Special Session 146: Nonlinear differential equations: control, delay, and boundary value problems

Which diffusion strategy is best for a prey and a predator moving with Fractional Laplacians?

Elisa Affili Universite de Rouen Normandie France Co-Author(s):    Serena Dipierro, Giovanni Giacomin, Enrico Valdinoci

Abstract: We consider a predator and a prey moving Fractional Laplacian, letting the predator and the prey free to choose the exponent characterising the distribution of the jump lengths. Which exponent should the predator use to maximise its chance to catch the prey? And which one should the prey select to minimise the possibility of meeting the predator? In this talk, we study a funtional that describe the chance of meeting taking into account both strategies. We focus in particular on the case of large initial distance and we find ranges for the parameter of fractional diffusion corresponding to the maximum (for the predator) and the minimum (for the prey) in dependence of the choice of the other species. These findings rely on very fine estimates of the fundamental solution of the fractional heat equation.

A structured model of vector-borne disease with within-host viral load and antibody dynamics

Paulo Amorim FGV - EMAp Brazil Co-Author(s):    M Soledad Aronna, Debora de Oliveira Medeiros

Abstract: We present an epidemiological model for vector-borne diseases, including witihin-host viral load and antibody dynamics, via structured (transport) equations. The within-host dynamics is present in the infected and recovered host compartments. The structure induces nonlinearities and nonlocalities in the formulation. We analyse the model, showing well-posedness, mass conservation, and study the characteristic curves of the transport equation. We deduce a simplified Uniform Host Response (UHR) model which includes delay-type terms. For both the full and UHR model, we determine the basic reproduction number $R_0$ and comment on its relation to linear stability of the disease-free equilibrium. We show with numerical experiments how the within-host dynamics influences the epidemiological outcomes. This is joint work with M.S. Aronna and D.O. Medeiros.

Controllability problem of a differential equation with memory

Sumit Arora Instituto de Matematicas, Universidad de Talca Chile Co-Author(s):    Sumit Arora and Rodrigo Ponce

Abstract: In this talk we addresses control problems governed by a semilinear evolution equation with memory kernel $\kappa\in\mathrm{L}^1_{loc}(\mathbb{R}^+)$. We discuss the existence of a mild solution and the approximate controllability of both linear and semilinear control systems. To this end, we introduce the concept of a resolvent family associated with the linear evolution equation with memory and its essential properties. Subsequently, we consider a linear-quadratic regulator problem to determine the optimal control that yields approximate controllability for the linear control system. Furthermore, sufficient conditions for the existence of a mild solution and the approximate controllability of a semilinear system in a reflexive Banach space having uniformly convex dual has been presented. Finally, we apply our theoretical findings to investigate the approximate controllability of the heat equation with singular memory. Recently, Pandolfi \cite{LP-2021} studied the approximate controllability of a general class of control systems with singular memory by applying boundary control. The considered system particularly includes systems with fractional derivatives and integrals as well as the standard heat equation. Pandolfi, L.: Controllability properties for equations with memory of fractional type, Appl. Math. Optim., 84(1) (2021) 325--353.

Periodic solutions for second order time and state-dependent delay equations with a non-asymptotic condition

Pierluigi Benevieri University of S\~ao Paulo Brazil Co-Author(s):

Abstract: We investigate the existence of periodic solutions, of a given period $\o$, to a class of non-autonomous second order differential equations with time and state-dependent delay. We assume that the associated linear part contains a friction term and the nonlinearity satifies a non-asymptotic condition.\r\nWe present some applications to superlinear problems and to a sunflower-type equation with variable delay. The approach is topological, based on Mawhin`s coincidence degree. This is a joint work with Pablo Amster

Shock wavefronts for parabolic equation with sign-changing diffusivity

Diego Berti Univeristy of Turin Italy Co-Author(s):

Abstract: This talk focuses on a reaction-diffusion equation in a one-dimensional space, where the diffusion is positive-negative-positive and the reaction term is bistable and changes sign where the diffusivity is negative. In this setting, continuous wavefronts are not allowed. We prove the existence of a family of shock wavefronts with profiles that have a jump discontinuity. We further investigate the properties of these profiles and their propagation speeds. Moreover, we discuss the application of the results to a model describing the movement of a population composed of both isolated and grouped individuals. This is joint work with Andrea Corli (University of Ferrara) and Luisa Malaguti (University Modena and Reggio Emilia).

