Variational inequalities, formulated on unknown dependent convex sets, are called quasi-variational inequalities (QVI). This paper is concerned with the abstract approach to a class of parabolic QVIs arising in many biochemical/mechanical problems, based on a compactness theorem for parabolic variational inequalities (cf. [9]). The prototype of our model for QVIs of parabolic type is formulated in a reflexive Banach space as the sum of the time-derivative operator under unknown convex constraints and a semimonotone operator, including a feedback system which selects a convex constraint. The main objective of this work is to specify a class of unknown-state dependent convex constraints and to give a precise formulation of QVIs.
We derive weighted Sobolev-Poincaré type inequalities in function spaces concerned with parabolic partial differential equations. We consider general weights depending on both space and time variables belonging to a Muckenhoupt class, so-called the parabolic $A_p$-class, where only the parabolic cubes are involved in the definition.
The carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type. Considering power law weights is a natural question in relation with symmetry breaking issues for Caffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a …
Let $G=(V,E)$ be a locally finite connected weighted graph, and $Ω$ be an unbounded subset of $V$. Using Rothe's method, we study the existence of solutions for the semilinear heat equation $\partial_tu+|u|^{p-1}\cdot u=Δu~(p\ge1)$ and the parabolic variational inequality \begin{eqnarray*} \int_{Ω^\circ} \partial_tu\cdot(v-u)\,dμ\ge \int_{Ω^\circ}(Δu+f)\cdot(v-u)\,dμ\qquad\mbox{for any }v\in \mathcal{H}, \end{eqnarray*} where $\mathcal{H}=\{u\in W^{1,2}(V):u=0\mbox{ on }V\backslashΩ^\circ\}$.
We are concerned with the nonexistence of sign-changing global weak solutions for a class of semilinear parabolic differential inequalities with convection terms in exterior domains. A weight function of the form $t^α|x|^σ$ is considered in front of the power nonlinearity. Two types of non-homogeneous boundary conditions are investigated: Neumann-type and Dirichlet-type boundary conditions. Using a unified approach, for each case, we establish sufficient criteria for the nonexistence of global weak solutions. When $α=0$, the critical exponent in the sense of Fujita is obtained. This exponent i…
In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincaré inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the Hölder regularity of parabolic functions for symmetric non-local Dirichlet forms.
In this paper we consider feedback stabilization for parabolic variational inequalities of obstacle type with time and space depending reaction and convection coefficients and show exponential stabilization to nonstationary trajectories. Based on a Moreau-Yosida approximation, a feedback operator is established using a finite (and uniform in the approximation index) number of actuators leading to exponential decay of given rate of the state variable. Several numerical examples are presented addressing smooth and nonsmooth obstacle functions.
This paper studies the quantitative unique continuation for a semi-linear parabolic-elliptic coupled system on a bounded domain. This system is a simplified version of the chemotaxis model introduced by Keller and Segel. With the aid of priori L^infty-estimates (for solutions of the system) built up in this paper, we treat the semi-linear parabolic equation in the system as a linear parabolic equation, and then use the frequency function method and the localization technique to build up two unique continuation inequalities for the system. As a consequence of the above-mentioned two inequalitie…
We discuss a parabolic version of the space of functions of bounded mean oscillation related to a doubly nonlinear parabolic partial differential equation. Parabolic John-Nirenberg inequalities, which give exponential decay estimates for the oscillation of a function, are shown in the natural geometry of the partial differential equation. Chaining arguments are applied to change the time lag in the parabolic John-Nirenberg inequality. We also show that the quasihyperbolic boundary condition is a necessary and sufficient condition for a global parabolic John-Nirenberg inequality. Moreover, we c…
This work aims to provide a comprehensive and unified numerical analysis for non linear system of parabolic variational inequalities (PVIs) subject to Dirichlet boundary condition. This analysis enables us to establish an existence of the exact solution to the considered model and to prove the convergence for the approximate solution and its approximate gradient. Our results are applicable for several conforming and non conforming numerical schemes.
We study master equations of the form $$(\partial_t+L)^su=f\quad\hbox{in}~\mathbb{R}\timesΩ$$ where $L$ is a divergence form elliptic operator and $Ω\subseteq\mathbb{R}^n$. These are nonlocal equations of order $2s$ in space and $s$ in time that take into account the values of $u$ everywhere in $Ω$ and for past times. We show parabolic interior and boundary Harnack inequalities and local parabolic Hölder continuity of solutions. To this end, we prove a characterization of fractional powers of parabolic operators $\partial_t+L$ with a degenerate parabolic extension problem.
In this article, we present two different approaches for obtaining quantitative inequalities in the context of parabolic optimal control problems. Our model consists of a linearly controlled heat equation with Dirichlet boundary condition $(u_f)_t-Δu_f=f$, $f$ being the control. We seek to maximise the functional $\mathcal J_T(f):=\frac12\int_{(0;T)\times Ω} u_f^2$ or, for some $ε>0$, $\mathcal J_T^ε(f):=\frac12\int_{(0;T)\times Ω} u_f^2+ε\int_Ωu_f^2(T,\cdot)$ and to obtain quantitative estimates for these maximisation problems. We offer two approaches in the case where the domain $Ω$ is a bal…
