We consider a viscoelastic body occupying a smooth bounded domain of $R^3$ under the effects of volumic traction forces. Inertial effects are considered: hence, the equation describing the evolution of displacements is of the second order in time. On
a part of the boundary of the domain, the body is anchored to a support and no displacement may occur; on a second part, the body can move freely; on a third portion of the boundary, the body is in adhesive contact with a solid support. The boundary forces acting there as a byproduct of elastic stresses are responsible for delamination, i.e., progressive failure of adhesive bonds. This phenomenon is mathematically represented by a boundary variable that represents the local fraction of active bonds and is assumed to satisfy a doubly nonlinear ODE. Following the lines of a new approach based on duality methods in Sobolev-Bochner spaces, we define a suitable concept of weak solution to the resulting system of equations. Correspondingly, we prove an existence result on finite time intervals of arbitrary length.
We introduce a fractional variant of the Cahn-Hilliard equation settled in a bounded domain $Omega$ of $R^N$ and complemented with homogeneous Dirichlet boundary conditions of solid type (i.e., imposed in the entire complement of $Omega$). After sett
ing a proper functional framework, we prove existence and uniqueness of weak solutions to the related initial-boundary value problem. Then, we investigate some significant singular limits obtained as the order of either of the fractional Laplacians appearing in the equation is let tend to 0. In particular, we can rigorously prove that the fractional Allen-Cahn, fractional porous medium, and fractional fast-diffusion equations can be obtained in the limit. Finally, in the last part of the paper, we discuss existence and qualitative properties of stationary solutions of our problem and of its singular limits.
This paper is concerned with the Cauchy-Dirichlet problem for a doubly nonlinear parabolic equation involving variable exponents and provides some theorems on existence and regularity of strong solutions. In the proof of these results, we also analyz
e the relations occurring between Lebesgue spaces of space-time variables and Lebesgue-Bochner spaces of vector-valued functions, with a special emphasis on measurability issues and particularly referring to the case of space-dependent variable exponents. Moreover, we establish a chain rule for (possibly nonsmooth) convex functionals defined on variable exponent spaces. Actually, in such a peculiar functional setting the proof of this integration formula is nontrivial and requires a proper reformulation of some basic concepts of convex analysis, like those of resolvent, of Yosida approximation, and of Moreau-Yosida regularization.
In this paper we analyze a nonlinear parabolic equation characterized by a singular diffusion term describing very fast diffusion effects. The equation is settled in a smooth bounded three-dimensional domain and complemented with a general boundary c
ondition of dynamic type. This type of condition prescribes some kind of mass conservation; hence extinction effects are not expected for solutions that emanate from strictly positive initial data. Our main results regard existence of weak solutions, instantaneous regularization properties, long-time behavior, and, under special conditions, uniqueness.
In this paper we derive, starting from the basic principles of Thermodynamics, an extended version of the nonconserved Penrose-Fife phase transition model, in which dynamic boundary conditions are considered in order to take into account interactions
with walls. Moreover, we study the well-posedness and the asymptotic behavior of the Cauchy problem for the PDE system associated to the model, allowing the phase configuration of the material to be described by a singular function.
We consider a phase-field model where the internal energy depends on the order parameter in a nonlocal way. Therefore, the resulting system consists of the energy balance equation coupled with a nonlinear and nonlocal ODE for the order parameter. Suc
h system has been analyzed by several authors, in particular when the configuration potential is a smooth double-well function. More recently, in the case of a potential defined on (-1,1) and singular at the endpoints, the existence of a finite-dimensional global attractor has been proven. Here we examine both the case of smooth potentials as well as the case of physically realistic (e.g., logarithmic) singular potentials. We prove well-posedness results and the eventual global boundedness of solutions uniformly with respect to the initial data. In addition, we show that the separation property holds in the case of singular potentials. Thanks to these results, we are able to demonstrate the existence of a finite-dimensional attractors in the present cases as well.
We study a Penrose-Fife phase transition model coupled with homogeneous Neumann boundary conditions. Improving previous results, we show that the initial value problem for this model admits a unique solution under weak conditions on the initial data.
Moreover, we prove asymptotic regularization properties of weak solutions.
P. Galenko et al. proposed a modified Cahn-Hilliard equation to model rapid spinodal decomposition in non-equilibrium phase separation processes. This equation contains an inertial term which causes the loss of any regularizing effect on the solution
s. Here we consider an initial and boundary value problem for this equation in a two-dimensional bounded domain. We prove a number of results related to well-posedness and large time behavior of solutions. In particular, we analyze the existence of bounded absorbing sets in two different phase spaces and, correspondingly, we establish the existence of the global attractor. We also demonstrate the existence of an exponential attractor.
The Penrose-Fife system for phase transitions is addressed. Dirichlet boundary conditions for the temperature are assumed. Existence of global and exponential attractors is proved. Differently from preceding contributions, here the energy balance equ
ation is both singular at 0 and degenerate at infinity. For this reason, the dissipativity of the associated dynamical process is not trivial and has to be proved rather carefully.
