We are interested in an inverse problem for the fractional Schrödinger equation with a time-independent potential on closed Riemannian manifolds. The model is given by

$$\begin{align*}i\partial_t^\alpha u+\big((-\Delta_g)^\beta+V\big)u=f, \end{align*}$$

with $\alpha ,\beta \in (0,1)$ and $V\in L^\infty (M)$ real-valued. Our main goal is to determine the fractional powers $\alpha ,\beta $, the Riemannian metric g (up to an isometry), and the potential V simultaneously from the knowledge of the associated source-to-solution map. For disjoint source and observation regions, the map determines $\alpha $, $\beta $, and the complete cross-spectral data of $(-\Delta _g)^\beta +V$ relative to these regions. For coinciding regions, if the potentials vanish there and the geometric and regularity hypotheses of Feizmohammadi et al. (2026, Invent. Math.) hold, the map determines $\alpha $, $\beta $, the metric (up to an isometry), and the potential simultaneously. Our argument builds on the work of Li (2026, Canad. Math. Bull., 1–15) by combining the high-frequency behavior of the resolvent, Weyl’s law for the eigenvalues of the operator $(-\Delta _g)^\beta +V$, the unique continuation property of fractional operators with potentials, and the fractional Calderón theorem of Feizmohammadi et al. (2026). A probabilistic formulation of the inverse problem is also provided, and we prove that the corresponding uniqueness conclusions hold when the equation is driven by additive temporal white noise with a prescribed spatial profile.

Ill-posed inverse problems occur everywhere in the sciences, including medical imaging, radar, astronomy, etc., yielding underdetermined or ill-posed linear (non-linear) reconstruction problems. There are now a myriad of techniques to design decoders/reconstruction methods that can tackle such problems, ranging from optimisation-based approaches, such as compressed sensing, to data-driven techniques such as deep learning (DL) and variants in between the two techniques. The variety of methods begs for a unifying approach to determine the existence of optimal decoders and fundamental accuracy bounds in order to facilitate a theoretical and empirical understanding of the performance of existing and future methods. Such a theory must allow for both single-valued and set-valued decoders, as underdetermined and ill-posed inverse problems typically have multiple solutions. Indeed, set-valued decoders arise due to non-uniqueness of minimisers in optimisation problems, such as in compressed sensing, and for DL-based decoders in generative adversarial models, such as diffusion models and ensemble models. In this work, we provide a framework for assessing the lowest possible reconstruction accuracy in terms of worst-case and average errors. The universal bounds only depend on the measurement model $F$, the model class $\mathscr{M}_1$ and the noise model $\mathscr{E}$. For linear $F$, these bounds depend on its kernel, and in the non-linear case, the concept of kernel is generalised for undersampled and ill-posed settings. Additionally, we provide set-valued variational solutions that obtain the lowest possible reconstruction error.

We study an inverse problem for a magnetic fractional Schrödinger equation with both a linear potential $q(x)$ and a nonlinear potential $a(x,u)$ analytic in u. The magnetic potential $A(x)$ is assumed to be known. We show that the exterior Dirichlet-to-Neumann map uniquely determines both the linear potential q and the full nonlinearity a. The proof relies on higher-order linearization and the Runge approximation property for the magnetic fractional Laplacian. Our work extends the linear magnetic Calderón problem to the nonlinear setting and provides a uniqueness result for recovering multiple unknown coefficients simultaneously.

Let $\mathbb {X}$ bea Riemannian symmetric space of non-compact type. For a real-valued $f \in L^1(\mathbb {X})$ with $\int _{\mathbb {X}} f \ge 0$, we prove that if f satisfies the iterated convolution inequality:

$$ \begin{align*} f \ge \sum_{n=2}^N a_n \left(*^n f\right),\:\:\text{ a.e. on } \mathbb{X}, \end{align*} $$

$N \ge 2$

$2 \le n \le N$

$a_n$

$\int _{\mathbb {X}} f \le t_{\mathcal {Q}}\:$

$\mathcal {Q}(t):=t-\sum _{n=2}^N a_n\:t^n$

$t_{\mathcal {Q}}$

$\mathcal {Q}'$

$(0,\infty )$

$\mathbb {R}^d$

$\mathbb R^d$

We study pathwise well-posedness of the stochastic heat equation (SHE) with a multiplicative noise on the circle. By combining the convolution Young and rough integration theory, introduced by Gubinelli and Tindel (2010), with the random tensor estimate approach to pathwise well-posedness of stochastic dispersive PDEs with multiplicative noises, introduced by Chapouto and the second and third authors (2026), we establish pathwise well-posedness of SHE in both the Young and rough cases, improving the results in Gubinelli and Tindel (2010). In particular, in the rough case (= the white-in-time case), our result covers the case of almost space-time white noise, thus establishing an optimal result within the framework of one-parameter rough paths.

