Abstract: We study the regularity properties of viscosity solutions to a class of degenerate normalized p-laplacian equations. In particular, we prove that the gradient of viscosity solutions are Hölder continuous, and we give the optimal exponent. Moreover, we also show that viscosity solutions to equations with very general degeneracy laws are differentiable. We apply our results to prove a particular case of the so-called C^p′-conjecture.

Abstract: In the past two decades, substantial progress has been made in understanding random dispersive PDEs with polynomial nonlinearities. However, the non-polynomial case remains largely open. In this talk, I will present recent progress on the well-posedness of the two-dimensional damped wave equation with either a sine or sinh nonlinearity, driven by additive space-time white noise. Our approach develops a new physical space framework for analyzing such random wave equations.

This talk is partially based on a joint work with J. Forlano (Monash Univ.)

Abstract: Schrödinger-type equations model a lot of natural phenomena and their solutions have interesting and important properties: one of them is the conservation of mass, which gives rise to the search for normalised solutions. In this talk, I will show a possible approach to find a solution with prescribed L^2 norm to a semilinear fractional Schrödinger equation with a strongly sublinear right-hand side g, i.e., g(t)/t blows up negatively as t converges to 0. Since this makes the corresponding energy functional non-well-defined in H^s, a direct use of variational methods is impossible. Instead, a family of approximating problems is considered, a corresponding family of ground-state solutions is obtained, and their convergence to a ground-state solution to the original problem is proved.

This is joint work with Marco Gallo (Brescia, Italy).

Abstract: In my latest work I studied a reaction diffusion problem with anisotropic diffusion and mixed Dirichlet-Neumann boundary conditions on the boundary of a bounded domain. First, I proved that the parabolic problem has a unique positive, bounded solution. Then, I showed that this solution converges, as the time tends to infinity, to the unique nonnegative solution of the elliptic associated problem. The existence of the unique positive solution to this problem depends on a principal eigenvalue of a suitable linearized problem with a sign-changing weights. Next, I studied the minimization of such eigenvalue with respect to the sign-changing weight, showing that there exists an optimal bang-bang weight, namely a piece-wise constant weight that takes only two values. Finally, I was able to completely solve the problem in dimension one. Here is the link to the preprint of the paper:
https://arxiv.org/abs/2504.07577

Abstract: We study the existence of least energy solutions to the system
-\Delta u + \lambda_1 u = u^3 + \beta uv^2 & \text{in } \Omega \\ -\Delta v + \lambda_2 v = v^3 + \beta u^2 v & \text{in } \Omega \\ \frac{\partial u}{\partial \nu} = \frac{\partial v}{\partial \nu} = 0 & \text{on } \partial \Omega,

 where $\Omega \subset \mathbb{R}^N$ is a $C^2$-bounded domain with $N \leq 4$ and $\nu$ is the outward normal at the boundary. Assuming suitable conditions on the parameters $\lambda_1, \lambda_2, \beta \in \mathbb{R}$, we prove the existence of non-constant least energy solutions. Our results can be adapted to cover some previously untreated cases for Dirichlet conditions.

 Our variational approach relies on the minimization of the energy functional over the set $X_\beta$ of  all non-trivial weak solutions of the system. Depending on the values of $\beta,\lambda_1,\lambda_2$, the minimizer is obtained either via a linking theorem, by minimizing over a suitable Nehari manifold, or by direct minimization on $X_\beta$.

Abstract: This talk deals with the emergence of cross-diffusion in the singular limit of a reaction-diffusion system with multiple time scales. A class of triangular cross-diffusion system is rigorously obtained at the singular limit of a fast reaction-diffusion system, as the time scale ε of the fast reaction goes to 0. We prove the convergence of the unique solution of the fast reaction-diffusion system towards a weak solution of the cross-diffusion system and we show a convergence rate. The main tool is a family of energy functionals that handles the fast reaction terms uniformly in ε.

Abstract: This paper investigates the profile decomposition of Palais-Smale sequences associated with a Brézis--Nirenberg-type problem involving a combination of mixed local nonlocal operator, given by 

- \Delta u + ( - Delta )^s u- \lambda u = |u|^{ 2^* - 2 }u \text{ in } \Omega,\\

u  =0 \text{ in } \mathbb R^N \setminus \Omega,

where $\Omega \subseteq \mathbb R^N$ is a smooth bounded domais with $N\geq 3$, $s \in (0,1)$, $\lambda \in \mathbb R$ is a real parameter and $2^* = \frac{N}{N-2}$ denotes the critical Sobolev exponent. As an application of the derived global compactness result, we further study the corresponding Coron-type problem.

Abstract: The detailed balance is a property of macroscopic systems that are obtained from an underlying time-reversible microscopic model. It states that each elementary process (for instance each chemical reaction) is in equilibrium with its reverse process. Even if, at the fundamental level, we expect chemical reactions to satisfy the so-called detailed balance condition, biochemical systems are often modeled by systems of equations for which detailed balance fails. This can be justified if the system is in contact with "reservoirs" that are out of equilibrium, as is usually the case in biological systems. In this talk, I will discuss the relation between the detailed balance property and the property of adaptation of chemical systems. In particular, a system satisfies the adaptation property if it responds to gradients instead of absolute values of signals. I will show that, unless a factorization assumption on the conservation laws holds, the property of adaptation cannot hold in a stable manner in closed systems (i.e., systems that satisfy the detailed balance property and do not exchange substances with the environment).

Abstract: We are concerned with the existence results of modified Schrödinger equation involving critical Stein-Weiss type exponential nonlinearity in bounded domain/unbounded domain. In particular, we will discuss the existence results of following modified Schrödinger equation 

-\Delta_N u-\Delta_N (u^2)u+V(x)|u|^{N-2}u=\left(\int_{\mathbb R^N}\frac{F(y,u)}{|u|^\beta|x-y|^\mu}dy\right)\frac{f(x,u)}{|x|^\beta\quad\text{ in }\mathbbR^N,

where $N\geq 2$, $0<\mu<N. The potential $V:\mathbb R^N\rightarrow\mathbb R$ is a continuous function satisfying  $0<V_0\leq V(x)$ for all $x\in \mathbb R^N$ and some suitable assumptions. The nonlinearity $f:\mathbb R^N\times\mathbb R$ is a continuous function with critical exponential growth in the sense of the Trudinger-Moser inequality and $F(x,s)=\int_0^s f(x,t) dt$ is the primitive of $f$.

Abstract: We derive reaction-diffusion equations from systems of kinetic Boltzmann equations for cell populations interacting in a host medium. We take the diffusive limit of the kinetic equations and obtain a macroscopic system with nonlinear diffusion and cross-diffusion effects. We apply our approach to study the evolution of bacterial communities living on a leaf surface, where they can interact with each other and with the host leaf, undergoing different reproductive or destructive processes and competing for resources. All these effects may lead populations to organize and evolve into bacterial clusters, resulting in diverse spatial patterns such as co-aggregation or segregation. The mathematical description of such dynamics may be obtained by means of a Turing instability analysis of the reaction-diffusion equations derived in the hydrodynamic limit of the kinetic system. Some numerical simulations are presented to illustrate different microbial colonization patterns.

This work is a collaboration with D. Cusseddu (Polytechnic University of Turin, Italy), M. Bisi, and R. Travaglini (University of Parma, Italy).