Page 222 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 222
TOPOLOGICAL METHODS IN DIFFERENTIAL EQUATIONS (MS-13)
[2] A. C´ wiszewski, R. Łukasiak, A Landesman-Lazer type for periodic parabolic problems
on RN at resonance, Nonlinear An. TMA 125 (2015), 608-625
[3] D. Fall, Y. You, Global attractors for the damped nonlinear wave equation in unbounded
domains, https://arxiv.org/pdf/1801.00104.pdf
[4] B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D 128
(1999), 41-52
Flow invariance of closed sets
Wojciech Kryszewski, wojciech.kryszewski@o.lodz.pl
Lodz Technical University, Poland
When studying the existence of solutions to systems of parabolic equations of the form ut +
Lu = f (x, u), u ∈ RM , x ∈ Ω ⊂ RN , subject to boundary conditions, where L is an elliptic
(vector valued) differential operator and f : Ω × RM → RM is a continuous map, such that
u(x) ∈ C for a.a. x ∈ Ω, where C ⊂ RM is a given closed set of state constraints, an important
hypotheses concern the so-called resolvent invariance of K, i.e. (I + λA)−1(K) ⊂ K for
sufficiently small λ > 0, where A is the sectorial operator corresponding to L, K := {u ∈
L2(Ω, RM ) | u(x) ∈ C, for a.a. x ∈ Ω} (being equivalent to the invariance of K with respect
to the semigroup generated by −A) and the so-called tangency of the nonlinear perturbation f .
We will discuss the sufficient and necessary conditions for the invariance stated in terms of the
coefficients of the operator L as well as in the language of Dirichlet bilinear form associated to
L. This topic is strictly related to the study of the viability and invariance questions for partial
differential equations.
This is a joint work with Jakub Siemianowski, Grzegorz Gabor and Aleksander C´ wiszewski.
Positive solutions for systems of Riemann-Liouville fractional differential
equations with coupled nonlocal boundary conditions
Rodica Luca Tudorache, rlucatudor@yahoo.com
“Gheorghe Asachi” Technical University of Iasi, Romania
Coauthor: Alexandru Tudorache
We investigate the existence and nonexistence of positive solutions for a system of Riemann-
Liouville fractional differential equations with p-Laplacian operators, nonnegative nonlinear-
ities and positive parameters, subject to coupled nonlocal boundary conditions which contain
various fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main existence
results we use the Guo-Krasnosel’skii fixed point theorem.
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[2] A. C´ wiszewski, R. Łukasiak, A Landesman-Lazer type for periodic parabolic problems
on RN at resonance, Nonlinear An. TMA 125 (2015), 608-625
[3] D. Fall, Y. You, Global attractors for the damped nonlinear wave equation in unbounded
domains, https://arxiv.org/pdf/1801.00104.pdf
[4] B. Wang, Attractors for reaction-diffusion equations in unbounded domains, Phys. D 128
(1999), 41-52
Flow invariance of closed sets
Wojciech Kryszewski, wojciech.kryszewski@o.lodz.pl
Lodz Technical University, Poland
When studying the existence of solutions to systems of parabolic equations of the form ut +
Lu = f (x, u), u ∈ RM , x ∈ Ω ⊂ RN , subject to boundary conditions, where L is an elliptic
(vector valued) differential operator and f : Ω × RM → RM is a continuous map, such that
u(x) ∈ C for a.a. x ∈ Ω, where C ⊂ RM is a given closed set of state constraints, an important
hypotheses concern the so-called resolvent invariance of K, i.e. (I + λA)−1(K) ⊂ K for
sufficiently small λ > 0, where A is the sectorial operator corresponding to L, K := {u ∈
L2(Ω, RM ) | u(x) ∈ C, for a.a. x ∈ Ω} (being equivalent to the invariance of K with respect
to the semigroup generated by −A) and the so-called tangency of the nonlinear perturbation f .
We will discuss the sufficient and necessary conditions for the invariance stated in terms of the
coefficients of the operator L as well as in the language of Dirichlet bilinear form associated to
L. This topic is strictly related to the study of the viability and invariance questions for partial
differential equations.
This is a joint work with Jakub Siemianowski, Grzegorz Gabor and Aleksander C´ wiszewski.
Positive solutions for systems of Riemann-Liouville fractional differential
equations with coupled nonlocal boundary conditions
Rodica Luca Tudorache, rlucatudor@yahoo.com
“Gheorghe Asachi” Technical University of Iasi, Romania
Coauthor: Alexandru Tudorache
We investigate the existence and nonexistence of positive solutions for a system of Riemann-
Liouville fractional differential equations with p-Laplacian operators, nonnegative nonlinear-
ities and positive parameters, subject to coupled nonlocal boundary conditions which contain
various fractional derivatives and Riemann-Stieltjes integrals. In the proof of the main existence
results we use the Guo-Krasnosel’skii fixed point theorem.
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