Bifurcation of periodic solutions from central force problems

Alberto Boscaggin University of Turin Italy Co-Author(s):    Walter Dambrosio and Guglielmo Feltrin

Abstract: I will discuss the existence of periodic solutions bifurcating from a manifold of periodic solutions of a central force problem in a Euclidean space of dimension two or three. The use of proper systems of coordinates will be the essential step in verifying the needed non-degeneracy condition. Joint works with Walter Dambrosio and Guglielmo Feltrin.

Evolutionary partial differential equations with finite memory: state space representation approach and the linear quadratic problem

Francesca Bucci Universit\\`a degli Studi di Firenze Italy Co-Author(s):    Francesca Bucci

Abstract: A functional-analytic reformulation of the heat equation with finite memory in a bounded domain, subjected to boundary control actions, brings about an integro-differential equation in a Hilbert space where the memory pertains to both the state and control variables; the resulting unbounded control operator occurs in the convolution term, besides the free dynamics generator. In the case of smooth kernels, an approach to the study of certain sought-after control-theoretic properties of the solutions -- such as, e.g., controllability -- utilizes MacCamy's trick along with the theory of Volterra integral equations of the second kind, and then operator semigroup theory. In this talk we will describe current work toward obtaining a representation formula for the solutions, based on a ``state space representation'' approach, inspired by the one introduced by C. Dafermos to deal with evolution equations with infinite memory and pursued since the 1970s on. The distinct tasks and technical challenges which need to be handled will be discussed, particularly but not exclusively with an eye to the study of the associated optimal control problem with quadratic functionals. (The talk is based on ongoing joint work with Paolo Acquistapace, Univ. di Pisa (Ret.).)

First and Second Order Nonlocal Evolution Equations Governed by Non Autonomous Forms

Vittorio Colao University of Calabria Italy Co-Author(s):

Abstract: We present existence results for semilinear non autonomous evolution equations with nonlocal initial conditions, covering both first and second-order problems. In the first-order setting, a finite-dimensional reduction combined with the Leray Schauder continuation principle yields solutions in L2, under demicontinuity and transversality assumptions on the nonlinearity. The second-order analysis relies on fundamental solution techniques and Schauder type fixedpoint arguments for wave equations. Superlinear nonlinearities are handled via maximal regularity and energy inequalities in uniformly smooth Banach spaces. Applications include vibrating membranes, population dynamics with memory effects, and parabolic problems with logistic or Nagumo type reactions.

On anisotropic fractional inverse problems

Simone Creo Sapienza Universita' di Roma Italy Co-Author(s):

Abstract: In this talk we study the well-posedness of an inverse problem modeling anisotropic diffusion, involving a fractional time derivative of Caputo type of order $\alpha\in(0,1)$. More precisely, we consider a fractional-in-time abstract Cauchy problem involving a finite sum of operators, multiplied by suitable positive conductivities, which are considered as unknowns in the inverse problem. We prove the uniqueness of the solution of the inverse problem under suitable additional conditions and we provide a conditioned existence result. Then, we study the behavior of the solution of the inverse problem as $\alpha\to 1^-$.

Random Dynamical Systems generated by Random Generalized Ordinary Differential Equations

Marcia Federson Universidade de Sao Paulo Brazil Co-Author(s):    Xiaoying Han and Antonio Veloso

Abstract: Within the framework of random generalized ordinary differential equations, which combine stochastic processes with generalized integration in the sense of Kurzweil, we establish new conditions ensuring the existence of solutions and, under suitable assumptions, their uniqueness, covering local, maximal, and global regimes. This setting connects deterministic generalized differential equations with random dynamical systems, providing a unified framework and new analytical tools for the study of dynamics under randomness.

On the splitting of Neumann eigenvalues in perforated domains

Veronica Felli University of Milano-Bicocca Italy Co-Author(s):

Abstract: We address the problem of splitting of eigenvalues of the Neumann Laplacian under singular domain perturbations. We consider a domain perturbed by the excision of a small spherical hole shrinking to an interior point. We establish that the splitting of multiple eigenvalues is a generic property: if the center of the hole is located outside a set of Hausdorff dimension N-1 and the radius is sufficiently small, multiple eigenvalues split into branches of lower multiplicity. The proof relies on the validity of an asymptotic expansion for the perturbed eigenvalues in terms of the scaling parameter.

