Pulling rate, thermal and capillary conditions setting for rod growth

The purpose of this paper is to elaborate a procedure, based on a mathematical model, for the setting of the pulling rate, capillary and thermal conditions, in order to grow a cylindrical rod with prescribed radius and length, by edge-defined film-fed growth (EFG) method. First, in the case of an axisymmetric meniscus, we use simultaneously the catching and the angle fixation conditions in order to find a formula, which describes the fluctuation of the angle between the horizontal axis Or and the tangent line to the free surface of the meniscus at the three phase point. This angle appears in the system of differential equations which describes the evolution of the radius of the rod. During the growth this angle can fluctuate due to the fluctuations of the crystal radius, or crystallization front level, or pressure, respectively. In the second part of the paper it is shown in which kind this formula together with the energy balance equation at the crystallization front level can be used for setting the pulling rate, the thermal and capillary conditions to grow a cylindrical rod with prescribed radius and length. Numerical illustration and simulation are presented for rods having thermo-physical properties similar to NdYAG and InSb. This type of results can be useful for the experiment planning, since personal computer simulation is less expensive than experiment. With this aim the present study was undertaken.     

Analysis of an elliptic-parabolic free boundary problem modelling the growth of non-necrotic tumor cord

We study a free boundary problem modelling the growth of a tumor cord in which tumor cells live around and receive nutrient from a central blood vessel. The evolution of the tumor cord surface is governed by Darcy's law together with a surface tension equation. The concentration of nutrient in the tumor cord satisfies a reaction-diffusion equation. In this paper we first establish a well-posedness result for this free boundary problem in some Sobolev-Besov spaces with low regularity by using the analytic semigroup theory. We next study asymptotic stability of the unique radially symmetric stationary solution. By making delicate spectrum analysis for the linearized problem, we prove that this stationary solution is locally asymptotically stable provided that the constant c representing the ratio between the diffusion time of nutrient and the birth time of new cells is sufficiently small.     

Lie group action and stability analysis of stationary solutions for a free boundary problem modelling tumor growth

In this paper we study asymptotic behavior of solutions for a free boundary problem modelling tumor growth. We first establish a general result for differential equations in Banach spaces possessing a Lie group action which maps a solution into new solutions. We prove that a center manifold exists under certain assumptions on the spectrum of the linearized operator without assuming that the space in which the equation is defined is of either DA(θ) or DA(θ,∞) type. By using this general result and making delicate analysis of the spectrum of the linearization of the stationary free boundary problem, we prove that if the surface tension coefficient γ is larger than a threshold value γ^* then the unique stationary solution is asymptotically stable modulo translations, provided the constant c is sufficiently small, whereas if γ<γ^* then this stationary solution is unstable.     

Existence and uniqueness of solution in Sobolev space for an unsteady crystal growth problem with zero surface tension

We study an initial value problem for a two-dimensional dendritic crystal growth model with zero surface tension. If the initial data is in Sobolev space H²(R), it is proved that an unique local solution exists in proper Sobolev space.     

Distributional solutions to a generalized wave equation for gravity waves on deep water

The equations governing the linearized small amplitude approximation for gravity waves on deep water can be reformulated by the introduction of a cross-surface differential operator, H, which acts like a square-root of the two-dimensional Laplacian. This yields a single scalar equation for the amplitude of the wave-like motion off a horizontal static surface resulting in a mixed initial and boundary value problem for the wave operator, ∂_tt + c²H. The pressure impulse response for an unperturbed static fluid will be calculated via a formal eigenfunction expansion and it will be shown that this yields a distributional solution. Then, the mixed problem will be generalized to allow for distributional data where the initial data is injected into the non-homogeneous term. By employing eigenfunction representations for distributions with compact support it will be shown that a formal eigenfunction expansion also yields a distributional solution to this generalized mixed problem.     

