Posters Abstracts


Eleonora Amoroso (University of Messina)
Parametric Robin problem involving the variable exponent double phase operator

Abstract In this poster we present some existence results for a parametric double phase problem with variable exponents and Robin boundary conditions with critical growth. In particular, exploiting a recent result on a new equivalent norm in the Musielak-Orlicz Sobolev space, we obtain the existence of two nontrivial weak solutions through critical point theory. This is a joint work with Valeria Morabito.

Francesco Bozzola (University of Parma)
The role of topology and capacity in some bounds for principal frequencies

Abstract The inradius of an open set is a geometric quantity naturally linked to its sharp Poincaré-Sobolev embedding constants. In general, providing two-sided estimates for these quantities in terms of the inradius is not always possible, unless some topological or capacitary assumptions comes into play. More precisely, for planar open sets with assigned topology we obtained an extension of a result due to Osserman and Taylor in the 70’s. As a byproduct, for these classes of open sets, we derived a reverse Cheeger’s inequality. We cover the superconformal case in any dimension and discuss the asymptotic optimality of the constants involved in the estimates, as well. Based on joint work with Lorenzo Brasco.

Giulia Duricchi (University of Firenze)
On the solutions for second order differential inclusions

Abstract The poster will be about new results, obtained in recent joint works with Tiziana Cardinali ([1], [2], [3], [4]), on the existence of solutions for problems driven by the following second order differential inclusions \begin{equation*} (p(t)x'(t))’ \in G(t)x(t)+F(t,x(t),x'(t)),\ t \in J \end{equation*} in which different conditions are present as initial conditions ([3]), periodic conditions with fixed initial velocity ([2]), nonlocal conditions ([1], [2]) and impulsive conditions ([4]), where the maps \(p\), \(G\) and \(F\) have different forms:
  • – \(p\equiv 1\), \(G(t)=A\), where \(A: D(A) \to X\) is a linear closed operator generating a strongly continuous cosine family ([4]);
  • – \(p\equiv 1\), \(G(t)=A(t)\), where \(\lbrace A(t) \rbrace_{t \in J}\), \(A(t):D(A) \to X\), is a family of linear operators, generating a fundamental system and \(F(t,x,y)=F(t,x)\) ([1], [2]);<\li>
  • – \(p:J \to \mathbb{R}\), \(G: J \to \mathcal{P}(\mathbb{R})\) is a suitable multimap and \(F(t,x,y)=F(t,x)\) ([3]).

  • [1] Cardinali T., Duricchi G., On nonlocal problems for semilinear second order differential inclusions without compactness. Electron. J. Qual. Theory Differ. Equ. No. 66 (2021), 1-32.
  • [2] Cardinali T., Duricchi G., Further study on second order nonlocal problems monitored by an operator: an approach without compactness. Electron. J. Qual. Theory Differ. Equ. No. 13 (2023), 1-34.
  • [3] Cardinali T., Duricchi G., Strong and mild solutions for Sturm-Liouville differential inclusions. Set-Valued Var. Anal. 32 No. 3 (2024).
  • [4] Cardinali T., Duricchi G., On the instant-controllability of a second order multidimensional differential equation subjected to damping term and impulses. Submitted.

Giovanni Giliberti (University of Modena and Reggio Emilia)
Delay evolution equations with nonlocal multivalued initial conditions