A paradigmatic superlinear boundary value problem: an overview of recent advances

Guglielmo Feltrin University of Udine Italy Co-Author(s):

Abstract: We first review the main features of the paradigmatic class of superlinear boundary value problems \begin{equation*} -u``= \lambda u + a(x) u^{p}, \qquad u(0)=u(L)=0, \end{equation*} where $p > 1$, $\lambda \in\mathbb{R}$ is regarded as a parameter, and $a(x)$ is a weight function. Next, we present a solution to a conjecture by R.~G\`{o}mez-Re\~{n}asco and J.~L\`{o}pez-G\`{o}mez (JDE, 2000) concerning the multiplicity of positive solutions. This is a joint work in collaboration with Juli\`{a}n L\`{o}pez-G\`{o}mez (Universidad Complutense de Madrid) and Juan Carlos Sampedro (Universidad de Cantabria).

Subharmonics for homoclinic trajectories via Melnikov theory

Matteo Franca University of Florence Italy Co-Author(s):

Abstract: Consider a $2$-dimensional system which has a trajectory $\gamma(t)$ homoclinic to the origin, which is assumed to be a critical point; Melnikov theory provides a condition sufficient for the persistence of $\gamma(t)$ to small non-autonomous perturbation of size $0<\varepsilon< \varepsilon_0$. Namely, a certain computable function $M(\tau)$ is required to have a non degenerate zero, say $M(\tau_0)=0 \ne M'(\tau_0)$. In this talk we want to show that, if we add a sign condition on $M'(\tau_0)$, i.e. $M'(\tau_0)>0$, then for any $k \in \mathbb{N}$ we find an $\varepsilon_k$ such that the perturbed system admits at least $k+1$ homoclinic trajectories, each performing exactly $j=1, \ldots , k+1$ loops. On the other hand if $M'(\tau_0)<0$ we just find persistence of the homoclinic performing a single loop. The talk is based on a joint preprint with M. Pospisil and A. Sfecci.

Structure of Optimal Solutions for Traffic Flow

Mauro Garavello University of Milano-Bicocca Italy Co-Author(s):    Fabio Ancona, Annalisa Cesaroni, Giuseppe Maria Coclite

Abstract: In the last years, starting with the seminal papers by Lighthill and Whitham \cite{LW} and Richards \cite{richards}, there was an increasing interest in conservation laws for the modeling of traffic flow in road networks, mainly justified by applications. Particularly, the problems of reducing congestions, car accidents, and pollution have been tackled with various approaches. We present, in this talk, a way for controlling traffic flow, through a control function acting at the level of junctions \cite{a-c-c-g-1}. We introduce the concept of solution, we show that the solution exists and that, in some cases, the input-output map is continuous. It is also addressed the problem of minimization of functionals describing traffic performance indexes \cite{a-c-c-g-2}. Finally, in the case of $1$-$1$ junctions, the structure of optimal solutions is presented \cite{a-c-c-g-3}. \bibitem{a-c-c-g-1} F. Ancona, A. Cesaroni, G. M. Coclite, M. Garavello. On the optimization of conservation law models at a junction with inflow and flow distribution controls. SIAM J. Control Optim. 56 (2018), no. 5, pp. 3370-3403. \bibitem{a-c-c-g-2} F. Ancona, A. Cesaroni, G. M. Coclite, M. Garavello. On optimization of traffic flow performance for conservation laws on networks. Minimax Theory Appl. 6 (2021), no. 2, pp. 205-226. \bibitem{a-c-c-g-3} F. Ancona, A. Cesaroni, G. M. Coclite, M. Garavello. On the structure of optimal solutions of conservation laws at a junction with one incoming and one outgoing arc. arXiv 2507.10090. \bibitem{LW} M. J. Lighthill, G. B. Whitham. On kinematic waves. II. A theory of traffic flow on long crowded roads. Proc. Roy. Soc. London. Ser. A. 229 (1955), pp. 317-345. \bibitem{richards} P.I. Richards. Shock waves on the highway. Operations Res. 4 (1956), pp. 42-51.

On semilinear non autonomous nonlocal equations

Maria Rosaria Lancia Sapienza Universit\`a di Roma Italy Co-Author(s):    Simone Creo

Abstract: I will explore non-autonomous fractional (in space and/or time) semilinear BVPS, possibly in irregular domains. The focus will be on well-posedness and regularity results for the mild solutions of the corresponding semilinear Cauchy problem, via the evolution family U(t, s). I will also present results on the ultracontractivity properties of the evolution family and discuss some applications and open problems.

