Stabilization of solutions of certain parabolic equations and systems

This paper concerns the investigation of the stabilization of solutions of the Cauchy problem for a system of equations of the form u/t = u + f_i(u, v); v/t = v + F₂(u, v). It is proved that under certain assumptions the behavior of solutions as t is determined by mutual arrangement of the set of initial conditions {(u, v): u = f₁(x), v =f ₂(x), xRⁿ} and the trajectories of the system of ordinary differential equations du/dt = F₁(u, v), dv/dt = F₂(u, v). The question of stabilization of the solutions of a single quasilinear parabolic equation is also considered.Translated from Matematicheskie Zametki, Vol. 3, No. 1, pp. 85–92, January, 1968.     

Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems

In this paper we investigate the diffusive zero-relaxation limit of the following multi-D semilinear hyperbolic system in pseudodifferential form:

We analyze the singular convergence, as , in the case which leads to a limit system of parabolic type. The analysis is carried out by using the following steps:

(i)

We single out algebraic ``structure conditions' on the full system, motivated by formal asymptotics, by some examples of discrete velocity models in kinetic theories.

(ii)

We deduce ``energy estimates ', uniformly in , by assuming the existence of a symmetrizer having the so-called block structure and by assuming ``dissipativity conditions' on .

(iii)

We assume a Kawashima type condition and perform the convergence analysis by using generalizations of compensated compactness due to Tartar and Gérard.

Finally, we include examples that show how to use our theory to approximate any quasilinear parabolic systems, satisfying the Petrowski parabolicity condition, or general reaction diffusion systems, including Chemotaxis and Brusselator type systems.

     

Comparison principles for systems of impulsive parabolic equations

For a finite system of impulse parabolic equations some comparison theorems and flow invariance results are obtained using the method of upper and lower solutions. These results can be used for the mathematical modelling of population dynamics systems with abrupt changes implied by diseases, harvesting or heavy immigration.     

Degenerate parabolic systems of vector order Kolmogorov-type equations

We define a new class of degenerate \(\overrightarrow {2b}\)-parabolic systems of Kolmogorov-type equations with coefficients depending only on time. We also construct a fundamental matrix of solutions to a Cauchy problem for systems of this class and study its main properties.     

Nonlinear reaction diffusion problems with ultra parabolic limiting equations

In this paper the nonlinear reaction diffusion problems with ultraparabolic equa- tions are considered.By using comparison theorem,the existence,uniqueness and asymptotic behavior of solution for the problem are studied.     

A random-walk interpretation of the incompressible navier-stokes equations

We introduce a stochastic process that roughly corresponds to a random walk of a single molecule in an incompressible fluid, and we consider the force (per unit volume) exerted by such a “molecule” on the rest of the fluid. Averaging this force over an ensemble of such molecules and taking an appropriate limit, we obtain the Navier-Stokes equations. The origin of time-irreversibility in these equations is discussed in terms of the model.     

Exact self-similar and two-phase solutions of systems of semilinear parabolic equations

Exact single-wave and two-wave solutions of systems of equations of the Newell—Whitehead type are presented. The Painlevé test and calculations in the spirit of Hirota are used to construct these solutions.Moscow Institute of Electronics and Mathematics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 2, pp. 189–199, November, 1994.     

Parametric bufferness in systems of parabolic and hyperbolic equations with small diffusion

We investigate the problem of parametric excitation of oscillations in systems of parabolic and hyperbolic equations with small coefficient of diffusion. We establish the phenomenon of parametric bufferness, i.e., the existence of an arbitrary fixed number of stable periodic solutions for a proper choice of the parameters of equations. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 1, pp. 22–35, January, 1998.     

Subsolutions and super solutions to systems of parabolic equations with applications to generalized Fujita-type systems

A constructive method for obtaining subsolutions and supersolutions to the Cauchy problem for systems of parabolic equations is discussed. Applications of the method to Fujita-type systems are considered leading to global existence and finite time blow-up results.     

A priori estimates for solutions to systems of nonlinear parabolic equations

A class of nondiagonal systems of nonlinear parabolic equations that can be reduced to a scalar parabolic equation in the phase space of a larger dimension is described. In view of such a reduction, it is possible to state the maximum principle for solutions to systems of nonlinear parabolic equations and derive a priori C^2+α-estimates for a solution to the Cauchy problem. Bibliography: 19 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 41–67.     

Monotone iterative methods for infinite systems of parabolic functional differential equations

Two monotone iterative methods for an infinite system of parabolic functional differential equations with initial boundary conditions are constructed: the method of direct iterations and the Chaplygin method. By using the first one the existence theorem is proved. Next, it is shown that the Chaplygin sequences converge quadratically to the unique solution of the original problem.