Papers

Upon fixing the adiabatic index, the spherically symmetric liquid Lane-Emden stars governed by the Euler-Poisson system with a "stiffened gas" equation of state $p=\rho^\gamma-1$ form a one-parameter family, naturally parametrised by the central density $\kappa=\bar\rho_\kappa(0)\in(1,\infty)$. In contrast to the gaseous case, the liquid free boundary breaks the self-similarity of the family and bends the mass-radius curve, creating extrema of the mass when $\gamma<2(d-1)/d$. We formulate a turning point principle in the spirit of Zel'dovich, Wheeler and Thorne, and of its rigorous relativistic counterpart from the works of Hadžić, Lin and Rein: the number of radial growing modes equals the negative Morse index of the linearised operator, is locally constant along the family, and can change only at the turning points of the mass-radius curve, the bending orientation there dictating whether a growing mode is gained or lost. We describe the large central density limit in which the gaseous tail (read off from a planar dynamical system) determines whether the mass-radius curve spirals, and hence whether the number of growing modes remains bounded or tends to infinity. As corollaries we recover and sharpen the known radial (in)stability results for liquid Lane-Emden stars and, in combination with existing non-radial analysis, obtain a turning point criterion for non-radial stability.

We construct global weak solutions of the inviscid surface quasi-geostrophic equation in $\mathbb R^2$ and in smooth bounded domains, for arbitrary initial data in the critical Lorentz space $L^{4/3,2}$. The solutions conserve the Hamiltonian $\|\Lambda^{-1/2}\theta(t)\|_{L^2}^2$ for all times. The second Lorentz exponent is determined by the sharp boundedness $\Lambda^{-1/2}:L^{4/3,2}\to L^2$; the corresponding estimate fails for $L^{4/3,q}$ when $q>2$. We use an approximation scheme that is tailored for Lorentz spaces. It smooths the advecting velocity and the initial data, and preserves order in distribution functions. Removing the approximation is made possible by uniform bounds for the Lorentz norms of high amplitude cutoffs of the solutions.

Let $\Omega$ be a bounded Lipschitz domain of $\mathbb R^N$, $N\geq 2$. In this paper, we study the asymptotic behavior of the first Robin eigenvalue of the $p$-Laplace operator as $\beta$ goes to $0$ and as $\beta$ goes to $+\infty$, deriving sharp asymptotic expansions of the eigenvalue; the expansion in the Dirichlet limit $\beta\to+\infty$ is obtained under the additional assumption that $\partial\Omega$ is of class $C^{1,1}$.

We study optimal control problems for a stationary reaction–diffusion model describing the spatial distribution of a tumor cell population with Gompertz growth. The control $m(x)$ represents a treatment term acting as a density-dependent removal rate and it is subject to $L^{1}-L^{\infty}$ constraints. When the intrinsic growth rate is constant, the uniform distribution of the treatment is shown to be the unique minimizer. For the maximization problem, we prove that every optimal control is of bang-bang type. In addition, we show that in the one dimensional case and for sufficiently large diffusion rates, the positivity set of optimal controls is an interval sticking to one of the extrema of the domain. Finally, numerical simulations complement the theoretical analysis and explore regimes that are not fully covered by the results proved in the paper. The computations confirm the bang-bang structure of maximizers, and illustrate how the shape of optimal controls and the associated states are affected by spatial heterogeneity in the growth rate, localized admissible treatment regions, and the diffusion coefficient. Moreover, they reveal a monotone dependence of the optimized total population on the diffusion coefficient: this is a new phenomenon with respect to the logistic setting.

We prove a local well-posedness criterion for the Vlasov–Poisson system on $\R^d_x\times\R^d_v$, $d\geq2$, under an anisotropic assumption on the initial distribution. The datum has finite mass, its weighted velocity supremum belongs to $L^p_x$ for some $p>d$, and it has an arbitrarily small positive Hölder regularity in the velocity variable, uniformly with respect to velocity and with the same spatial $L^p$ control. The main estimate is a nonlinear mixing bound \[ [\rho(t)]_{C^\alpha_x}\lesssim t^{-d/p-\epsilon}C(f_0), \qquad \epsilon>0~~\text{small}. \] Thus the density is integrable in time with values in a positive spatial Hölder class, and the corresponding electric field belongs to $L^1_tC^{1,\alpha}_x$. We construct a solution by a Schauder fixed point on the density and prove uniqueness by a Loeper-type stability estimate.

Based on a 1981 work by G. Talenti, which established a comparison principle for smooth solutions of the Monge-Ampère equation alongside related two-dimensional a priori estimates, this paper explores the connection to the Blaschke-Santaló inequality and its reverse form by K. Mahler (1939). More specifically, in this work, a quantitative version of one of Talenti's a priori estimates is proved. We exploit the interplay between Talenti's analytical framework and Mahler/Blaschke-Santaló inequalities via convex geometry, demonstrating how this link can be utilized to gain sharp insights into the Alexandrov-Bakelman-Pucci (ABP) maximum principle for elliptic PDEs.

In this paper we study an equation driven by a non-local operator of order "near zero" with mixed Dirichlet-Neumann boundary conditions, modelling a biological system consisting of a population self-competing for resources in a closed environment and subject to an extremely heavy-tailed dispersal process. We establish several results concerning the existence of states in which the population can survive requiring different assumptions on the non-linear reaction term and on the various configurations of the sets where the Dirichlet or Neumann conditions hold.

