Milan Kučera - Academia.edu (original) (raw)

Papers by Milan Kučera

We consider a classical reaction-diffusion system undergoing Tur-ing instability and augment it b... more We consider a classical reaction-diffusion system undergoing Tur-ing instability and augment it by an additional unilateral source term. We investigate its influence on the Turing instability and on the character of resulting patterns. The nonsmooth positively homogeneous unilateral term τ v − has favourable properties in spite of the fact that the standard linear stability analysis cannot be performed. We illustrate the importance of the nonsmooth-ness by a numerical case study, which shows that the Turing instability can considerably change if we replace this term by its arbitrarily precise smooth approximation. However, the nonsmooth unilateral term and all its approximations yield qualitatively same patterns although not necessarily developing from arbitrarily small disturbances of the spatially homogeneous steady state. Further, we show that inserting the unilateral source into a classical system breaks the approximate symmetry and regularity of the classical patterns and yield...

arXiv (Cornell University), Jul 23, 2018

A reaction-diffusion system exhibiting Turing's diffusion driven instability is considered. The e... more A reaction-diffusion system exhibiting Turing's diffusion driven instability is considered. The equation for an activator is supplemented by unilateral terms of the type s − (x)u − , s + (x)u + describing sources and sinks active only if the concentration decreases below and increases above, respectively, the value of the basic spatially constant solution which is shifted to zero. We show that the domain of diffusion parameters in which spatially non-homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms. It is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources. The case of mixed (Dirichlet-Neumann) boundary conditions as well as that of pure Neumann conditions is described.

Czechoslovak Mathematical Journal, 1982

Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents ... more Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.

Journal of Mathematical Analysis and Applications, 2008

We consider a reaction-diffusion system with implicit unilateral boundary conditions introduced b... more We consider a reaction-diffusion system with implicit unilateral boundary conditions introduced by U. Mosco. We show that global continua of stationary spatially nonhomogeneous solutions bifurcate in the domain of parameters where bifurcation in the case of classical boundary conditions is excluded. The problem is formulated as a quasivariational inequality and the proof is based on the Leray-Schauder degree.

Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digi... more Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use. This paper has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://project.dml.cz COMMENT AT IONES MATHEMATICAE TJNIVERSITATIS CAROLINAE 18,1 (1977) A NEW METHOD FOR THE OBTAINING OF EIGENVALUES OF VARIATIONAL INEQUALITIES OF THE SPECIAL TXPE

Differential and Integral Equations

The existence of smooth families of solutions bifurcating from the trivial solution for a two-par... more The existence of smooth families of solutions bifurcating from the trivial solution for a two-parameter bifurcation problem for a class of variational inequalities is proved. As an example, a model of an elastic beam compressed by a force lambda\lambdalambda and supported by a unilateral connected fixed obstacle at the height hhh is studied. In the language of this example, we show that nontrivial solutions touching the obstacle on connected intervals bifurcate from the trivial solution and form smooth families parametrized by lambda\lambdalambda and hhh. In particular, the corresponding contact intervals depend smoothly on lambda\lambdalambda and hhh.

Nonlinear Analysis: Theory, Methods & Applications, 2015

The equation ∆u + λu + g(λ, u)u = 0 is considered in a bounded domain in R 2 with a Signorini con... more The equation ∆u + λu + g(λ, u)u = 0 is considered in a bounded domain in R 2 with a Signorini condition on a straight part of the boundary and with mixed boundary conditions on the rest of the boundary. It is assumed that g(λ, 0) = 0 for λ ∈ R, λ is a bifurcation parameter. A given eigenvalue of the linearized equation with the same boundary conditions is considered. A smooth local bifurcation branch of non-trivial solutions emanating at λ 0 from trivial solutions is studied. We show that to know a direction of the bifurcating branch it is sufficient to determine the sign of a simple expression envolving the corresponding eigenfunction u 0. In the case when λ 0 is the first eigenvalue and the branch goes to the right, we show that the bifurcating solutions are asymptotically stable in W 1,2-norm. The stability of the trivial solution is also studied and an exchange of stability is obtained.