Abstract We consider the nonlinear delay differential evolution equation in a Banach space \( X \) \[ \begin{cases} x'(t)=Ax(t)+f(t,x_t)\qquad &t\in[0,T]\\ x(t) \in g(x)(t)\qquad &t\in[-r,0]. \end{cases} \] where the linear operator \( A \) generates a \( C_0 \)-semigroup of contractions and the function \( x_t \), defined by \( x_t(s)=x(t+s) \) for \( s\in[-r,0] \), is the delay term. We prove the existence of mild solutions satisfying the nonlocal, multivalued, Cauchy condition defined by the multimap \( g:C([-r,T],X)\multimap C([-r,0],X) \), whether the semigroup generated by \( A \) is compact or not. Our approach involves a suitable degree argument. We then apply our results to a transport equation in the form \[ \begin{cases} u_t(t,y)+a\cdot\nabla u(t,y)= \Phi\big(\int_{\mathbb{R}^n}|u(t,\xi)|^pd\xi\big)\cdot &\\ \qquad\qquad\cdot\, h\big(\int_{t-r}^t\int_{\mathbb{R}^n}|u(s,\xi)|^p\,\mathrm{d}\xi\,\mathrm{d}s\big)\cdot \ell(t,u(t,y)) &[0,T]\times\mathbb{R}^n\\ u(t,y) \in\{ u_0(t,y)+ &\\ \qquad\qquad+ \sum_{i=1}^N\mu_i\int_t^0u(t_i+s,y)\,\mathrm{d}s:\mu_i\in A_i,\, i=1,…,N\} &[-r,0]\times\mathbb{R}^n \end{cases} \] where \(a\in \mathbb{R}^n\), the functions \(\Phi\), \( h :\mathbb{R}\to\mathbb{R} \) are continuous and bounded and the map \( \ell:[0,T]\times\mathbb{R}\to\mathbb{R} \) satisfies suitable assumptions. Here, the multivalued condition is defined by the function \( u_0:[-r,0]\times \mathbb{R}^n\to\mathbb{R} \), the fixed instants \(T\geq t_{N}>\dots>t_{2}>t_{1}\) and the variable coefficients \(\mu_i\in A_i:=[\lambda_i-\epsilon_i;\lambda_i+\epsilon_i]\), with \(\lambda_i\in\mathbb{R}\) and \(\epsilon_i>0\) for \(i=1,…,N\).

Ahmad Makki (University of Modena and Reggio Emilia)
On the generalized coupled Allen-Cahn/Cahn-Hilliard equation

Abstract This poster presents an investigation of the coupled Allen-Cahn/Cahn-Hilliard equations, approached from both theoretical and numerical perspectives. Theoretically, we establish the existence and uniqueness of solutions for the coupled system and analyze the asymptotic behavior, demonstrating the presence of a finite-dimensional global attractor. On the numerical side, we implement a semi-discretization in space using the finite element method and derive error estimates between the exact and approximate solutions. By applying the backward Euler scheme for time discretization, we develop a fully discrete system and prove its stability. Numerical simulations are provided to illustrate the solution’s behavior under dynamic boundary conditions.

Rim Mheich (Université de Poitiers)
On the Cahn-Hilliard equation with general proliferation term

Abstract The aim in this poster is to study the well-posedness of the generalized logarithmic nonlinear Cahn-Hilliard equation with regularization and proliferation terms. We are interested in studying the asymptotic behavior, in terms of finite-dimensional attractors, of the dynamical system associated with the problem and majorate the rate of convergence between the solutions of the Cahn-Hilliard equation and the regularized one. Additionally, we present some further regularity results and subsequently prove a strict separation property of the solution. Finally, we provide some numerical simulations to compare the solution with and without the regularization term, and more.

Emanuele Pastorino (Politecnico di Milano)

Unpredictable behavior of a partially damped system of PDEs modeling suspension bridges

Abstract We consider a nonlinear nonlocal coupled system of beam-wave equations, governing the dynamics of a degenerate plate modeling the behavior of suspension bridges. In line with the physical observations, the beam equation is both damped and forced, whereas the wave equation is “isolated”. Since the resulting overall damping is degenerate, the full dynamical system is only partially dissipative. This leads to an unpredictable behavior for the solutions of the system, which prevents to forecast the general behavior of bridges, encompassing the instability of the unique stationary solution and the richness of the \(\omega\)-limit set, which contains infinitely many periodic solutions. We also discuss some striking differences between the sole beam equation and the coupled system, which may be the cause of some not fully understood paradoxes and erroneous conclusions.

Wahid Ullah (University of Trieste)

Periodic solutions of coupled Hamiltonian systems: Twist with lower/upper solutions

Abstract The Hamiltonian systems considered are obtained by weakly coupling two systems having completely different behaviours. The first one satisfies the usual twist assumptions taylored for the application of the Poincaré–Birkhoff Theorem, while the second one presents the existence of some well-ordered generalized lower and upper solutions.

Poster Preparation Guidelines:

  • Format: Posters should be prepared in A0 portrait format
  • Printing: Presenters are responsible for bringing their own pre-printed posters to the event

Submission Details:

  • Please complete the abstract submission form below
  • Deadline: September 5, 2024