A singular limit is considered for a system of Cahn-Hilliard equations with a degenerate mobility matrix near the deep quench limit. Via formal asymptotics, this singular limit is seen to give rise to geometric motion in which the interfaces between the various pure phases move by motion by minus the surface Laplacian of mean curvature. These interfaces may couple at triple junctions whose evolution is prescribed by Young's law, balance of fluxes, and continuity of the chemical potentials. Short time existence and uniqueness is proven for this limiting geometric motion in the parabolic Hölder space ${ C}_{t, \, p}^{1 + \frac{\alpha}{4}, \, 4 + \alpha}, \, 0 <\alpha < 1$, via parameterization of the interfaces.

We consider linear heat equations in a bounded domain of $\mathbb R^{d}$ with Dirichlet boundary conditions. We analyze the problem of controllability when the control acts on a (small) open subset of the domain. It is by now well known that the system is approximately controllable, null-controllable and also finite-approximately controllable. This last property means that there exist controls by means of which we can simultaneously guarantee the approximate controllability and the exact controllability of a projection of the solution over a finite dimensional subspace. In this paper we obtain explicit bounds of the cost of approximate controllability, i.e., of the minimal norm of a control needed to control the system approximately. We also address the problem of simultaneous finite-approximate controllability. The methods we use combine global Carleman estimates, energy estimates for parabolic equations and the variational approach to approximate controllability. In the case of the constant coefficient heat equation, following a different approach, we are able to obtain better bounds. We also show that, in this particular case, the estimates are sharp. As a consequence of our estimates, we can determine the speed of convergence of the limiting process in which the approximate control is obtained through a sequence of penalized optimal control problems.

We prove the existence of solutions for a filtration problem related to an incompressible flow governed by a nonlinear Darcy's law. Generalized boundary conditions are allowed. Bounded and unbounded domains are considered.

This article deals with the existence of solutions of $$\left\{\begin{array}{rll} \Delta u-\beta(y,c) \frac{\partial u} {\partial x_1} +f(u) & =0 & \hbox{ in }\Sigma\\ \frac{\partial u}{\partial \nu} & =0 & \hbox{ on }\partial\Sigma\\ u(-\infty,\cdot)=0,\ u(+\infty,\cdot) & =1 & \end{array}\right.$$ where $\Sigma=\{(x_1,y)\in\mathbb R \times\omega\}$ is an infinite cylinder with outward unit normal $\nu$ and whose section $\omega\subset\mathbb R^{n-1}$ is a bounded convex domain. The unknowns are the real parameter $c$ and the function $u$ (which respectively represent the speed and the profile of a travelling wave). The function $\beta$ and the nonlinear term $f:[0,1]\to \mathbb R$ are given. We investigate the case where the function $f$ changes sign several times. We prove that there exists a travelling front $(c,u)$ provided that the speeds of the travelling waves for simpler problems can be compared. The proof uses the sliding method and the theory of sub- and supersolutions. This result generalizes for higher dimensions a one-dimensional result of Fife and McLeod.

We consider nonautonomous reaction-diffusion equations on $\mathbb R$, $$u_t-u_{xx}=f(t,u),\ x\in \mathbb R,\ t>0,$$ and investigate solutions in ${C_0(\mathbb R )}$, that is, solutions that decay to zero as $|x|$ approaches infinity. The nonlinearity is assumed to be a $C^1$ function which is $\tau$-periodic in $t$ and satisfies $\ f(t,0)=0\ $ and $ f_u(t,0) <0$ \ \ ($t\in \mathbb R$). Our two main results describe the structure of $\tau$-periodic solutions and asymptotic behavior of general solution with compact trajectories: (1) Each nonzero $\tau$-periodic solution is of definite sign and is even in $x$ about its unique peak (which is independent of $t$) . Moreover, up to shift in space, ${C_0(\mathbb R )}$ contains at most one $\tau$-periodic solution of a given sign. (2) Each solution with trajectory relatively compact in ${C_0(\mathbb R )}$ converges to a single $\tau$-periodic solution. In the proofs of these properties we make extensive use of nodal and symmetry properties of solutions. In particular, these properties are crucial ingredients in our study of the linearization at a periodic solution and in our description of center and stable manifolds of periodic solutions. The paper also contains a section discussing various sufficient conditions for existence of nontrivial periodic solutions.