Convergence results for varying measures under convexity conditions and applications

Valeria Marraffa Dipartimento di Matematica e Informatica Italy Co-Author(s):    A.R. Sambucini

Abstract: Some limit theorems of the type $$\int_{\Omega}f_n\,dm_n \rightarrow \int_{\Omega}f \,dm$$ are presented for scalar, (vector), (multi)-valued sequences of $m_n$-integrable functions $f_n$. Conditions for the convergence of sequences of measures $(m_n)_n$ and of their integrals $(\int f_n dm_n)_n$ in a measurable space $\Omega $ are of interest in many areas of pure and applied mathematics.\ Sufficient conditions in order to obtain some kind of Vitali`s convergence theorems for a sequence of (multi)functions $(f_n)_n$ integrable with respect to a sequence of measures $(m_n)_n$ are considered.\ We consider the asymptotic properties of $(\int_{\Omega} f_n d m_n)_n$ with respect to varying measures, which are setwisely convergent in measurable spaces.\ Consequently, a continuous dependence result for a wide class of differential equations with many interesting applications, namely measure differential equations (including Stieltjes differential equations, generalized differential problems, impulsive differential equations with finitely or countably many impulses and also dynamic equations on time scales) is provided. \begin{thebibliography}{9} \bibitem{1} L. Di Piazza, V. Marraffa, K. Musia{\l}, A.R. Sambucini, {\emph Convergence for varying measures}, {\it J. Math. Anal. Appl.}, Vol. 518, N.2, Paper N. 126782, (2023). \bibitem{3} L. Di Piazza, V. Marraffa, K. Musia{\l}, A.R. Sambucini, {\emph Convergence for varying measures in the topological case}, {\it Annali di Matematica Pura e Applicata, (4) 203 (2024) 71-86}. \bibitem{2}V. Marraffa, B. Satco, {\emph Convergence Theorems for Varying Measures Under Convexity Conditions and Applications}, {\it Mediterr. J. Math.}, (2022), 19:274. \bibitem{4} V. Marraffa, A.R. Sambucini, {\emph Vitali theorems for varying measure}, {\it Symmetry} 2024, 16(8), 972. \end{thebibliography}

Global positive bounded solutions for second order nonlinear equations with regularly varying operator

Serena Matucci Department of Mathematics and Computer Sciences Italy Co-Author(s):    Zuzana Do\v{s}l\`{a} and Mauro Marini

Abstract: A second order nonlinear differential equation with inhomogeneous differential operator $\Phi,$ which is regularly varying at zero, is considered. The operator $\Phi$ can be viewed as an extension of the $p$-Laplacian operator and arises in many physical problems, as we will illustrate by several examples. In particular, the existence of global positive bounded solutions on the half-line with the Neumann type boundary conditions is studied by means of an abstract fixed point theorem and certain properties of an associated half-linear equation. The results do not require the explicit form of the inverse operator of $\Phi$, and are completed by an asymptotic analysis of these solutions near infinity.

An interpolation approach to $L^{\infty}$ a priori estimates for elliptic problems with nonlinearity on the boundary

Nsoki Mavinga Swarthmore College USA Co-Author(s):    Maya Chhetri, Rosa Pardo

Abstract: In this talk, we present an explicit $L^\infty(\Omega)$ a priori estimate for weak solutions to semilinear elliptic equations with nonlinearity on the boundary, in terms of the powers of their $H^1(\Omega)$ norms. We combine an appropriate version of a Moser iteration argument along with elliptic regularity and Gagliardo--Nirenberg interpolation inequality to prove our result. We illustrate our result with an application to subcritical problems satisfying Ambrosetti-Rabinowitz condition.

Some uniqueness and multiplicity results in predator-prey models with saturation

Eduardo Mu\~noz-Hern\`andez Complutense University of Madrid Spain Co-Author(s):    Kousuke Kuto and Juli\`an L\`opez-G\`omez

Abstract: In this talk we present some results on uniqueness and multiplicity results in diffusive predator-prey models with saturation. Depending on the saturation amplitude, we will see how to obtain either uniqueness or multiplicity of coexistence states in the system. This talk is based on joint works with Juli\`an L\`opez-G\`omez (Complutense University of Madrid) and Kousuke Kuto (Waseda University).