Navier—Stokes Equations and Related Nonlinear Problems, 1995

Let Ω be a bounded domain in R N with a lipschitzian boundary 2202;Ω, let Γ D , Γ N , Γ U be open... more Let Ω be a bounded domain in R N with a lipschitzian boundary 2202;Ω, let Γ D , Γ N , Γ U be open (in ∂Ω) disjoint subsets of ∂Ω, meas(∂Ω \ Γ D ⋃ Γ N ⋃Γ U ) = 0, f, g real differentiable functions on R ū, v, positive constants such that f(ū, v) = g(ū, v) = 0.

Applications of Mathematics, 2016

Nonlinear Analysis, 1978

Publisher Summary This chapter discusses the eigenvalue problem for variational inequalities and ... more Publisher Summary This chapter discusses the eigenvalue problem for variational inequalities and a new version of the Ljusternik–Schnirelmann theory. It seems reasonable to formulate for variational inequalities problems analogous to those studied in the usual, linear or nonlinear, theory of operators. Furthermore, it might be of interest to examine the structure of the set of all eigenvalues of a variational inequality. The classical Ljusternik–Schnirelmann theory is based on the notion of the category of a set defined for compact subsets of the symmetric manifold Mr. In the case of variational inequalities, it becomes necessary to investigate only subsets of a nonsymmetric set Mr ∩ K, which represents a serious difficulty.

Physical review. E, 2017

We consider a reaction-diffusion system undergoing Turing instability and augment it by an additi... more We consider a reaction-diffusion system undergoing Turing instability and augment it by an additional unilateral source term. We investigate its influence on the Turing instability and on the character of resulting patterns. The nonsmooth positively homogeneous unilateral term τv^{-} has favorable properties, but the standard linear stability analysis cannot be performed. We illustrate the importance of the nonsmoothness by a numerical case study, which shows that the Turing instability can considerably change if we replace this term by its arbitrarily precise smooth approximation. However, the nonsmooth unilateral term and all its approximations yield qualitatively similar patterns although not necessarily developing from small disturbances of the spatially homogeneous steady state. Further, we show that the unilateral source breaks the approximate symmetry and regularity of the classical patterns and yields asymmetric and irregular patterns. Moreover, a given system with a unilate...

A bifurcation problem for variational inequalities U(t) ∈ K, (U̇(t)−BλU(t)−G(λ, U(t)), Z − U(t)) ... more A bifurcation problem for variational inequalities U(t) ∈ K, (U̇(t)−BλU(t)−G(λ, U(t)), Z − U(t)) 0 for all Z ∈ K, a.a. t 0 is studied, where K is a closed convex cone in κ , κ 3, Bλ is a κ×κ matrix, G is a small perturbation, λ a real parameter. The main goal of the paper is to simplify the assumptions of the abstract results concerning the existence of a bifurcation of periodic solutions developed in the previous paper and to give examples in more than three dimensional case.

Czechoslovak Mathematical Journal

Czechoslovak Math. J, 1986

Collection of Czechoslovak Chemical Communications, 1995

Liquid phase hydration of cyclohexene over faujasite, mordenite and ZSM-5 catalysts at 10 MPa and... more Liquid phase hydration of cyclohexene over faujasite, mordenite and ZSM-5 catalysts at 10 MPa and temperatures 150 - 210 °C in a batch system was studied. Influence of the type and Si/Al ratio of the zeolite on the conversion of cyclohexene and the selectivity to cyclohexanol formation was determined. Performance of the catalyst was also investigated through adsorption measurements of water and cyclohexene. Mordenite with the Si/Al ratio 70 provided the best results. The selectivity to cyclohexanol was 99% at the conversion of cyclohexene 27% at 200 °C after 4 h. All the catalysts were regenerated by calcination at 450 °C.