_t u + \partial _\nu u=0$. Finally, for many boundary conditions, global existence and blow up phenomena for solutions of the nonlinear parabolic problem in an unbounded domain $\Omega$ are investigated by using some standard super-solutions and some weighted $L^1-$norms.

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