Convergence of asymptotic systems in Cohen-Grossberg neural network models with unbounded delays

Jos\`{e} J. Oliveira University of Minho Portugal Co-Author(s):    A. Elmwafy, Jos\\`e J. Oliveira, and C\\`esar M. Silva

Abstract: In this presentation, we present sufficient conditions for the convergence of asymptotic systems in non-autonomous Cohen-Grossberg neural network models that incorporate both infinite discrete time-varying and distributed delays. The main stability criterion is obtained by imposing conditions under which the non-delay terms asymptotically dominate the delay terms. As an applications, we provide sufficient conditions ensuring that all solutions of a non-periodic neural network model with unbounded delays converge to a periodic function as time goes to infinity. A numerical example is presented to illustrate the effectiveness of the new results.

Optimal strategies for differential problems with feedback controls and distributed delay

Paola Rubbioni University of Perugia Italy Co-Author(s):

Abstract: The topic of this talk concerns our recent studies on optimal strategies for differential equations with distributed delay. By employing topological methods and tools from multivalued analysis, we establish the existence of controlled trajectories that minimize or maximize a cost functional associated with the system. The framework allows for the presence of impulses and feedback controls. The results apply to models arising in the natural and applied sciences, such as population dynamics, driven by differential equations with Volterra type distributed delays involving fading memory kernels or with functional unbounded delay. Bibliography: [1] Benedetti I., Rubbioni P.; Impulsive delay differential inclusions applied to optimization problems, arXiv:2512.21275, pp. 1-20 [2] Rubbioni, P.; Optimal Solutions for a Class of Impulsive Differential Problems with Feedback Controls and Volterra-Type Distributed Delay: A Topological Approach. Mathematics 2024, 12, 2293 [3] Rubbioni P.; Existence of optimal periodic strategies in a model with nonlocal spatiotemporal dispersal, J. Math. Anal. Appl. 556 (2026) 130095

Limiting behavior of principal eigenvalues for a class of mixed boundary value problems as the measure of the support domain goes to zero

Alejandro Sahuquillo Universidad Complutense de Madrid Spain Co-Author(s):    Juli\`{a}n L\`{o}pez-G\`{o}mez

Abstract: In this paper we characterize the limiting behavior of the principal eigenvalue, $\sigma_1[-\Delta,\beta,\Omega]$, of the boundary value problem (1.1) as the Lebesgue measure of the underlying domain, $\Omega$, tends to zero. Naturally, the domains $\Omega$ are assumed to be included on a fixed open set $D$ such that $\beta\in\mathcal{C}(D)$, and they satisfy $\bar\Omega\subset D$. Our main result establishes that, in the classical case when $\inf_{D}\beta >0$, $$ \lim_{|\Omega|\downarrow 0}\sigma_1[-\Delta,\beta,\Omega] =+\infty, $$ whereas $$ \lim_{|\Omega|\downarrow 0}\sigma_1[-\Delta,\beta,\Omega] =-\infty\;\;\hbox{if}\;\; \sup_{D}\beta < 0, $$ which is a surprising result at the light of the classical existing theorems for Dirichlet boundary conditions. Furthermore, in the special case when $\beta>0$ is a constant, we can prove that $$ \lim_{R\downarrow 0}\left( R \sigma_1[-\Delta,\beta,B_R]\right)=\beta \frac{\mathrm{Area}(\partial B_1)}{|B_1|}, $$ where, we are denoting $B_\varrho:=\{x \in \mathbb{R}^N \;:\;|x|< \varrho \}$ for all $\varrho>0$. This is a joint work with Juli\`{a}n L\`{o}pez-G\`{o}mez.

Boundary value problems with Stieltjes derivative

Bianca Satco Stefan cel Mare University of Suceava Romania Co-Author(s):

Abstract: Let $g:[0,T]\to \mathbb{R}$ be a left-continuous nondecreasing function and $\;\mu_g\;$ the Lebesgue-Stieltjes measure generated by $\;g$ on $[0, T]$. The aim of the talk is to present an existence result (\cite{ms cmj}) for first order set-valued problems with a very general boundary condition \begin{equation*} \left\{ \begin{array}{l} u'_g(t) \in F(t,u(t)),\; \mu_g{\rm -a.e.\;} t\in [0,T]\ L(u(0),u(T))=0 \end{array} \right. \end{equation*} involving the Stieltjes derivative with respect to $g$ (see \cite{pouso}). The multifunction $F : [0,T] \times \mathbb{R} \to \mathcal{P}(\mathbb{R}) $ is of Carath\'{e}odory type with convex compact values and $L:\mathbb{R}^2\to \mathbb{R}$ is continuous and nonincreasing with respect to its second argument. The applied method is that of lower and upper solutions inspired from \cite{mf}. The announced result generalizes several already known outcomes since the theory of Stieltjes differential equations encompasses various classical problems: ordinary differential equations (when $g$ is the identity map), impulsive differential problems (if $g$ can be written as a sum of the identical function with a sum of Heaviside functions) and dynamic equations on time scales. Moreover, for particular maps $L$ some existence results for Stieltjes differential inclusions with initial value conditions ($L(x,y)=x-u_0$) or with periodic conditions ($L(x,y)=x-y$) are covered.