We consider an infinite sequence of radial wave equations obtained by the separation of variables in the spherical coordinates from the 3-dimensional damped wave equation with spatially nonhomogeneous spherically symmetric coefficients. Our main objects of interest are the nonselfadjoint operators in the energy spaces of 2-component initial data, which are the dynamics generators for the systems governed by the aforementioned equations and nonselfadjoint boundary conditions on the sphere $r=a$. Our main result is the fact that the sets of the root vectors (generalized eigenvectors) of these operators form Riesz bases in the corresponding energy spaces. This result has several applications. The first one is the fact that the aforementioned operators are spectral in the sense of N. Dunford, and, therefore, we have a new nontrivial class of spectral operators. Another application is a precise estimate on the rate of the energy decay, which is equal to the spectral abscissa of the corresponding semigroup. Finally, we use the results of our spectral analysis to formulate the solutions of several problems in control theory of systems governed by damped wave equations (the proofs are given in another work).

We consider the asymptotic derivation of a parabolic equation with nonlocal diffusivity and with a semi-linear nonlocal term that may also be discontinuous. From a reaction-diffusion system where the diffusivity of the second equation is arbitrarily large, using only energy estimates, we obtain a shadow system as an intermediate step for the limit equation. In particular, we obtain the existence of weak solutions and we give a rigorous derivation of a class of diffusion equations that have been used in the literature to model threshold phenomena, for instance, in porous-medium combustion or in localized patterns of excitable media.

This work deals with travelling fronts solutions of some reaction-diffusion equations in an infinite cylinder in dimension $\ge 2$. The problem is set in $\Sigma=\{(x_1,y)\in{\mathbb R}\times\omega\}$ where $\omega\subset{\mathbb R}^{N-1}$ is a bounded and smooth domain with outward normal $\nu$. The equations, with unknowns $c\in {\mathbb R}$ and $u\in C^2(\overline{\Sigma})$, are $$ (P) \qquad\qquad \left\{\begin{array}{rl} \Delta u-(c+\alpha(y))\ \partial_{1}u+f(u)=0 & \hbox{ in }\Sigma={\mathbb R}\times\omega\\ \displaystyle{\frac{\partial u }{\partial\nu}}=0 & \hbox{ on }\partial\Sigma= {\mathbb R}\times\partial\omega\\ u(-\infty,\cdot)=0\hbox{ and }u(+\infty,\cdot)=1 & \end{array} \right. $$ The function $\alpha \in C^0(\overline{\omega})$ is given. The nonlinearity $f$ is assumed to be of the ``bistable type": it changes sign once in $(0,1)$. Berestycki and Nirenberg [8] proved that if $\omega$ is convex then the problem has a solution. Here, by using the invariance by translation and the sliding method, we construct an example of a non-convex domain $\omega$ and of a function $\alpha$ for which we prove that $(P)$ has no solutions. This is in sharp contrast with other types of nonlinearities for which solutions exist whatever $\omega$ may be.

Existence and stability of spherically symmetric stationary interfaces in a two-phase boundary problem are studied in ${\bf R}^{N}$ $(N \geq 2)$. We show that there exist two such solutions: a large ball and a small one. The linearized eigenvalue problem shows that the large ball is unstable with some fastest growing mode. We specify the mode precisely.

The steady-state hydrodynamic equations for isothermal states including the quantum Bohm potential are analyzed. The one-dimensional equations for the electron current density and the particle density are coupled self-consistently to the Poisson equation for the electric potential. The quantum correction can be interpreted as a dispersive regularization of the classical hydrodynamic equations. Physically motivated Dirichlet and Neumann boundary conditions for the electron density are prescribed. The existence and uniqueness of strong solutions for sufficiently small current densities are proven. Furthermore, the classical limit (vanishing scaled Planck constant) and the zero-space-charge limit (vanishing scaled Debye length) are performed. The proofs are based on a transformation of variable for the electron density, yielding a fourth-order, elliptic equation for the new variable. As a by-product of the classical limit, the existence of subsonic solutions to the hydrodynamic system is obtained. Finally, numerical examples are presented showing that for ``large'' current densities, fast oscillations in the particle density occur as the scaled Planck constant tends to zero.