Singular Parabolic Problems Research Articles

Global solutions of singular parabolic equations arising from electrostatic MEMS

We study dynamic solutions of the singular parabolic problem ( P) { u t − Δ u = λ ∗ | x | α ( 1 − u ) 2 , ( x , t ) ∈ B × ( 0 , ∞ ) , u ( x , 0 ) = u 0 ( x ) ⩾ 0 , x ∈ B , u ( x , t ) = 0 , x ∈ ∂ B , where α ⩾ 0 and λ ∗ > 0 are two parameters, and B is the unit ball { x ∈ R N : | x | ⩽ 1 } with N ⩾ 2 . Our interest is focussed on ( P) with λ ∗ : = ( 2 + α ) ( 3 N + α − 4 ) 9 , for which ( P) admits a singular stationary solution in the form S ( x ) = 1 − | x | 2 + α 3 . This equation models dynamic deflection of a simple electrostatic Micro-Electro-Mechanical-System (MEMS) device. Under the assumption u 0 ≨︀ S ( x ) , we address the existence, uniqueness, regularity, stability or instability, and asymptotic behavior of weak solutions for ( P) . Given α ∗ ∗ : = 4 − 6 N + 3 6 ( N − 2 ) 4 , in particular we show that if either N ⩾ 8 and α > α ∗ ∗ or 2 ⩽ N ⩽ 7 , then the minimal compact stationary solution u λ ∗ of ( P) is stable and while S ( x ) is unstable. However, for N ⩾ 8 and 0 ⩽ α ⩽ α ∗ ∗ , ( P) has no compact minimal solution, and S ( x ) is an attractor from below not from above. Furthermore, the refined asymptotic behavior of global solutions for ( P) is also discussed for the case where N ⩾ 8 and 0 ⩽ α ⩽ α ∗ ∗ , which is given by a certain matching of different asymptotic developments in the large outer region closer to the boundary and the thin inner region near the singularity.

Estimates for the Quenching Time of a Parabolic Equation Modeling Electrostatic MEMS

The singular parabolic problem ut = ∆u − λf(x) (1+u)2 on a bounded domain Ω of R with Dirichlet boundary conditions, models the dynamic deflection of an elastic membrane in a simple electrostatic Micro-Electromechanical System (MEMS) device. In this paper, we analyze and estimate the quenching time of the elastic membrane in terms of the applied voltage —represented here by λ. As a byproduct, we prove that for sufficiently large λ, finitetime quenching must occur near the maximum point of the varying dielectric permittivity profile f(x).

A Class of Free Boundary Problems with Onset of a New Phase

A class of diffusion driven Free Boundary Problems is considered which is characterized by the initial onset of a phase and by an explicit kinematic condition for the evolution of the free boundary. By a domain fixing change of variables it naturally leads to coupled systems comprised of a singular parabolic initial boundary value problem and a Hamilton-Jacobi equation. Even though the one dimensional case has been thoroughly investigated, results as basic as well-posedness and regularity have so far not been obtained for its higher dimensional counterpart. In this paper a recently developed regularity theory for abstract singular parabolic Cauchy problems is utilized to obtain the first well-posedness results for the Free Boundary Problems under consideration. The derivation of elliptic regularity results for the underlying static singular problems will play an important role.

On the uniqueness of bounded solutions to singular parabolic problems

We provide criteria for uniqueness or nonuniqueness of bounded solutions for a wide class of second order parabolic problems with singular coefficients.

A nonlocal singular parabolic problem modelling gravitational interaction of particles

Existence and nonexistence of solutions (both stationary and evolution) for a parabolic-elliptic system describing a cloud of self-interacting particles surrounding a fixed mass are studied. In particular, sharp conditions for local-in-time and global-in-time existence, as well as finite-time blow up are given.

The use of linear approximation scheme for solving the Stefan problem

This paper deals with the linear approximation scheme to approximate a singular parabolic problem: the two-phase Stefan problem on a domain consisting of two components with imperfect contact. The results of some numerical experiments and comparisons are presented. The method was used to determine the temperature of steel in the process of continuous casting.

Existence of classical solutions for singular parabolic problems

Let L u ≡ u x x + b u x / x − u t Lu \equiv {u_{xx}} + b{u_x}/x - {u_t} with b b a constant less than 1. Its Green’s function corresponding to first boundary conditions is constructed by eigenfunction expansion. With this, a representation formula is established to obtain existence of a classical solution for the linear first initial-boundary value problem. Uniqueness of a solution follows from the strong maximum principle. Properties of Green’s function and of the solution are also investigated.