Well-posedness and stability for an elliptic-parabolic free boundary problem modeling the growth of multi-layer tumors

In this paper we study a free boundary problem modeling the growth of multi-layer tumors. This free boundary problem contains one parabolic equation and one elliptic equation, defined on an unbounded domain in R2 of the form 0 〈 y 〈p（x,t）, where p（x,t） is an unknown function. Unlike previous works on this tumor model where unknown functions are assumed to be periodic and only elliptic equations are evolved in the model, in this paper we consider the case where unknown functions are not periodic functions and both elliptic and parabolic equations appear in the model. It turns out that this problem is more difficult to analyze rigorously. We first prove that this problem is locally well-posed in little H61der spaces. Next we investigate asymptotic behavior of the solution. By using the principle of linearized stability, we prove that if the surface tension coefficient y is larger than a threshold value y〉0, then the unique flat equilibrium is asymptotically stable provided that the constant c representing the ratio between the nutrient diffusion time and the tumor-cell doubling time is sufficiently small.     

Finite difference solution of a singular boundary value problem for the p-Laplace operator

We are concerned about a singular boundary value problem for a second order nonlinear ordinary differential equation. The differential operator of this equation is the radial part of the so-called N-dimensional p-Laplacian (where p?>?1), which reduces to the classical Laplacian when p?=?2. We introduce a finite difference method to obtain a numerical solution and, in order to improve the accuracy of this method, we use a smoothing variable substitution that takes into account the behavior of the solution in the neighborhood of the singular points.     

Local existence and non-relativistic limits of shock solutions to a multidimensional piston problem for the relativistic Euler equations

The multidimensional piston problem is a special initial-boundary value problem. The boundary conditions are given in two conical surfaces: one is the boundary of the piston, and the other is the shock whose location is to be determined later. In this paper, we are concerned with spherically symmetric piston problem for the relativistic Euler equations. A local shock front solution with the state equation p = a ² ρ, a is a constant and has been established by the Newton iteration. To overcome the difficulty caused by the free boundary, we introduce a coordinate transformation to fix it and employ the linear iteration scheme to establish a sequence of approximate solutions to the auxiliary problems by iteration. In each step, the value of the solution of the previous problem is taken as the data to determine the solution of the next problem. We obtain the existence of the original problem by establishing the convergence of these sequences. Meanwhile, we establish the convergence of the local solution as c → ∞ to the corresponding solution of the classical non-relativistic Euler equations.     

Existence and uniqueness of solutions for acoustic scattering over infinite obstacles

The paper considers the solution of the boundary value problem (BVP) consisting of the Helmholtz equation in the region D with a rigid boundary condition on ∂D and its reformulation as a boundary integral equation (BIE), over an infinite cylindrical surface of arbitrary smooth cross-section. A boundary integral equation, which models three-dimensional acoustic scattering from an infinite rigid cylinder, illustrates the application of the above results to prove existence of solution of the integral equation and the corresponding boundary value problem.     

Homogenization of monotone operators under conditions of coercitivity and growth of variable order

We obtain a homogenization procedure for the Dirichlet boundary-value problem for an elliptic equation of monotone type in the domain Ω ? ? ^d . The operator of the problem satisfies the conditions of coercitivity and of growth with variable order p _? (x) = p(x/?); furthermore, p(y) is periodic, measurable, and satisfies the estimate 1 < α ≤ p(y) ≤ β < ∞, while the parameter ? > 0 tends to zero. Here α and β are arbitrary constants. Taking Lavrent’ev’s phenomenon into account, we consider solutions of two types: H- and W-solutions. Each of the solution types calls for a distinct homogenization procedure. Its justification is carried out by using the corresponding version of the lemma on compensated compactness, which is proved in the paper.     

Bifurcation for a free boundary problem modeling the growth of multi-layer tumors

In this paper we study bifurcations for a free boundary problem modeling the growth of multi-layer tumors under the action of inhibitors. An important feature of this problem is that the surface tension effect of the free boundary is taken into account. By reducing this problem into an abstract bifurcation equation in a Banach space, overcoming some technical difficulties and finally using the Crandall–Rabinowitz bifurcation theorem, we prove that this problem has infinitely many branches of bifurcation solutions bifurcating from the flat solution.     

LOCAL CLASSICAL SOLUTION OF FREE BOUNDARY PROBLEM FOR A COUPLED SYSTEM

This paper considers a two-phase free boundary problem for coupled system including one parabolic equation and two elliptic equations. The problem comes from the discussion of a growth model of self-lnaintaining protocell in multidimensional case. The local classical solution of the problem with free boundary F : y = g(x,t) between two domains is being seeked. The local existence and uniqueness of the problem will be proved in multidimensional case.     

