Solonnikov parabolic systems with quasihomogeneous structure

We consider a new class of systems of equations that combine the structures of Solonnikov and éidel’man parabolic systems. We prove a theorem on the reduction of a general initial-value problem to a problem with zero initial data and a theorem on the correct solvability of an initial-value problem in a model case. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1501–1510, November, 2006.     

On mixed problem for parabolic variational inequalities

Summary We give an existence-uniqueness result for the parabolic variational inequality (1.3) and we apply this result to mixed problem for parabolic variational inequalities and to parabolic quasi-variational inequalities with constraints on the boundary.

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Sunto. Si dà un risultato di esistenza ed unicità per la disequazione variazionale parabolica (1.3) e si applica tale risultato al problema misto per disequazioni variazionali paraboliche e a disequazioni quasi variazionali paraboliche con vincolo sulla frontiera.

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Entrata in Redazione il 7 giugno 1976.

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Politecnico di Milano, Istituto di Matematica (Analisi); lavoro eseguito nell'ambito del gruppo G.N.A.F.A. del C.N.R.     

On double degenerate quasilinear parabolic variational inequalities

We deal with the existence of weak solutions of double degenerate quasilinear parabolic inequalities with a Signorini-Dirichlet-Neumann type mixed boundary condition, which may degenerate in certain subset of the boundary or on a segment in the interior of the domain and in time. The main tools in our study are the maximM monotone property of the derivative operator with zero-initial valued conditions and the theory of pseudomonotone perturbations of maximal monotone mappings.     

On nonlinear parabolic variational inequalities containing nonlocal terms

We investigate nonlinear parabolic variational inequalities which contain functional dependence on the unknown function. Such parabolic functional differential equations were studied e.g. by L. Simon in [8] (which was motivated by the work of M. Chipot and L. Molinet in [4]), where the following equation was considered: (1) $$ \begin{array}{*{20}c} {D_t u(t,x) - \sum\limits_{i = 1}^n {D_i \left[ {a_i (t,x,u(t,x),Du(t,x);u)} \right]} } \\ { + a_0 (t,x,u(t,x),Du(t,x);u) = f(t,x)} \\ {(t,x) \in Q_T = (0,T) \times \Omega ,a_i :Q_T \times R^{n + 1} \times L^p (0,T;V) \to R,} \\ \end{array} $$ where V denotes a closed linear subspace of the Sobolev-space W ^1,p (Ω) (2 ≦ p < ∞). In the above mentioned paper existence of weak solutions of the above equation is shown. These results were extended to systems of functional differential equations in [2]. In the following, we extend these existence results to variational inequalities by using the (less known) results of [6]. Finally, we show some examples.     

On a class of singular nonlinear parabolic variational inequalities

Summary We study a class of singular or degenerate parabolic variational inequalities, containing some nonlinear operators. We prove an existence and uniqueness result for weak solutions, in the framework of suitable Banach weighted spaces.

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Sunto Si studia una classe di disequazioni variazionali paraboliche singolari o degeneri, contenenti operatori non lineari. Si dimostra un risultato di esistenza e unicità per soluzioni deboli, nell'ambito di opportuni spazi di Banach con peso.

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This work was supported in part by the «Istituto di Analisi Numerica del C.N.R.» (Pavia, Italy), the G.N.A.F.A. of the C.N.R. and the Ministero della Pubblica Istruzione (Italy) (through 60% and 40% grants).     

Global solvability of the Cauchy-Dirichlet problem for nondiagonal parabolic systems with variational structure in the case of two spatial variables

The Cauchy-Dirichlet problem for quasilinear parabolic systems of second-order equations is considered in the case of two spatial variables. Under the condition that the corresponding elliptic operator has variational structure, the global in time solvability is established. The solution is smooth almost everywhere and the number of singular points is finite. Sufficient conditions that guarantee the absence of singular points are given. Bibliography: 23 titles. Translated fromProblemy Matematicheskogo Analiza No. 16, 1997, pp. 3–40.     

