The normalization of systems of parabolic equations with small diffusion in a neighborhood of a cycle

Problems concerning the behavior of solutions of parabolic boundary value problems in a neighborhood of a cycle in near-critical cases are considered. As a consequence of smallness of the diffusion coefficients the critical cases are infinite-dimensional. Steady-state regimes in a neighborhood of a cycle are studied by means of the formalism of normalization methods.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 9, pp. 1155–1161, September, 1991.     

Asymptotic finite solutions of degenerate quasilinear parabolic equations with small diffusion

Translated from Matematicheskie Zametki, Vol. 50, No. 2, pp. 77–88, August, 1991.     

Identifying unknown terms in hyperbolic and parabolic equations

We recover unknown source terms in nonlinear hyperbolic differential equations and in nonlinear parabolic integro-differential equations in one space variable under the assumption of knowing a first integral (in the hyperbolic case) or the value of the solution at a point inside the domain (in the parabolic case). For this class of problems we prove existence results in classes of smooth solutions. Moreover, for linear hyperbolic and parabolic differential equations in one space variable we recover some characteristic parameters. Conferenza tenuta il giorno 29 Novembre 1999     

On parabolic equations with a small parameter

A singularly perturbed parabolic equation with a nonlinear right-hand side of a special form is examined. A numerical analytical study of such equations is performed.     

Collapse of the solutions of parabolic and hyperbolic equations with nonlinear boundary conditions

It is shown that the solutions of linear and quasilinear equations of parabolic and hyperbolic type may collapse because of the presence of nonlinearities in the boundary conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 75–83, 1983.     

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 generalised cole-hopf transformation for nonlinear parabolic and hyperbolic equations

Summary The Cole-Hopf transformation has been generalized to generate a large class of nonlinear parabolic and hyperbolic equations which are exactly linearizable. These include model equations of exchange processes and turbulence. The methods to solve the corresponding linear equations have also been indicated.

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Sommaire La transformation de Cole et de Hopf a été généralisée en vue d'engendrer une classe d'équations nonlinéaires paraboliques et hyperboliques qui peuvent être rendues linéaires de façon exacte. Elles comprennent des équations modèles de procédés d'échange et de turbulence. Les méthodes pour résoudre les équations linéaires correspondantes ont également été indiquées.

     

Semilinear hyperbolic systems and equations with singular initial data

We study the weak limits (t, .)of solutions to semilinear strictly hyperbolic systems and wave equations with initial datau (0, .) approximating a distribution, 0 < 1. We propose an optimal link between the singularity of and the growth of the nonlinear term in order that exists. In this way we extend some of the results in [3], [10], [13].     

A study of the stability of solutions of linear parabolic equations with nearly constant coefficients and small diffusion

We study the stability of the solutions of boundary value problems for a certain class of Petrovskii-parabolic systems with sufficiently small diffusion coefficients. The dimension of the set of critical cases in the stability problem turns out to be infinite. We develop an efficient algorithm for studying stability. As examples we consider parabolic boundary value problems with delay and rapidly oscillating coefficients, the problem of parametric resonance under a double-frequency perturbation, and problems with variable leading terms and variable domain of definition. Bibliography: 21 titles.Translated fromTrudy Seminara imeni I. G. Petrovskogo, No. 15, pp. 128–155, 1991.     

Stability of an abstract system of coupled hyperbolic and parabolic equations

In this paper, we provide a complete stability analysis for an abstract system of coupled hyperbolic and parabolic equations $$\begin{array}{ll}\;\;u_{tt} = -Au + \gamma A^{\alpha} \theta,\\ \quad \theta_t = -\gamma A^{\alpha}u_t - kA^{\beta}\theta,\\ u(0) = u_0, \quad u_t(0) = v_0, \quad \theta(0) = \theta_0\end{array}$$ where A is a self-adjoint, positive definite operator on a Hilbert space H. For ${(\alpha,\beta) \in [0,1] \times [0,1]}$ , the region of exponential stability had been identified in Ammar-Khodja et al. (ESAIM Control Optim Calc Var 4:577–593,1999). Our contribution is to show that the rest of the region can be classified as region of polynomial stability and region of instability. Moreover, we obtain the optimality of the order of polynomial stability.     

Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations

We consider compositions of stochastic processes that are governed by higherorder partial differential equations. The processes studied include compositions of Brownian motions, stable-like processes with Brownian time, Brownian motion whose time is an integrated telegraph process, and an iterated integrated telegraph process. The governing higher-order equations that are obtained are shown to be either of the usual parabolic type or, as in the last example, of hyperbolic type.