Uniqueness for an elliptic-parabolic problem with Neumann boundary condition

We consider the problem in a smooth boundary domain , as well as the corresponding evolution equation . For the stationary equation we show existence results, then we adapt the techniques of doubling of variables to the case of the homogeneous Neumann boundary conditions and obtain the appropriate L ¹ -contraction principle and uniqueness. Subsequently, we are able to apply the nonlinear semigroup theory and prove the L ¹ -contraction principle for the associated evolution equation.     

Linear and nonlinear heat equations in $L^q_\delta$ spaces and universal bounds for global solutions

We develop a theory of both linear and nonlinear heat equations in the weighted Lebesgue spaces , where is the distance to the boundary. In particular, we prove an optimal estimate for the heat semigroup, and we establish sharp results on local existence-uniqueness and local nonexistence of solutions for semilinear heat equations with initial values in those spaces. This theory enables us to obtain new types of results concerning positive global solutions of superlinear parabolic problems. Namely, under certain assumptions, we prove that any global solution is uniformly bounded for by a universal constant, independent of the initial data. In all previous results, the bounds for global solutions were depending on the initial data. Received March 15, 2000 / Accepted October 18, 2000 / Published online February 5, 2001     

Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy

In this paper we investigate the asymptotic behavior of the nonlinear Cahn–Hilliard equation with a logarithmic free energy and similar singular free energies. We prove an existence and uniqueness result with the help of monotone operator methods, which differs from the known proofs based on approximation by smooth potentials. Moreover, we apply the Lojasiewicz–Simon inequality to show that each solution converges to a steady state as time tends to infinity.     

Sharp estimates for approximations to a nonlinear backward heat equation

A nonlinear backward heat problem for an infinite strip is considered. The problem is ill-posed in the sense that the solution (if it exists) does not depend continuously on the data. In this paper, we use the Fourier regularization method to solve the problem. Some sharp estimates of the error between the exact solution and its regularization approximation are given.     

Nonlinear mean ergodic theorems for nonexpansive semigroups in Banach spaces

We prove two nonlinear ergodic theorems for noncommutative semigroups of nonexpansive mappings in Banach spaces. Using these results, we obtain some nonlinear ergodic theorems for discrete and one-parameter semigroups of nonexpansive mappings. Dedicated to Professors Albrecht Dold and Ed Fadell     

Stability estimates for parabolic problems with Wentzell boundary conditions

Of concern is the uniformly parabolic problem

     

Ginzburg-Landau equation with magnetic effect in a thin domain

We study the Ginzburg-Landau equation with magnetic effect in a thin domain in , where the thickness of the domain is controlled by a parameter . This equation is an Euler equation of a free energy functional and it has trivial solutions that are minimizers of the functional. In this article we look for a nontrivial stable solution to the equation, that is, a local minimizer of the energy functional. To prove the existence of such a stable solution in , we consider a reduced problem as and a nondegenerate stable solution to the reduced equation. Applying the standard variational argument, we show that there exists a stable solution in near the solution to the reduced equation if is sufficiently small. We also present a specific example of a domain which allows a stable vortex solution, that is, a stable solution with zeros. Received: 11 May 2001 / Accepted: 11 July 2001 /Published online: 19 October 2001     

Universal bounds for global solutions of a diffusion equation with a nonlocal reaction term

We consider the nonlinear heat equation with nonlocal reaction term in space , in smoothly bounded domains. We prove the existence of a universal bound for all nonnegative global solutions of this equation. Moreover, in contrast with similar recent results for equations with local reaction terms, this is shown to hold for all p>1. As an interesting by-product of our proof, we derive for this equation a smoothing effect under weaker assumptions than for corresponding problem with local reaction.     

On second-order periodic elliptic operators in divergence form

We consider second-order, strongly elliptic, operators with complex coefficients in divergence form on . We assume that the coefficients are all periodic with a common period. If the coefficients are continuous we derive Gaussian bounds, with the correct small and large time asymptotic behaviour, on the heat kernel and all its H?lder derivatives. Moreover, we show that the first-order Riesz transforms are bounded on the -spaces with . Secondly if the coefficients are H?lder continuous we prove that the first-order derivatives of the kernel satisfy good Gaussian bounds. Then we establish that the second-order derivatives exist and satisfy good bounds if, and only if, the coefficients are divergence-free or if, and only if, the second-order Riesz transforms are bounded. Finally if the third-order derivatives exist with good bounds then the coefficients must be constant. Received in final form: 28 February 2000 / Published online: 17 May 2001     

Viscosity methods for common solutions of equilibrium and variational inequality problems via multi-step iterative algorithms and common fixed points

In this paper, we present a new multi-step iterative method. We prove the strong convergence of the method to a common fixed point of a finite number of nonexpansive mappings that also solves a suitable equilibrium problem.     

A global Lipschitz continuity result for a domain-dependent Neumann eigenvalue problem for the Laplace operator

Let Ω be an open connected subset of Rⁿ of finite measure for which the Poincaré-Wirtinger inequality holds. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset φ(Ω) of Rⁿ, where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then, we show Lipschitz-type inequalities for the reciprocals of the eigenvalues delivered by the Rayleigh quotient. Then, we further assume that the imbedding of the Sobolev space W^1,2(Ω) into the space L²(Ω) is compact, and we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.     

A nonlinear parabolic equation backward in time: Regularization with new error estimates

Consider a nonlinear backward parabolic problem in the form

     

Maximal dissipativity of Kolmogorov operators with Cahn-Hilliard type drift term

We prove the existence of invariant measures μ for Kolmogorov operators LF associated with semilinear stochastic partial differential equations with Cahn-Hilliard type drift term. Based on gradient estimates on the pseudo-resolvent associated with LF and a priori estimates for the moments of μ we prove maximal dissipativity of LF in the space L¹(μ).     

Exponential convergence for a periodically driven semilinear heat equation

We consider a semilinear heat equation in one space dimension, with a periodic source at the origin. We study the solution, which describes the equilibrium of this system and we prove that, as the space variable tends to infinity, the solution becomes, exponentially fast, asymptotic to a steady state. The key to the proof of this result is a Harnack type inequality, which we obtain using probabilistic ideas.     

Stability of parabolic problems with nonlinear Wentzell boundary conditions

Of concern is the nonlinear uniformly parabolic problem

     

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 theorems for monotone operators in reflexive Banach spaces

In this paper, we obtain an existence theorem for single-valued monotone operators in a reflexive Banach space. Using this result, we prove a fixed point theorem for nonexpansive mappings in a Hilbert space and an existence theorem for maximal monotone operators in a Banach space. Received: 3 July 2006 Revised: 15 January 2007     

Enclosure theorems for generalized mean curvature flows

We define a generalized notion of mean curvature for regular hypersurfaces in . This enables us to introduce a new class of geometric curvature flows for which we prove enclosure theorems, using methods of Dierkes [D] and Hildebrandt [H]. In particular, we obtain “neck-pinching” results that generalize previous observations by Ecker [E] concerning the classical mean curvature flow. Received: 8 October 2001 / Accepted: 1 March 2002 / Published online: 23 May 2002     

Strong convergence theorems for a family of quasi-asymptotic pseudo-contractions in Hilbert spaces

In the present paper, we propose two kinds of new algorithms for a family of quasi-asymptotic pseudo-contractions in real Hilbert spaces. By using the proposed algorithms, we prove several strong convergence theorems for a family of quasi-asymptotic pseudo-contractions. The results of this paper are interesting extensions of those known results.