Page 221 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 221
TOPOLOGICAL METHODS IN DIFFERENTIAL EQUATIONS (MS-13)
References
[1] A. Bressan, G. Guerra, and W. Shen. Vanishing viscosity solutions for conservation laws
with regulated flux. Journal of Differential Equations, 266(1):312 – 351, 2019.
[2] H. Brézis and A. Pazy. Convergence and approximation of semigroups of nonlinear oper-
ators in Banach spaces. J. Functional Analysis, 9:63–74, 1972.
[3] M. G. Crandall and T. M. Liggett. Generation of semi-groups of nonlinear transformations
on general Banach spaces. Amer. J. Math., 93:265–298, 1971.
[4] G. Guerra and W. Shen. Vanishing Viscosity and Backward Euler Approximations for
Conservation Laws with Discontinuous Flux. SIAM J. Math. Anal., 51(4):3112–3144,
2019.
Multiple solutions for the fractional p-Laplacian via degree theory
Antonio Iannizzotto, antonio.iannizzotto@unica.it
Università di Cagliari, Italy
Coauthor: Silvia Frassu
We study a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degener-
ate fractional p-Laplacian with a jumping reaction, crossing the first eigenvalue. By means of
Browder’s degree for (S)+-mappings and fractional spectral theory (in particular, monotonic-
ity of weighted eigenvalues with respect to the weight), we prove existence of two nontrivial
solutions.
Compactness properties of operator of translation along trajectories in
evolution equations
Władysław Klinikowski, wklin@mat.umk.pl
Nicolaus Copernicus University in Torun´, Poland
Our aim is to present some results about compactness properties of operator of translation along
trajectories (which is also known as Poincaré operator) which is associated with some evolution
equation. Fixed points of this operator are periodic solutions of connected evolution equation.
In order to apply some kind of topological degree we study just compactness properties of oper-
ator of translation. We consider two types of evolution equations: first is linked with parabolic
problems and second with hyperbolic problems. In case of parabolic equations we discuss re-
sults which come from A. C´ wiszewski and R. Łukasiak ([1], [2]). Next we present approach
(but not yet with specific results) to hyperbolic problems which is a part of collaboration with
A. C´ wiszewski. Our method uses so-called „tail estimates" (which were firstly introduced by
B. Wang in [4]) and is based on work of D. Fall and Y. You (see [3], but note that they looked
for global attractors).
References
[1] A. C´ wiszewski, R. Łukasiak, Forced periodic solutions for nonresonant parabolic equa-
tions on RN , https://arxiv.org/pdf/1404.0256v2.pdf
219
References
[1] A. Bressan, G. Guerra, and W. Shen. Vanishing viscosity solutions for conservation laws
with regulated flux. Journal of Differential Equations, 266(1):312 – 351, 2019.
[2] H. Brézis and A. Pazy. Convergence and approximation of semigroups of nonlinear oper-
ators in Banach spaces. J. Functional Analysis, 9:63–74, 1972.
[3] M. G. Crandall and T. M. Liggett. Generation of semi-groups of nonlinear transformations
on general Banach spaces. Amer. J. Math., 93:265–298, 1971.
[4] G. Guerra and W. Shen. Vanishing Viscosity and Backward Euler Approximations for
Conservation Laws with Discontinuous Flux. SIAM J. Math. Anal., 51(4):3112–3144,
2019.
Multiple solutions for the fractional p-Laplacian via degree theory
Antonio Iannizzotto, antonio.iannizzotto@unica.it
Università di Cagliari, Italy
Coauthor: Silvia Frassu
We study a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degener-
ate fractional p-Laplacian with a jumping reaction, crossing the first eigenvalue. By means of
Browder’s degree for (S)+-mappings and fractional spectral theory (in particular, monotonic-
ity of weighted eigenvalues with respect to the weight), we prove existence of two nontrivial
solutions.
Compactness properties of operator of translation along trajectories in
evolution equations
Władysław Klinikowski, wklin@mat.umk.pl
Nicolaus Copernicus University in Torun´, Poland
Our aim is to present some results about compactness properties of operator of translation along
trajectories (which is also known as Poincaré operator) which is associated with some evolution
equation. Fixed points of this operator are periodic solutions of connected evolution equation.
In order to apply some kind of topological degree we study just compactness properties of oper-
ator of translation. We consider two types of evolution equations: first is linked with parabolic
problems and second with hyperbolic problems. In case of parabolic equations we discuss re-
sults which come from A. C´ wiszewski and R. Łukasiak ([1], [2]). Next we present approach
(but not yet with specific results) to hyperbolic problems which is a part of collaboration with
A. C´ wiszewski. Our method uses so-called „tail estimates" (which were firstly introduced by
B. Wang in [4]) and is based on work of D. Fall and Y. You (see [3], but note that they looked
for global attractors).
References
[1] A. C´ wiszewski, R. Łukasiak, Forced periodic solutions for nonresonant parabolic equa-
tions on RN , https://arxiv.org/pdf/1404.0256v2.pdf
219
