Existence and nonexistence of global solutions of degenerate and singular parabolic systems.
Existence and nonexistence results for reaction-diffusion equations in product of cones
Problems of existence and nonexistence of global nontrivial solutions to quasilinear evolution differential inequalities in a product of cones are investigated. The proofs of the nonexistence results are based on the test-function method developed, for the case of the whole space, by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established using the method of supersolutions.
Existence of solutions for a variational unilateral system.
Existence of solutions to nonlocal and singular variational elliptic inequality via Galerkin method.
Fefferman's SAK principle in one dimension
In this article we give a complete proof in one dimension of an a priori inequality involving pseudo-differential operators: if and are symbols in such that , then for all we have the estimate for all in the Schwartz space, where is the usual norm. We use microlocalization of levels I, II and III in the spirit of Fefferman’s SAK principle.
Function classes pertaining to differential inequalities of parabolic type in unbounded regions
Functional differential inequalities with unbounded delay
Classical solutions of functional partial differential inequalities with initial boundary conditions are estimated by maximal solutions of suitable problems for ordinary functional differential equations. Uniqueness of solutions and continuous dependence on given functions are obtained as applications of the comparison result. A theorem on weak functional differential inequalities generated by mixed problems is proved. Our method is based on an axiomatic approach to equations with unbounded delay....
Generalized method of lines for first order partial functional differential equations
Classical solutions of initial boundary value problems are approximated by solutions of associated differential difference problems. A method of lines for an unknown function for the original problem and for its partial derivatives with respect to spatial variables is constructed. A complete convergence analysis for the method is given. A stability result is proved by using differential inequalities with nonlinear estimates of the Perron type for the given operators. A discretization...
Hardy-Poincaré type inequalities derived from p-harmonic problems
We apply general Hardy type inequalities, recently obtained by the author. As a consequence we obtain a family of Hardy-Poincaré inequalities with certain constants, contributing to the question about precise constants in such inequalities posed in [3]. We confirm optimality of some constants obtained in [3] and [8]. Furthermore, we give constants for generalized inequalities with the proof of their optimality.
Hysteresis memory preserving operators
The recent development of mathematical methods of investigation of problems with hysteresis has shown that the structure of the hysteresis memory plays a substantial role. In this paper we characterize the hysteresis operators which exhibit a memory effect of the Preisach type (memory preserving operators). We investigate their properties (continuity, invertibility) and we establish some relations between special classes of such operators (Preisach, Ishlinskii and Nemytskii operators). For a general...
Hysteresis operators - a new approach to evolution differential inequalities
Impulsive nonlocal nonlinear parabolic differential problems.
Infinite systems of hyperbolic differential-functional inequalities.
Infinite systems of strong parabolic differential-functional inequalities.
Initial-boundary value problems for impulsive parabolic functional differential equations
Theorems on differential inequalities generated by an initial-boundary value problem for impulsive parabolic functional differential equations are considered. Comparison results implying uniqueness criteria are proved.
Instantaneous blow-up of solutions to a class of hyperbolic inequalities.
Iterative methods for the Darboux problem for partial functional differential equations.
-inequalities for the laplacian and unique continuation
We prove an inequality of the typeThis is then used to derive the unique continuation property for the differential inequality under suitable local integrability assumptions on the function .
Local behavior and global existence of positive solutions of auλ≤−Δu≤uλ
Monotone iteration for infinite systems of parabolic equations with functional dependence
We consider the initial value problem for an infinite system of differential-functional equations of parabolic type. General operators of parabolic type of second order with variable coefficients are considered and the system is weakly coupled. The solutions are obtained by the monotone iterative method. We prove theorems on weak partial differential-functional inequalities as well the existence and uniqueness theorems in the class of continuous bounded functions and in the class of functions satisfying...