We introduce a fractional Bessel process with constant drift, defined as a time-changed Bessel process via the inverse of a stable subordinator, independent of the base process. This construction yields a model capable of capturing subdiffusive behavior and long-range dependence, relevant in various complex systems. We derive an explicit spectral representation of its transition density, extending the non-fractional setting. Using this representation, we establish several analytical properties of the process, including its stationary distribution and correlation structure. These results provide new insights into the behavior of fractional diffusions and offer analytical tools for applications in queueing theory, mathematical finance, and related domains. In particular, we demonstrate their applicability through a concrete problem in queueing theory.

We use the relation obtained by Guillarmou and Guillopé [‘The determinant of the Dirichlet-to-Neumann map for surfaces with boundary’, Int. Math. Res. Not. IMRN 2007(22) (2007), Article no. rnm099] and Chang and González [‘Fractional Laplacian in conformal geometry’, Adv. Math. 226(2) (2011), 1410–1432], between the generalised eigenvalue problem for asymptotically hyperbolic (AH) manifolds and the conformal Laplacian, to obtain a new inverse scattering result: on an AH manifold of dimension $n+1$ with constant scalar curvature $-n(n+1)$, the scattering matrix at energy $(n+1)/2$ determines the jet of the metric on the boundary, up to a diffeomorphism and conformal factor.

In this paper, we study a nonlinear free boundary problem modelling the growth of radially symmetric tumours. The tumour consists of a central necrotic core, an intermediate quiescent layer and an outer proliferating shell. The evolution of tumour layers and the movement of the tumour boundary are totally governed by external nutrient supply and conservation of mass. The three-layer structure generates three free boundaries with discontinuous nutrient consumption rates and cell growth rates. We develop a nonlinear analysis method to clarify the interactive relationships among free boundaries. By carefully studying the dependence of the critical-state tumour growth rate on the external nutrient concentration, we reveal the evolutionary mechanism in tumour growth and the mutual transformation of its internal structures. The existence and uniqueness of the radial stationary solution is proved, and its globally asymptotic stability towards different dormant tumour states is established.

We formulate an inverse problem for an uncoupled space–time fractional Schrödinger equation on closed manifolds. Our main goal is to determine the fractional powers and the Riemannian metric (up to an isometry) simultaneously from the knowledge of the associated source-to-solution map. Our argument relies on the asymptotic behavior of Mittag-Leffler functions, Weyl’s law for the eigenvalues of the Laplace–Beltrami operator, the unique continuation property of the space-fractional operator and the disjoint data metric determination result for the wave equation. We also provide a probabilistic formulation of our inverse problem.

We study the long-time behaviour of solutions to a free boundary model in river networks with small water flow speeds. First, we establish the well-posedness of our model. Under suitable assumptions, we then establish a spreading–vanishing dichotomy result for solutions to our model. By introducing a parameter $\sigma $ in the initial data, we derive sharp threshold behaviours.

We describe a new method to reconstruct the permittivity distribution, of an object to image, from the remotely measured electromagnetic field. We propose to use the remote fields measured before and after injecting locally in the medium plasmonic nanoparticles. Such a technique is known in the framework of imaging using contrast agents where, in optical imaging, the nanoparticles play the role of these contrast agents. The plasmonic nanoparticles are known to enjoy resonant effects, as enhancing the applied incident field, while excited at certain particular frequencies called plasmonic resonances. These resonant frequencies encode the values of the unknown permittivity at the location of the injected nanoparticles. The imaging methods we propose mainly use this resonant effect. We show that the imaging functional build up from contrasting the fields before and after injecting the nanoparticles, measured at one single back-scattered direction, and in an explicit band of incident frequencies, reaches its maximum values, in terms of the incident frequency, precisely at the mentioned plasmonic resonances. Such a behaviour allows us to recover these plasmonic resonances from which we recover the point-wise values of the permittivity distribution. In this work, we describe the method and provide the mathematical justification of this resonant effect and its use for the optical inversion using plasmonic nanoparticles as contrast agents.

With a mere usage of well-established properties of paradifferential operators, the conjugacy equations in several model KAM problems are converted to parahomological equations solvable by standard fixed point argument. Such discovery greatly simplifies KAM proofs, renders the traditional KAM iteration steps unnecessary, and may suggest a systematic scheme of finding quasi-periodic solutions of realistic magnitude.

In this paper, we use a degenerate reaction-diffusion system with free boundaries to model the spatial spread of rabies among foxes, whose population is divided into three sub-populations: susceptible ($S$), infected ($E$), and rabid ($I$). Based on established biological observations, susceptible and infected foxes are assumed to be territorial (random diffusion is ignored for $E$ and $S$ in the model), whereas rabid foxes disperse randomly (random diffusion is assumed for $I$), causing the spread of the disease. While $S$ evolves over the entire real line $\mathbb{R}$, $E$ and $I$ are found only in the infected region represented by an interval $[g(t), h(t)]$, which expands with moving fronts $x = g(t)$ and $x = h(t)$ as time $t$ increases. We show that this system admits a unique global solution and then analyse its dynamics and establish a spreading-vanishing dichotomy in certain natural parameter regimes. Moreover, we supply some simple sufficient conditions for the vanishing and spreading of the rabies, respectively. For example, we show that if the corresponding ODE system has basic reproduction number $\mathcal{R}_0 \gt 1$, then a spreading-vanishing dichotomy holds, and the outcome depends on the initial size of the infected region, while if a certain quantity $\mathcal R_0^* \in (\mathcal R_0,\infty)$ is no bigger than 1, then the rabies will always vanish. The degenerate nature of the model, combined with the evolving infected region, causes considerable difficulties in the mathematical treatment, both in proving the well-posedness and in understanding the long-time dynamics. This paper appears to be the first to treat a free boundary model where one reaction-diffusion equation is coupled with two ordinary differential equations.

