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1
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The principal spectrum for linear nonautonomous parabolic PDEs of second order: Space-independent case

100%
Mierczyński J.
Colloquium Mathematicum
|
2001
|
tom 90
|
nr 1
95-100
EN
We show that the study of the principal spectrum of a linear nonautonomous parabolic PDE of second order $u_t = Δu + a(t)u$ on a bounded domain, with the Dirichlet or Neumann boundary conditions, reduces to the investigation of the spectrum of the linear nonautonomous ODE v̇ = a(t)v.
2
Content available remote

Uniqueness for a class of cooperative systems of ordinary differential equations

100%
Mierczyński J.
Colloquium Mathematicum
|
1994
|
tom 67
|
nr 1
21-23
3
Content available remote

Three-dimensional cooperative irreducible systems with a first integral

100%
Mierczyński J.
Colloquium Mathematicum
|
1998
|
tom 76
|
nr 2
181-190
4
Content available remote

Estimates for Principal Lyapunov Exponents: A Survey

100%
Mierczyński J.
Nonautonomous Dynamical Systems
|
2014
|
tom 1
|
nr 1
EN
This is a survey of known results on estimating the principal Lyapunov exponent of a timedependent linear differential equation possessing some monotonicity properties. Equations considered are mainly strongly cooperative systems of ordinary differential equations and parabolic partial differential equations of second order. The estimates are given either in terms of the principal (dominant) eigenvalue of some derived time-independent equation or in terms of the parameters of the equation itself. Extensions to other differential equations are considered. Possible directions of further research are hinted.
5
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Smoothness of unordered curves in two-dimensional strongly competitive systems

100%
Mierczyński J.
Applicationes Mathematicae
|
1998-1999
|
tom 25
|
nr 4
449-455
EN
It is known that in two-dimensional systems of ODEs of the form $\dotx^i=x^if^i(x)$ with ${\partial f^i}/{\partial x^j} < 0$ (strongly competitive systems), boundaries of the basins of repulsion of equilibria consist of invariant Lipschitz curves, unordered with respect to the coordinatewise (partial) order. We prove that such curves are in fact of class $C^1$.
6
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A class of strongly cooperative systems without compactness

100%
Mierczyński J.
Colloquium Mathematicum
|
1991
|
tom 62
|
nr 1
43-47
7
Content available remote

Invariant manifolds for one-dimensional parabolic partial differential equations of second order

100%
Mierczyński J.
Colloquium Mathematicum
|
1998
|
tom 75
|
nr 2
285-314
8
Content available remote

On peaks in carrying simplices

100%
Mierczyński J.
Colloquium Mathematicum
|
1999
|
tom 81
|
nr 2
285-292
EN
A necessary and sufficient condition is given for the carrying simplex of a dissipative totally competitive system of three ordinary differential equations to have a peak singularity at an axial equilibrium. For systems of Lotka-Volterra type that result translates into a simple condition on the coefficients.
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