On the growth of solutions of certain second-order nonlinear differential equations

In this paper, we determine the growth of real-valued solutions of certain second-order algebraic differential equations. Our main result, together with a result of G. Valiron, shows that if y₀ is an entire function which has only real, nonnegative coefficients in its power series around the origin, and which is a solution of a quadratic second-order algebraic differential equation, then y₀ satisfies a growth estimate of the form,y₀(x) ? exp(exp x^c), where c is a constant, for all sufficiently large x. The determination of the growth of such solutions was an open problem since the Valiron-Wiman theory fails to provide any information on growth, if the equation possesses a solution of infinite order of growth.     

The time-periodic diffusive competition models with a free boundary and sign-changing growth rates

To understand the spreading of invasive and native species, in this paper we consider the diffusive competition models with a free boundary in the heterogeneous time-periodic environments, in which the variable intrinsic growth rates of these two species change signs and may be very negative in some large regions. We study the spreading–vanishing dichotomy, long-time dynamical behavior of solution, sharp criteria for spreading and vanishing, and estimates of the asymptotic spreading speed of the free boundary. Moreover, we establish the existence of positive solutions to a T-periodic boundary value problem of the diffusive competition system with sign-changing growth rates in the half line.     

On a time nonlocal problem for inhomogeneous evolution equations

Under consideration is some problem for inhomogeneous differential evolution equation in Banach space with an operator that generates a C ₀-continuous semigroup and a nonlocal integral condition in the sense of Stieltjes. In case the operator has continuous inhomogeneity in the graph norm. We give the necessary and sufficient conditions for existence of a generalized solution for the problem of whether the nonlocal data belong to the generator domain. Estimates on solution stability are given, and some conditions are obtained for existence of the classical solution of the nonlocal problem. All results are extended to a Sobolev-type linear equation, the equation in Banach space with a degenerate operator at the derivative. The time nonlocal problem for the partial differential equation, modeling a filtrating liquid free surface, illustrates the general statements.     

Explosion en temps fini de la solution d'un problème d'évolution de surface de film mince

In this Note, we are interested in the evolution of a surface of a crystal structure, constituted by an elastic substrate and a thin film. If the crystal is constrained, some morphological instabilities may appear. To study these instabilities, we made use of the model developped in Phys. Rev. B 47 (1993) 9760–9777. There, the map f of the free surface of the film satisfies a parabolic partial differential equation, depending on the elastic displacement of the substrate. For simplicity, the substrate is assumed to be linearly elastic and the structure to be infinite in one direction. Then, under some formal asymptotic assumptions, a formal expansion of the displacement can be determined after some appropriate scalings, allowing to derive a simplified parabolic nonlinear equation as in Lods et al. (Asymptotic Anal. 33 (2003) 67–91). We give here some results about the finite-time blow-up and the existence and uniqueness of the solution in an appropriate space. To validate the theoretical results, we also performed some numerical simulations using a pseudo-spectral method and also compute the initial-profile dependent critical value of the parameter θ involved in the nonlinear equation. To cite this article: M. Boutat et al., C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

On the second solution to a critical growth Robin problem

We investigate the existence of the second mountain-pass solution to a Robin problem, where the equation is at critical growth and depends on a positive parameter λ. More precisely, we determine existence and nonexistence regions for this type of solutions, depending both on λ and on the parameter in the boundary conditions.     

Locally one-dimensional scheme for a loaded heat equation with Robin boundary conditions

The third boundary value problem for a loaded heat equation in a p-dimensional parallelepiped is considered. An a priori estimate for the solution to a locally one-dimensional scheme is derived, and the convergence of the scheme is proved.     

Asymptotic expansion of solutions to the drift-diffusion equation with large initial data

We consider the large-time behavior of the solution to the initial value problem for the Nernst-Planck type drift-diffusion equation in whole spaces. In the Lp-framework, the global existence and the decay of the solution were shown. Moreover, the second-order asymptotic expansion of the solution as t→∞ was derived. We also deduce the higher-order asymptotic expansion of the solution. Especially, we discuss the contrast between the odd-dimensional case and the even-dimensional case.     

Necessary and sufficient conditions for a generalized solution to the initial-boundary value problem for the wave equation to belong to W p 1 with p ≥ 1

We establish necessary and sufficient conditions on the boundary function under which a generalized solution to the initial-boundary value problem for the wave equation with boundary conditions of the first kind belongs to W _p ¹ .