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Aleksander Ćwiszewski

Publications and source records attributed to Aleksander Ćwiszewski.

9 recordsLinked to original sources
arXiv2026

Standing waves for Schrodinger equations with Kato class potentials and $L^\infty$-bounded nonlinearities

We establish the existence of standing waves for a nonlinear Schrodinger equation with potentials belonging to the Kato class and an $L^\infty$-bounded nonlinearity, whose Lipschitz constant is smaller than the distance from zero to the essential spectrum of the linear part. We consider both the nonresonant and resonant cases. Our approach is based on the Conley index theory applied to study invariant sets of the associated parabolic semiflow. Using properties of the Schrodinger semigroup with Kato-class potentials, which follow from its Feynman-Kac representation, we derive a priori estimates for bounded solutions in the $L^\infty$ and $L^2$ norms, as well as regularity bounds in Sobolev spaces. As a consequence, we obtain conditions ensuring the existence of connecting orbits between stationary solutions, which in turn yield the existence of nontrivial standing waves.

math.AP↗
arXiv2024

Effects of Marine Reserve Creation in Single Species and Prey-Predator Models

Single species fisheries and prey-predator models with marine protected areas (MPA) are studied. The single species case is considered when the fishing effort is around the species extinction threshold and the influence of implementing MPA on catch quantity are studied. In the prey-predator fishery model, the situation with the fishing effort close to the predator extinction value is considered and the effects of implementing MPA are discussed with the focus on MPA sizes assuring an acceptable catch level (i.e. food security) and the sustainability of both the predator and prey populations.

q-bio.PE↗
arXiv2023

Standing Waves for Schrödinger Equations with Kato-Rellich potentials

We show the existence of standing waves for the nonlinear Schrödinger equation with Kato-Rellich type potential. We consider both resonant with the nonlinearity satisfying one of Landesman-Lazer type or sign conditions and non-resonant case where the linearization at infinity has zero kernel. The approach relies on the geometric and topological analysis of the parabolic semiflow associated to the involved elliptic problem. Tail estimates techniques and spectral theory of unbounded linear operators are used to exploit subtle compactness properties necessary for use of the Conley index theory due to Rybakowski.

math.AP↗
arXiv2020

Invariance and Strict Invariance for Nonlinear Evolution Problems with Applications

Sufficient conditions for the invariance of evolution problems governed by perturbations of (possibly nonlinear) $m$-accretive operators are provided. The conditions for the invariance with respect to sublevel sets of a constraint functional are expressed in terms of the Dini derivative of that functional, outside the considered sublevel set in directions determined by the governing $m$-accretive operator. An approach for non-reflexive Banach spaces is developed and some result improving a recent paper [P. Cannarsa, G. Da Prato, H. Frankowska, Invariance of quasi-dissipative systems in Banach spaces. J. Math. Anal. App. 457 (2018), 1173-1187] is presented. Applications to nonlinear obstacle problems and age-structured population models are presented in spaces of continuous functions where advantages of that approach are taken. Moreover, some new abstract criteria for the so-called strict invariance are derived and their direct applications to problems with barriers are shown.

math.AP↗
arXiv2020

Period Estimates for Autonomous Evolution Equations with Lipschitz Nonlinearities

We derive an estimate for the minimal period of autonomous strongly damped hyperbolic problems. Our result corresponds to the works by Yorke, Busenberg et al. for ordinary differential equations as well as Robinson and Vidal-Lopez for parabolic problems. A general approach is developed for treating both hyperbolic and parabolic problems. An example of application to a class of beam equations is provided.

math.AP↗
arXiv2018

Bifurcation from infinity for elliptic problems on $R^N$

In the paper the asymptotic bifurcation of solutions to a parameterized stationary semilinear Schrödinger equation involving a potential of the Kato-Rellich type is studied. It is shown that the bifurcation from infinity occurs if the parameter is an eigenvalue of the hamiltonian lying below the asymptotic bottom of the bounded part of the potential. Thus the bifurcating solution are related to bound states of the corresponding Schrödinger equation. The argument relies on the use of the (generalized) Conley index due to Rybakowski and resonance assumptions of the Landesman-Lazer or sign-condition type.

math.AP↗
arXiv2015

Periodic solutions for nonlinear hyperbolic evolution systems

We shall deal with the periodic problem for nonlinear perturbations of abstract hyperbolic evolution equations generating an evolution system of contractions. We prove an averaging principle for the translation along trajectories operator associated to the nonlinear evolution system, expressed in terms of the topological degree. The abstract results shall be applied to the damped hyperbolic partial differential equation.

math.DS↗
arXiv2015

Krasnosel'skii type formula and translation along trajectories method for evolution equations

The Krasnosel'skii type degree formula for the equation $\dot u = - Au + F(u)$ where $A:D(A)\to E$ is a linear operator on a separable Banach space $E$ such that $-A$ is a generator of a $C_0$ semigroup of bounsed linear operators of $E$ and $F:E\to E$ is a locally Lipschitz $k$-set contraction, is provided. Precisely, it is shown that if $V$ is an open bounded subset of $E$ such that $0\not\in (-A+F)(\partial V \cap D(A))$, then the topological degree of $-A+F$ with respect to $V$ is equal to the fixed point index of the operator of translation along trajectories for sufficiently small positive time. The obtained degree formula is crucial for the method of translation along trajctories. It is applied to the nonautonomous periodic problem and an average principle is derived. As an application a first order system of partial differential equations is considered.

math.DS↗