Asymptotic behaviour of large solutions of quasilinear elliptic problems

The paper deals with the large solutions of the problems $\triangle u=u^p$ and $\triangle u= e^u.$ They blow up at the boundary. It is well-known that the first term in their asymptotic behaviour near the boundary is independent of the geometry of the boundary. We determine the second term which depends on the mean curvature of the nearest point on the boundary. The computation is based on suitable upper and lower solutions and on estimates given in [4]. In the last section these estimates are used together with the P-function to establish the asymptotic behaviour of the gradients.     

Stefan-like problems with curvature

We study a two-phase free boundary problem in which the speed of the free boundary depends also on its curvature. It is assumed that the free boundary is Lipschitz and it is proved that the solution as well as the free boundary are classical.     

Local well‐posedness for volume‐preserving mean curvature and Willmore flows with line tension

We show the short‐time existence and uniqueness of solutions for the motion of an evolving hypersurface in contact with a solid container driven by the volume‐preserving mean curvature flow (MCF) taking line tension effects on the boundary into account. Difficulties arise due to dynamic boundary conditions and due to the contact angle and the non‐local nature of the resulting second order, nonlinear PDE. In addition, we prove the same result for the Willmore flow with line tension, which results in a nonlinear PDE of fourth order. For both flows we will use a curvilinear cordinate system due to Vogel to write the flows as graphs over a fixed reference hypersurface.     

Pseudo-differential operators with involutive double characteristics

Suppose G is a bounded C²domain in IR ⁿ, n ? 2 . We exam¬ine the regularity at the boundary of solutions to a class of quasi-linear elliptic equations having continuous boundary values ? . If ? has a modulus of continuity β , we give a modulus of continunity for the solution which depends on β and the generalized mean curvature of ?G . When the order of non-uniformity of the equation is between 0 and 1 , no curvature condition on ?G is needed.     

Asymptotics for a free boundary model in price formation

We study the asymptotics for a large time of solutions to a one-dimensional parabolic evolution equation with non-standard measure-valued right hand side, that involves derivatives of the solution computed at a free boundary point. The problem is a particular case of a mean-field free boundary model proposed by Lasry-Lions on price formation and dynamic equilibria.The main step in the proof is based on the fact that the free boundary disappears in the linearized problem, thus it can be treated as a perturbation through semigroup theory. This requires a delicate choice for the function spaces since higher regularity is needed near the free boundary. We show global existence for solutions with initial data in a small neighborhood of any equilibrium point, and exponential decay towards a stationary state. Moreover, the family of equilibria of the equation is stable, as follows from center manifold theory.     

Existence analysis for a simplified transient energy‐transport model for semiconductors

A simplified transient energy‐transport system for semiconductors subject to mixed Dirichlet–Neumann boundary conditions is analyzed. The model is formally derived from the non‐isothermal hydrodynamic equations in a particular vanishing momentum relaxation limit. It consists of a drift‐diffusion‐type equation for the electron density, involving temperature gradients, a nonlinear heat equation for the electron temperature, and the Poisson equation for the electric potential. The global‐in‐time existence of bounded weak solutions is proved. The proof is based on the Stampacchia truncation method and a careful use of the temperature equation. Under some regularity assumptions on the gradients of the variables, the uniqueness of solutions is shown. Finally, numerical simulations for a ballistic diode in one space dimension illustrate the behavior of the solutions. Copyright © 2012 John Wiley & Sons, Ltd.     

Smooth zero-contact-angle solutions to a thin-film equation around the steady state

In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the “complete wetting” regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide “minimal” conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary.     

Existence and regularity for a nonlinear boundary flow problem of population genetics

In this paper we consider a heat equation with nonlinear boundary condition occurring in population genetics, the selection–migration problem for alleles in a region, considering flow of genes throughout the boundary. Such a problem determines a gradient flow in a convenient phase space and then the dynamics for large times depends heavily on the knowledge of the equilibrium solutions. We address the questions of the existence of a nontrivial equilibrium solution and its regularity.     

Controlling Inflation: The Infinite Horizon Case

This paper studies the two-dimensional singular stochastic control problem over an infinite time-interval arising when the Central Bank tries to contain the inflation by acting on the nominal interest rate. It is shown that this problem admits a variational formulation which can be differentiated (in some sense) to lead to a stochastic differential game with stopping times between the conservative and the expansionist tendencies of the Bank. Substantial regularity of the free boundary associated to the differential game is obtained. Existence of an optimal policy is established when the regularity of the free boundary is strengthened slightly, and it is shown that the optimal process is a diffusion reflected at the boundary. Accepted 22 May 1998     

On a quasilinear parabolic–elliptic chemotaxis system with logistic source

This paper deals with a quasilinear parabolic–elliptic chemotaxis system with logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. For the case of positive diffusion function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, if the diffusion function is zero at some point, or a positive diffusion function and the logistic damping effect is rather mild, we proved that the weak solutions are global existence. Finally, it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source.     

