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We consider a reaction-diffusion system of the activator-inhibitor type with unilateral boundary conditions leading to a quasivariational inequality. We show that there exists a positive eigenvalue of the problem and we obtain an instability of the trivial solution also in some area of parameters where the trivial solution of the same system with Dirichlet and Neumann boundary conditions is stable. Theorems are proved using the method of a jump in the Leray-Schauder degree.
The paper deals with a model for water freezing in a deformable elastoplastic container. The mathematical problem consists of a system of one parabolic equation for temperature, one integrodifferential equation with a hysteresis operator for local volume increment, and one differential inclusion for the water content. The problem is shown to admit a unique global uniformly bounded weak solution.
We consider a two dimensional elastic body submitted to unilateral contact conditions, local friction
and adhesion on a part of his boundary. After discretizing the variational formulation with respect
to time we use a smoothing technique to approximate the friction term by an auxiliary problem. A shifting
technique enables us to obtain the existence of incremental solutions with bounds independent of the
regularization parameter. We finally obtain the existence of a quasistatic solution...
One proves, in the case of piecewise smooth coefficients, that the time derivative of the solution of the so called dam problem is a measure, extending the result proved by the same authors in the case of Lipschitz continuous coefficients.
This paper addresses the Cauchy problem for the
gradient flow equation in a Hilbert space
In this paper we consider a two-phase one-dimensional free boundary problem for the heat equation, arising from a mathematical model for a Bingham-like fluid with a visco-elastic core. The main feature of this problem consists in the very peculiar structure of the free boundary condition, not allowing to use classical tools to prove well posedness. Existence of classical solution is proved using a fixed point argument based on Schauder's theorem. Uniqueness is proved using a technique based on a...
We consider an abstract formulation for a class of parabolic quasi-variational inequalities or quasi-linear PDEs, which are generated by subdifferentials of convex functions with various nonlocal constraints depending on the unknown functions. In this paper we specify a class of convex functions on a real Hilbert space H, with parameters 0 ≤ t ≤ T and v in a set of functions from [-δ₀,T], 0 < δ₀ < ∞, into H, in order to formulate an evolution equation of the form
, 0 < t < T, in H.
Our...
A conceptual numerical strategy for rate-independent processes in the
energetic formulation is proposed and its convergence is proved under various
rather mild data qualifications. The novelty is that we obtain convergence of
subsequences of space-time discretizations even in case where the limit
problem does not have a unique solution and we need no
additional assumptions on higher regularity of the limit solution.
The variety of general perspectives thus
obtained is illustrated on several...
A Navier-Stokes type equation corresponding to a non-linear relationship between the stress tensor and the velocity deformation tensor is studied and existence and uniqueness theorems for the solution, in the 3-dimensional case, of the Cauchy-Dirichlet problem, for a bounded solution and for an almost periodic solution are given. An inequality which in some sense is the limit of the equation is also considered and existence theorems for the solution of the Cauchy-Dirichlet problems and for a periodic...
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