We investigate the qualitative properties of positive solutions to mixed local–nonlocal equations with indefinite nonlinearities, emphasizing the interaction between classical and fractional Laplacians. We first establish maximum principles and prove strict monotonicity along the $x_1$-direction for mixed elliptic operators. By combining a mollified first eigenfunction with a suitable sub-solution, we derive nonexistence results for the mixed operator ${(-\Delta )}^s - \Delta $ via a contradiction argument. These results are further extended to the parabolic setting, incorporating both the Marchaud-type fractional time derivative and the classical first-order derivative, revealing new qualitative features under dual nonlocality. A key aspect of our approach is a careful adaptation of the method of moving planes to the mixed local–nonlocal context. By addressing the distinct scaling behaviors of local and nonlocal terms, the method yields monotonicity and Liouville-type results without standard decay assumptions and provides a framework potentially applicable to a broader class of mixed elliptic and parabolic problems.

For $h>1,$ we consider the reaction-diffusion equation:

$$ \begin{align*} \Delta_\infty^hu(x)=f(x,u(x),Du(x)),\quad x\in\Omega, \end{align*} $$

$\Delta _\infty ^h$

$f\in C(\Omega \times \mathbb {R}\times \mathbb {R}^n)$

$ 0\leq f(x,\delta t,p)\leq \Lambda (x)\delta ^{\gamma }f(x,t,p),$

$\Lambda (x)\in C(\overline {\Omega }), \, \gamma \in [0,h), \,t>0$

$\delta>0 $

$C^{({h+1})/({h-\gamma })}$

$\partial \{u>0\}\cap \Omega .$

$\gamma =h,$

We consider a toy model of rate-independent droplet motion on a surface with contact angle hysteresis based on the one-phase Bernoulli free boundary problem. We introduce a notion of solutions based on an obstacle problem. These solutions jump “as late and as little as possible”, a physically natural property that energy solutions do not satisfy. When the initial data is star-shaped, we show that obstacle solutions are uniquely characterized by satisfying the local stability and dynamic slope conditions. This is proved via a novel comparison principle, which is one of the main new technical results of the paper. In this setting we can also show the (almost) optimal $C^{1,1/2-}$-spatial regularity of the contact line. This regularity result explains the asymptotic profile of the contact line as it de-pins via tangential motion similar to de-lamination. Finally we apply our comparison principle to show the convergence of minimizing movements schemes to the same obstacle solution, again in the star-shaped setting.

We consider an infinite-dimension SIS model introduced by Delmas, Dronnier, and Zitt, with a more general incidence rate, and study its equilibria. Unsurprisingly, there exists at least one endemic equilibrium if and only if the basic reproduction number is larger than 1. When the pathogen transmission exhibits one way propagation, it is possible to observe different possible endemic equilibria. We characterize in a general setting all the equilibria, using a decomposition of the space into atoms, given by the transmission operator. We also prove that the proportion of infected individuals converges to an equilibrium, which is uniquely determined by the support of the initial condition. We extend those results to infinite-dimensional SIS models with reservoir or with immigration.

In this work, we study a class of elliptic problems involving nonlinear superpositions of fractional operators of the form

\begin{equation*}A_{\mu,p}u := \int_{[0,1]} (-\Delta)_{p}^{s} u \, d\mu(s),\end{equation*}

$\mu$

$[0,1]$

$p$

$p$

$p$

$p$

We consider the following problem

\begin{equation*}\varepsilon^{2 s} P_{\hat{h}}^s u+u=u^p, \quad u \gt 0 \quad \text {on } \,(M, \hat{h}),\end{equation*}

$(X, g^{+})$

$(M,[\hat{h}])$

$s\in (0,1)$

$P_{\hat{h}}^s$

$1 \lt p \lt \frac{n+2s}{n-2s}$

$\varepsilon$

$C^1$

$H$

$0 \lt s \lt {1}/{2}$

$C^1$

${1}/{2}\le s \lt 1$

We consider the existence of normalized solutions to non-linear Schrödinger equations on non-compact metric graphs in the L2-supercritical regime. For sufficiently small prescribed mass (L2 norm), we prove existence of positive solutions on two classes of graphs: periodic graphs and non-compact graphs with finitely many edges and suitable topological assumptions. Our approach is based on mountain pass techniques. A key point to overcome the serious lack of compactness is to show that all solutions with small mass have positive energy. To complement our analysis, we prove that this is no longer true, in general, for large masses. To the best of our knowledge, these are the first results with an L2-supercritical non-linearity extended on the whole graph and unravelling the role of topology in the existence of solutions.