Homogenization of a fluid problem with a free boundary

The stationary Stokes equations with a free boundary are studied in a perforated domain. The perforation consists of a periodic array of cylinders of size and distance O(ε). The free boundary is given as the graph of a function on a two‐dimensional perforated domain. We derive equations for the two‐scale limit of solutions. The limiting equation is a free boundary system. It involves a nonlinear eliptic operator corresponding to the nonlinear mean‐curvature expression in the original equations. Depending on the equation for the contact angle, the pressure is in general unbounded. © 2000 John Wiley & Sons, Inc.     

Regularity of flat free boundaries in two-phase problems for the p-Laplace operator

In this paper we continue the study in Lewis and Nyström (2010) [19], concerning the regularity of the free boundary in a general two-phase free boundary problem for the p-Laplace operator, by proving regularity of the free boundary assuming that the free boundary is close to a Lipschitz graph.     

A free boundary problem modeling the cell cycle and cell movement in multicellular tumor spheroids

This paper deals with a mathematical model describing the cell cycle dynamics and chemotactic driven cell movement in a multicellular tumor spheroid. Tumor cells consist of two types of cells: proliferating cells and quiescent cells, which have different chemotactic responses to an extracellular nutrient supply. The model is a free boundary problem for a nonlinear system of reaction-diffusion-advection equations, where the free boundary is the outer boundary of the spheroid. The free boundary condition is quite novel due to different velocity of two types of cells. The global existence and uniqueness of solutions to the model is proved. The proof is based on a fixed point argument, together with the Lp-theory for parabolic equations with the third boundary condition.     

Non‐linear initial boundary value problems of hyperbolic–parabolic type. A general investigation of admissible couplings between systems of higher order. Part 2: abstract quasilinear theory

This is the second part of an article that is devoted to the theory of non‐linear initial boundary value problems. We consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2 at hand, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (3.4) using the so‐called energy method. In the above sense the regularity assumptions (A6) and (A7) about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to non‐linear initial boundary value problems. In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

The global attractor of a competitor–competitor–mutualist reaction–diffusion system with time delays

The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor–mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction–diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state–state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation,II

This paper is a continuation of our recent paper 8 . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L² scale with additional L¹ regularity for the data. In order to deal with this L² regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.     

Non‐linear initial boundary value problemsof hyperbolic–parabolic type. A general investigationof admissible couplings between systems of higher order. Part 1: linear theory

In this article (which is divided in three parts) we investigate the non‐linear initial boundary value problems (1.2) and (1.3). In both cases we consider coupled systems where each system is of higher order and of hyperbolic or parabolic type. Our goal is to characterize systematically all admissible couplings between systems of higher order and different type. By an admissible coupling we mean a condition that guarantees the existence, uniqueness and regularity of solutions to the respective initial boundary value problem. In part 1 at hand, we develop the underlying theory of linear hyperbolic and parabolic initial boundary value problems. Testing the PDEs with suitable functions we obtain a priori estimates for the respective solutions. In particular, we make use of the regularity theory for linear elliptic boundary value problems that was previously developed by the author. In part 2, we prove the local in time existence, uniqueness and regularity of solutions to the quasilinear initial boundary value problem (1.2) using the so‐called energy method. In the above sense, the regularity assumptions about the coefficients and right‐hand sides define the admissible couplings. In part 3, we extend the results of part 2 to the non‐linear initial boundary value problem (1.3). In particular, the assumptions about the respective parameters correspond to the previous regularity assumptions and hence define the admissible couplings now. Moreover, we exploit the assumptions about the respective parameters for the case of two coupled systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Further regularity results for almost cubic nonlinear Schrödinger equation

We present three results related with the regularity of solutions of the almost cubic NLS. In the first one, following Ozawa’s idea, we establish mass and energy conservation for the solutions without regularizing the initial datum. Our second result is the H^s well-posedness for the Cauchy problem for 0<s<1. Finally, we show that the same solutions are also in some Bourgain spaces for possibly a smaller time interval. In all of our results, the non-local nonlinear term in the equation is shown to act like a cubic nonlinearity on the appropriate Sobolev and Besov spaces.