Invariant manifolds for parabolic equations under perturbation of the domain

We study the effect of domain perturbation on invariant manifolds for semilinear parabolic equations subject to the Dirichlet boundary condition. Under the Mosco convergence assumption on the domains, we prove the upper and lower semicontinuity of both the local unstable invariant manifold and the local stable invariant manifold near a hyperbolic equilibrium. The continuity results are obtained by keeping track of the construction of invariant manifolds in [P.W. Bates, C.K.R.T. Jones, Invariant manifolds for semilinear partial differential equations, in: Dynamics Reported, Vol. 2, in: Dynam. Report. Ser. Dynam. Systems Appl., vol. 2, Wiley, Chichester, 1989, pp. 1–38].     

Persistence of bounded solutions to degenerate Sobolev-Galpern equations

In this paper the persistence of bounded solutions to degenerate evolution equations of Sobolev-Galpern type is discussed. In order to define the evolution operator well, we study the existence and uniqueness of solutions to its linear form. On this basis we discuss exponential dichotomies of the evolution operator and give the Fredholm alternative result for bounded solutions of nonhomogeneous linear degenerate equations. This result enables us to give a condition for the persistence of bounded solutions of a general degenerate nonlinear autonomous equation under a nonautonomous perturbation.     

Uniqueness and non-uniqueness of bounded solutions to singular nonlinear parabolic equations

We investigate well-posedness of initial-boundary value problems for a class of nonlinear parabolic equations with variable density. At some part of the boundary, called singular boundary, the density can either vanish or diverge or not need to have a limit. We provide simple conditions for uniqueness or non-uniqueness of bounded solutions, depending on the behaviour of the density near the singular boundary.     

Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains

We consider solutions of some nonlinear parabolic boundary value problems in radial bounded domains whose initial profile satisfies a reflection inequality with respect to a hyperplane containing the origin. We show that, under rather general assumptions, these solutions are asymptotically (in time) foliated Schwarz symmetric, that is, all elements in the associated omega limit set are axially symmetric with respect to a common axis passing through the origin and nonincreasing in the polar angle from this axis. In this form, the result is new even for equilibria (i.e., solutions of the corresponding elliptic problem) and time periodic solutions.     

Existence and stability of bounded solutions for a system of parabolic equations

In this paper we study the existence and the stability of bounded solutions of the following non-linear system of parabolic equations with homogeneous Dirichlet boundary conditions:

     

Strong full bounded solutions of nonlinear parabolic equations with nonlinear boundary conditions

We study the existence of (generalized) bounded solutions existing for all times for nonlinear parabolic equations with nonlinear boundary conditions on a domain that is bounded in space and unbounded in time (the entire real line). We give a counterexample which shows that a (weak) maximum principle does not hold in general for linear problems defined on the entire real line in time. We consider a boundedness condition at minus infinity to establish (one-sided) L^∞-a priori estimates for solutions to linear boundary value problems and derive a weak maximum principle which is valid on the entire real line in time. We then take up the case of nonlinear problems with (possibly) nonlinear boundary conditions. By using comparison techniques, some (delicate) a priori estimates obtained herein, and nonlinear approximation methods, we prove the existence and, in some instances, positivity and uniqueness of strong full bounded solutions existing for all times.     

Asymptotic behavior of periodic solutions of parabolic equations with weakly nonlinear perturbation

Translated from Matematicheskie Zametki, Vol. 57, No. 3, pp. 369–376, March, 1995.     

Convergence and decay rate to equilibrium of bounded solutions of quasilinear parabolic equations

We study the convergence and decay rate to equilibrium of bounded solutions of the quasilinear parabolic equation

ut−diva(x,∇u)+f(x,u)=0     

Transformation of approximate solutions of linear parabolic equations into asymptotic solutions of quasilinear parabolic equations

Moscow Institute of Electronics and Mathematics. Translated from Matematicheskie Zametki, Vol. 56, No. 6, pp. 122–126, December, 1994.     

A semilinear parabolic system in a bounded domain

Consider the system

Existence of statistical solutions for Karman systems of equations in a bounded domain

Translated from Matematicheskie Zametki, Vol. 57, No. 6, pp. 827–841, June, 1995.     

Estimates of solutions of parabolic equations degenerating on the boundary of a domain

For a parabolic equation, degenerating as a power on the boundary of a domain, a theorem on the a priori boundary estimates of solutions is established. This theorem is applied for the study of the behavior of the constructed solution of the Dirichlet problem near the boundary.Translated from Matematicheskie Zametki, Vol. 22, No. 4, pp. 553–560, October, 1977.In conclusion the author thanks M. I. Matiichuk for the formulation of the problem.     

Estimates of solutions of impulsive parabolic equations under Neumann boundary condition

In this paper, we prove the relation v(t)?u(t,x)?w(t), where u(t,x) is the solution of an impulsive parabolic equations under Neumann boundary condition ∂u(t,x)/∂ν=0, and v(t) and w(t) are solutions of two impulsive ordinary equations. We also apply these estimates to investigate the asymptotic behavior of a model in the population dynamics, and it is shown that there exists a unique solution of the model which converges to the periodic solution of an impulsive ordinary equation asymptotically.     

BLOW-UP AND GLOBAL EXISTENCE FOR A COUPLED SYSTEM OF DEGENERATE PARABOLIC EQUATIONS IN A BOUNDED DOMAIN

This article deals with the conditions that ensure the blow-up phenomenon or its absence for solutions of the system ut=△ul up1vq1 and vt=△vm up2vq2 with homogeneous Dirichlet boundary conditions. The results depend crucially on the sign of the difference p2q1-(l-P1)(m-q2), the initial data, and the domainΩ.