The asymptotics of constrained control in optimal singular parabolic problems

A complete asymptotic solution is constructed and justified for the optimal singular parabolic problems with constrained control and a completely degenerate differential part of an operator.

Computational error analysis of periodic solutions for singular semilinear parabolic problems

By using the monotone method, a theoretical and computational method is given to find, to the degree of accuracy desired, approximate solutions of a class of singular semilinear parabolic problems. So that the error between the actual solution and its approximation is within a given error tolerance, the number of iterations is determined. Since each iterate is in terms of an infinite series, the number of terms to be retained in each iterate is determined so that its error from the exact iterate is within a given error tolerance. An improved rate of convergence is then given to show that it is possible to reduce the number of terms retained in each iterate. An algorithm is also described to obtain numerical solutions. For illustration of the computational methods developed, a numerical example is given.

Computational methods for time-periodic solutions of singular semilinear parabolic problems

Existence and uniqueness of time-periodic solutions for a class of singular semilinear parabolic problems are established by the method of alternating bounds and the Picard method respectively. For each method, the number of iterations and the number of terms retained in each successive iterate (which is in terms of an infinite series) are determined such that the difference between the resulting approximation and the actual solution is within the degree of accuracy desired. Under additional assumptions, it is shown that the rates of convergence can be improved so that the number of terms retained in each iterate may be reduced without sacrificing accuracy.

Galerkin approximations for singular linear elliptic and semilinear parabolic problems

We prove the existence of exact solutions and convergence of numerical (Galerkin) approximations for the semilinear degenerate parabolic problem ut+Lu = F(t,u) and for the degenerate elliptic problem Lu = f where L is a highly singular linear elliptic operator. The class of operators L contains in particular the operators of generalized bi-axially symmetric potential theory.

Critical lengths for global existence of solutions for coupled semilinear singular parabolic problems

Critical lengths for global existence of solutions for coupled semilinear singular parabolic problems

Finite element approximations of singular parabolic problems

AbstractAn efficient implementation of finite element methods for free boundary parabolic problems in general two dimensional space domains is presented. The Stefan problem, the Hele‐Shaw problem and the porous medium equation are included. Backward differences or linearization techniques are used for the time discretization of the problem. The performances of these schemes are discussed with several numerical tests.

An Efficient Linear Scheme to Approximate Parabolic Free Boundary Problems: Error Estimates and Implementation

This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of $O({h^{1/2}})$ is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.

An efficient linear scheme to approximate parabolic free boundary problems: error estimates and implementation

This paper deals with a fully discrete scheme to approximate multidimensional singular parabolic problems; two-phase Stefan problems and porous medium equations are included. The algorithm consists of approximating at each time step a linear elliptic partial differential equation by piecewise linear finite elements and then making an element-by-element algebraic correction to account for the nonlinearity. Several energy error estimates are derived for the physical unknowns; a sharp rate of convergence of O ( h 1 / 2 ) O({h^{1/2}}) is our main result. The crucial point in implementing the scheme is the efficient resolution of linear systems involved. This topic is discussed, and the results of several numerical experiments are shown.

Error estimates for multidimensional singular parabolic problems

We consider a rather general class of singular parabolic problems; multidimensional two-phase Stefan problem and porous-medium equation involving additional nonlinearities are included. We analyze an approximation scheme consisting in a regularization procedure and discretization by means ofC °-piecewise linear finite elements in space and backward-differences in time. We prove severalL β-error estimates for the various unknowns, wherep depends on the problem. As a by-product of our technique, we obtain an optimalL 2-error estimate for enthalpy.

Generalized axially symmetric heat potentials and singular parabolic initial boundary value problems

Generalized axially symmetric heat potentials and singular parabolic initial boundary value problems

Singular perturbations for a singular parabolic initial boundary value problem

The first Fourier problem is considered for a class of singular parabolic equations with a small parameter multiplying the time derivative term. Boundary layer corrections for t close to t=0 are constructed by a singular perturbation procedure and the asymptotic correctness of the approximation is established.

Nonlinear problem of a shock-tube interaction-region boundary layer.

Description of a numerical method that dispenses with the recourse to linearization or 'momentum-integral-correction' schemes in solving nonlinear problems of boundary layer flow in the interaction region of a shock tube or of similar singular parabolic problems of shock-induced unsteady boundary layers. The accuracy of the numerical results obtainable is shown to be satisfactory.