Heat Equation In Non-cylindrical Domains Research Articles

The stability and stabilization of heat equation in non-cylindrical domain

This paper is addressed to a study of the stability and stabilization of heat equation in non-cylindrical domain. Special solutions of the system are first given by the method of the undetermined function and the similarity variables, which indicate that the system is not exponentially stable. Then the stability and stabilization of the system are obtained by the energy estimate and the backstepping method.

On a Volterra equation of the second kind with ‘incompressible’ kernel

Solving the boundary value problems of the heat equation in noncylindrical domains degenerating at the initial moment leads to the necessity of research of the singular Volterra integral equations of the second kind, when the norm of the integral operator is equal to 1. The paper deals with the singular Volterra integral equation of the second kind, to which by virtue of ‘the incompressibility’ of the kernel the classical method of successive approximations is not applicable. It is shown that the corresponding homogeneous equation when $|\lambda|>1$ has a continuous spectrum, and the multiplicity of the characteristic numbers increases depending on the growth of the modulus of the spectral parameter $|\lambda|$ . By the Carleman-Vekua regularization method (Vekua in Generalized Analytic Functions, 1988) the initial equation is reduced to the Abel equation. The eigenfunctions of the equation are found explicitly. Similar integral equations also arise in the study of spectral-loaded heat equations (Amangaliyeva et al. in Differ. Equ. 47(2):231-243, 2011).

Large time and space size behaviour of the heat equation in non-cylindrical domains

In this note we consider a linear parabolic problem defined on a non-cylindrical unbounded domain Q. If Ω t denotes the section of Q above t, the Ω t size goes to +∞, when t → +∞, i.e. the sections Ω t become unbounded in some directions when the time t becomes large. Here a model problem is studied, but the technique used can be applied for a wide class of problems, as nonlinear ones, defined on more general domains Q as those introduced by Lions [11]. An asymptotically exponential convergence of the solutions of such problems towards the solution of an elliptic problem defined on a lower dimensional domain is established.

L p -solvability of the Dirichlet problem for the heat equation in noncylindrical domains with isolated characteristic points at the boundary

We study the solvabitlity of the Dirichlet problem for the heat operator in weighted Sobolev L p -spaces in noncylindrical paraboloid type domains with isolated characteristic points at the boundary. For any p > 1 we find a necessary and sufficient L p -solvability condition and establish an L p -estimate. The results are formulated in terms of Muckenhoupt type conditions on the weight. Bibliography: 10 titles.

The [formula omitted] Neumann problem for the heat equation in non-cylindrical domains

We consider the L p ( ∂ Ω ) Neumann problem for the heat equation when 1 < p < 2 in a time varying domain Ω . Using a perturbation technique and previous L 2 ( ∂ Ω ) Neumann results, the authors obtain a generalization of earlier work in caloric cylinders.

Finite approximate controllability for semilinear heat equations in noncylindrical domains

We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.

The heat equation and reflected Brownian motion in time-dependent domains.: II. Singularities of solutions

We study the space-time Brownian motion and the heat equation in non-cylindrical domains. The paper is mostly devoted to singularities of the heat equation near rough points of the boundary. Two types of singularities are identified—heat atoms and heat singularities. A number of explicit geometric conditions are given for the existence of singularities. Other properties of the heat equation solutions are analyzed as well.

-solubility of the Dirichlet problem for the heat equation in non-cylindrical domains

The Dirichlet problem for the heat equation is considered in bounded and unbounded domains of paraboloid type with isolated characteristic points at the boundary. Necessary and sufficient conditions in terms of the weight ensuring the unique solubility of this problem in weighted Sobolev -spaces are found. In particular, a criterion for the solubility of the problem in the classical Sobolev space is established in the case when the domain is a ball.

The $L^{p}$ regularity problem for the heat equation in non-cylindrical domains

We consider the Dirichlet problem for the heat equation in domains with a minimally smooth, time-varying boundary. Our boundary data is taken to belong to a parabolic Sobolev space having a tangential (spatial) gradient, and $1/2$ of a time derivative, in $L^{p}$, $1 \lt p \lt 2 + \epsilon$. We obtain sharp $L^{p}$ estimates for the parabolic non-tangential maximal function of the gradient of our solutions.

The L p Neumann problem for the heat equation in non-cylindrical domains

The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow> <mml:mi>p</mml:mi> </mml:msup></mml:math> Neumann problem for the heat equation in non-cylindrical domains