Symmetry breaking for biharmonic H\\`{e}non type problems with exponential nonlinearities

Cristina Tarsi Universita` degli Studi di Milano Italy Co-Author(s):    Marta Calanchi

Abstract: Let \( B \subset \mathbb{R}^4 \) be the unit ball. We discuss different maximization problems for exponential-type nonlinearities with weights of H\`{e}non type, $$ F_m(u)=\int_{B} \left(e^{\sigma u^{2}}-\sum_{k=0}^m \frac{\sigma^k u^{2k}}{k!}\right)|x|^\alpha\, \quad m\geq 0, $$ by varying $u$ in $H_{\mathcal N}^2 (B)=H_{0}^1(B)\cap H^2(B)$, in $H_0^2(B)$ and in their radial subspaces. It is well known that, as a consequence of Adams` type inequalities, all the suprema are finite if $\sigma \leq 32\pi^2$. We first prove that, in the radial framework, the finiteness of the suprema is enlarged up to $\sigma_\alpha=32\pi^2(1+\frac{\alpha}4)$. Then, for $\sigma \leq 32\pi^2$ and $m\geq 1$, we prove that radial symmetry of maximizers in the whole spaces $H_{\mathcal N}^2 (B)$ and $H_0^2 (B)$ is broken for large values of the weight exponent $\alpha$. These results extend classical ones in the second-order setting to the biharmonic context in the limiting dimension $N=4$. This is a joint work with Marta Calanchi (Universit\`{a} degli Studi di Milano, Italy).

Bifurcation approach for periodic solutions in a superlinear BVP

Andrea Tellini Universidad Politécnica de Madrid Spain Co-Author(s):    Eduardo Mu\~{n}oz-Hern\`{a}ndez, Juan Carlos Sampedro

Abstract: I will show how a bifurcation approach can be applied to a periodic superlinear boundary value problem also for eigenvalues with even multiplicity. This approach allows us to obtain existence and multiplicity results for local and global branches of periodic solutions with a certain nodal behavior. The optimality of such results will be also discussed with the aid of numerical simulations. These results have been obtained in collaboration with Eduardo Mu\~{n}oz-Hern\`{a}ndez (Universidad Complutense de Madrid, Spain) and Juan Carlos Sampedro (Universidad de Cantabria, Spain).

The degenerate quenching problem

Rafayel Teymurazyan KAUST Saudi Arabia Co-Author(s):    D.J. Ara\`ujo, J.M. Urbano

Abstract: We study minimizers of non-differentiable functionals modeled on the degenerate quenching problem. Our main result establishes the finiteness of the (n-1)-dimensional Hausdorff measure of the free boundary. The proof is based on optimal gradient decay estimates obtained from an intrinsic Harnack-type inequality, along with a detailed analysis in a flatness regime, where minimizers enjoy improved regularity. Our arguments provide an alternative proof of classical results of Phillips and, although developed in the degenerate setting, also offer insights relevant to the singular case.

Standing waves for the wave equation with hyperbolic boundary conditions

Enzo Vitillaro Dipartimento di Matematica e Informatica Universit\`a degli Studi di Perugia Italy Co-Author(s):

Abstract: We deal with standing waves for the wave equation with hyperbolic boundary conditions, posed in a bounded domain with regular boundary. These problems possess a wide literature, including the papers on Arch. Rat. Mech. Anal. (2017), J.D.E. (2018) and DCDS-S (2021) by the author. Standing waves solutions of this evolution problem, in the linear setting, turn out to be eigenfunctions for a doubly elliptic problem, which involves the Laplace operator inside the domain and the Laplace--Beltrami one at the boundary. In the talk we show how these eigenfunctions constitute a Hilbert basis of a suitable space, and we study some of their properties.