Existence of solutions of nonlinear degenerate systems of parabolic variational inequalities

Systems of degenerated parabolic inequalities with an operator of gradient type are investigated. A Galerkintype argument is applied to approximate these systems by a sequence of time dependent variational inequalities in finite-dimensional spaces. Bibliography:1 title. Dedicated to N. N. Uraltseva on her jubilee Published inZapiski Nauchnykh Seminarov POMI, Vol. 221, 1995, pp. 243–252. Translated by M. Fuchs.     

Wiener estimates for a class of systems of parabolic variational inequalities

Summary Wiener estimates at a point for parabolic diagonal systems of parabolic variational inequalities with obstacle are proved by a Green function method.This paper was written while the first Author was visiting the Department of Mathematics of Linköping University.     

On global stability of parabolic systems with delay

A semilinear differential equation with delay in a Hilbert space is considered. Conditions for global stability are established. An application to parabolic systems with delay is discussed.     

Quasilinear parabolic variational inequalities with multi-valued lower-order terms

In this paper, we provide an analytical frame work for the following multi-valued parabolic variational inequality in a cylindrical domain \({Q = \Omega \times (0, \tau)}\) : Find \({{u \in K}}\) and an \({{\eta \in L^{p'}(Q)}}\) such that $$\eta \in f(\cdot,\cdot,u), \quad \langle u_t + Au, v - u\rangle + \int_Q \eta (v - u)\,{\rm d}x{\rm d}t \ge 0, \quad \forall \, v \in K,$$ where \({{K \subset X_0 = L^p(0,\tau;W_0^{1,p}(\Omega))}}\) is some closed and convex subset, A is a time-dependent quasilinear elliptic operator, and the multi-valued function \({{s \mapsto f(\cdot,\cdot,s)}}\) is assumed to be upper semicontinuous only, so that Clarke’s generalized gradient is included as a special case. Thus, parabolic variational–hemivariational inequalities are special cases of the problem considered here. The extension of parabolic variational–hemivariational inequalities to the general class of multi-valued problems considered in this paper is not only of disciplinary interest, but is motivated by the need in applications. The main goals are as follows. First, we provide an existence theory for the above-stated problem under coercivity assumptions. Second, in the noncoercive case, we establish an appropriate sub-supersolution method that allows us to get existence, comparison, and enclosure results. Third, the order structure of the solution set enclosed by sub-supersolutions is revealed. In particular, it is shown that the solution set within the sector of sub-supersolutions is a directed set. As an application, a multi-valued parabolic obstacle problem is treated.     

Some nonlinear parabolic variational inequalities

In this paper we study initial value problems for nonlinear parabolic variational inequalities involving time-dependent subdifferentials of convex functions on a Hilbert space. We shall show the existence of a solution by a semi-discretisation method with respect to the time.     

On homogenization of periodic parabolic systems

We study homogenization in the small period limit for a periodic parabolic Cauchy problem in ^d and prove that the solutions converge in L ₂(^d) to the solution of the homogenized problem for each t > 0. For the L₂(^d)-norm of the difference, we obtain an order-sharp estimate uniform with respect to the L ₂(^d)-norm of the initial value.Translated from Funktsionalnyi Analiz i Ego Prilozheniya, Vol. 38, No. 4, pp. 86–90, 2004Original Russian Text Copyright © by T. A. SuslinaSupported by RFBR grant No. 02-01-00798.     

On systems of second-order variational inequalities

Letu be a solution of an elliptic (linear or nonlinear) variational inequality with obstacle. Under natural smoothness conditions put upon the data, it is shown that the second derivatives ofu lie in a certain Morrey space and hence, in the case of two independent variables, the solutionu has a Hölder continuous gradient.     

On the structure of parabolic subgroups

The main aim of this paper is to show that the Jacobson and Brown-McCoy radicals of rings graded by free groups are homogeneous. As an application we get some information on the structure of the Jacobson radical of monomial rings. In particular we give a positive answer to a question posed in [12]. We extend also a result of [13] on the Brown-McCoy radical of polynomial rings in non-commutative variables. Actually this and the question of [12] motivated our studies.     

Stochastic variational inequalities of parabolic type

Existence and uniqueness of strong solutions of stochastic partial differential equations of parabolic type with reflection (e.g., the solutions are never allowed to be negative) is proved. The problem is formulated as a stochastic variational inequality and then compactness is used to derive the result, but the method requires the space dimension to be one.This research was supported by NSERC under Grant No. 8051.