On symmetry properties of parabolic equations in bounded domains

We study symmetry properties of nonnegative bounded solutions of fully nonlinear parabolic equations on bounded domains with Dirichlet boundary conditions. We propose sufficient conditions on the equation and domain, which guarantee asymptotic symmetry of solutions.     

Regional controllability of semilinear degenerate parabolic equations in bounded domains

In this paper we study controllability properties of semilinear degenerate parabolic equations. Due to degeneracy, classical null controllability results do not hold in general. Thus we investigate results of ‘regional null controllability’, showing that we can drive the solution to rest at time T on a subset of the space domain, contained in the set where the equation is nondegenerate.     

Asymptotic behavior of solutions to the Dirichlet problem for parabolic equations in domains with singularities

We study solutions of the Dirichlet problem for a second-order parabolic equation with variable coefficients in domains with nonsmooth lateral surface. The asymptotic expansion of the solution in powers of the parabolic distance is obtained in a neighborhood of a singular point of the boundary. The exponents in this expansion are poles of the resolvent of an operator pencil associated with the model problem obtained by freezing the coefficients at the singular point. The main point of the paper is in proving that the resolvent is meromorphic and in estimating it. In the one-dimensional case, the poles of the resolvent satisfy a transcendental equation and can be expressed via parabolic cylinder functions.Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 12–23, January, 1996.This work was partially supported by the Russian Foundation for Basic Research under grant No. 242 93-01-16035.     

Weak solutions of parabolic equations in non-cylindrical domains

In their classical work, Ladyzhenskaya and Uraltseva gave a definition of weak solution for parabolic equations in cylindrical domains. Their definition was broad enough to guarantee the solvability of all such problems but narrow enough to guarantee the uniqueness of these solutions. We give here some alternative definitions which are appropriate to non-cylindrical domains, and we prove the unique solvability of such problems.

     

Asymptotic behavior of solutions of degenerating parabolic equations

Translated from Issledovaniya po Prikladnoi Matematike, No. 17, pp. 166–173, 1990.     

Asymptotic behavior of the solutions of certain parabolic equations

Some laws in physics describe the change of a flux and are represented by parabolic equations of the form (*) \documentclass{article}\pagestyle{empty}\begin{document}$$\frac{{\partial u}}{{\partial t}}=\frac{\partial}{{\partial x_j }}(\eta \frac{{\partial u}}{{ax_j}}-vju),$$\end{document} j≤m, where η and v_j are functions of both space and time. We show under quite general assumptions that the solutions of equation (*) with homogeneous Dirichlet boundary conditions and initial condition u(x, 0) = u_o(x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § R^m the spatial domain, while the constant c depends additionally on which norm is considered. For the solutions of equation (*) with homogeneous Neumann boundary conditions and initial condition u₀(x) ≥ 0 we derive bounds d₁u₁ ≤ u(x, t) ≤ d₂u₂, Where d_i, i = 1, 2, depend on bounds for η, v and G, and the u_i are bounds on the initial condition u₀.     

Asymptotic expansion of solutions of quasilinear parabolic problems in perforated domains

An asymptotic expansion is constructed for solutions of quasilinear parabolic problems with Dirichlet boundary conditions in domains with a fine-grained boundary. It is proved that the sequence of remainders of this expansion in the space W ₂ ^1.1/2 strongly converges to zero.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1542–1566, November, 1993.     

Asymptotic symmetry of solutions of nonlinear partial differential equations

For a large class of partial differential equations on exterior domains or on ?^N we show that any solution tending to a limit from one side as x goes to infinity satisfies the property of “asymptotic spherical symmetry”. The main examples are semilinear elliptic equations, quasilinear degenerate elliptic equations, and first-order Hamilton-Jacobi equations.     

Asymptotic symmetry of singular solutions of semilinear elliptic equations

For dimensions 3?n?6, we derive lower bound for positive solution of

     

Energy decay of solutions of maxwell's equations in bounded domains

In this paper the long time asymptotic behavior of solutions of Maxwell's equations in a bounded domain is investigated. A possibly nonlinear damping term arising from the electric conductivity is considered, whith vanishes on some part of the domain     

Persistence of bounded solutions of parabolic equations under domain perturbation

The aim of this paper is to study the behavior of bounded solutions of parabolic equations on the whole real line under perturbation of the underlying domain. We give the convergence of bounded solutions of linear parabolic equations in the L ² and the L ^p -settings. For the L ^p -theory, we also prove the H?lder regularity of bounded solutions with respect to time. In addition, we study the persistence of a class of bounded solutions which decay to zero at t → ±∞ of semilinear parabolic equations under domain perturbation.     

Singular solutions for semi-linear parabolic equations on nonsmooth domains

We prove the existence of positive singular solutions for the semi-linear parabolic equation on Ω=D×]0,∞[, where p>1,D is a bounded NTA-domain in Rn, n?2, and μ is in a general class of signed Radon measures on D covering the elliptic Kato class of potentials adopted by Zhang and Zhao. A new proof of the result based on a simple fixed point theorem is also given.     

Existence of bounded solutions for nonlinear elliptic equations in unbounded domains

In this paper we study the existence of bounded weak solutions for some nonlinear Dirichlet problems in unbounded domains. The principal part of the operator behaves like the p-laplacian operator, and the lower order terms, which depend on the solution u and its gradient u, have a power growth of order p–1 with respect to these variables, while they are bounded in the x variable. The source term belongs to a Lebesgue space with a prescribed asymptotic behaviour at infinity.     

Nonconvergent bounded solutions of semilinear heat equations on arbitrary domains

We consider the Dirichlet problem for the semilinear heat equation

(0.1)     

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 analysis for blow-up solutions in parabolic equations involving variable exponents

In this article, we consider non-negative solutions of the homogeneous Dirichlet problems of parabolic equations with local or nonlocal nonlinearities, involving variable exponents. We firstly obtain the necessary and sufficient conditions on the existence of blow-up solutions, and also obtain some Fujita-type conditions in bounded domains. Secondly, the blow-up rates are determined, which are described completely by the maximums of the variable exponents. Thirdly, we show that the blow-up occurs only at a single point for the equations with local nonlinearities, and in the whole domain for nonlocal nonlinearities.