Motivated by the Onsager statistical mechanics description of turbulent Euler flows with point singularities, we refine the blow up analysis for sequences of solutions of a class of perturbed singular Liouville equations which share the phenomenon of "blow up and concentration without quantization". The problem is delicate because we are dealing with the exact threshold value above which one meets the well known "concentration without quantization" phenomenon, as recently pushed forward in [C.S. Lin, G. Tarantello, C. R. Math. Acad. Sci. Paris (2016)] and in [Y. Lee, C.S. Lin, G. Tarantello, W. Yang, Comm. PDE. (2017)]. First of all we need a new sharp Harnack type inequality for this particularly rich singular limit. However this is not enough, since the growth of the conformal factor inherited by the singularity prevents the use of classical quantization arguments. We solve also this issue with different strategies for "fast" and "slow" blow up, by a careful adaptation of arguments based on the Pohozaev identity, elliptic estimates and "Sup+CInf" inequalities in the same spirit of [C.C. Chen, C.S. Lin, Comm. An. Geom. (1998)].

Phase separation phenomena occurring in complex fluids are known to be sensitive to hydrodynamic effects: when a binary fluid mixture undergoes spinodal decomposition, the emerging pattern of domains is no longer dictated by interfacial dynamics alone, but is reshaped by advection. In several applications of interest, advection effects may even cause the onset of turbulence, further affecting the segregation process on a wide range of spatial scales. Motivated by this, we propose a Cahn-Hilliard model driven by transport noise able to capture the intrinsic stochasticity of turbulence. The model is analyzed in its thermodynamically-relevant framework, namely employing a singular Flory-Huggins potential and a possibly degenerate mobility, and the noise is considered both in Ito and Stratonovich form. In this work, as a first-step investigation, we establish well-posedness of the system in two and three spatial dimensions and its thermodynamical consistency.

We formulate a proof on non-Archimedean analogue of the Caffarelli-Silvestre extension for fractional Laplacian on the Bruhat-Tits tree $T_p$ in terms of energy relations after identifying $\partial T_p$ with $\mathbb{P}^1(\mathbb{Q}_p)$, which reproduces the generalised Vladimirov derivative of the rescaled boundary function.

In this paper, which is a continuation of our recent paper [1], we give two new definitions of a weak solution of the one-dimensional, elliptic, nonlinear singular problem which is formally written as $$ \begin{cases} \displaystyle -\frac{d}{dx}\left(a(x) \frac{d u}{dx}\right) = - \frac{d \phi (u) }{dx}- \frac{d g(x) }{dx}& \text{in}\;(0,L),\\ \newline u(0)=u(L)=0\,, & \end{cases} $$ where $\phi:\mathbb{R}\mapsto\mathbb{R}\cup\{+\infty\}$ is a singular function whose model is given by $\displaystyle\phi(s)=\phi_\gamma (s)=\frac{1}{|s|^{\gamma}}$ with $\gamma>0$. In these two definitions the solutions belong respectively to the space $u\in W_0^{1,1}(0,L)$ with $\phi(u)\in L^1(0,L)$, and to the space $u\in W_0^{1,m}(0,L)$ with $\phi(u)\in L^m(0,L)$ for $m>1$. In these frameworks we state and prove results of non-existence, existence, and non-isolation of the solutions.

We study the concentration of the field between two nearly touching high-index dielectric resonators in three dimensions. The model is a scalar Helmholtz transmission problem in the resonant regime, wherein the wavelength inside the resonators of the same order as their typical diameter. The material contrast enters only a lower-order term of the equation and not its principal part. As a consequence, the gradient of the field stays bounded independently of the distance separating the particles, and does not blow up. Nevertheless, simulations and experiments show that the gradient still concentrates in the gap: as the particles approach, it grows like the inverse of their separation over an intermediate range of distances and then saturates once they are very close. We explain this pre-asymptotic effect through a weak-coupling regime, in which the resonant modes of the pair are perturbations of the modes of each isolated particle. When such a mode keeps a nonzero contrast between the two facing boundaries, a mean value argument across the gap accounts for the growth of the gradient; this requires strengthening the standard spectral perturbation estimates from an average to a pointwise control. The growth stops once the interaction between the particles is no longer weak and the perturbation theory fails. Numerical experiments confirm the transition from amplification to saturation.

We give a new criterion to show optimality of Hardy weights for general operators on graphs via the supersolution construction. For Laplacians on graphs without killing terms this always gives rise to an optimal Hardy weight via the Green's function without any further assumptions. Furthermore, in contrast to earlier results, our result is not restricted to locally finite graphs. This allows us in particular to obtain optimal Hardy weights for the fractional Laplacian on general graphs. For the fractional Laplacian on the Euclidean lattice, we then obtain an optimal Hardy weight with the constant and asymptotics as it is expected from the continuous setting.

In this paper, we study a stochastic parabolic-parabolic Keller-Segel system driven by nonlocal, nonlinear multiplicative noise in a two-dimensional bounded domain. Under suitable assumptions, we establish global existence and pathwise uniqueness of a strong solution for arbitrary initial data, without imposing any smallness conditions. This sharply contrasts with the deterministic two-dimensional Keller-Segel system, which typically requires smallness assumptions on initial mass. The main analytical challenges stem from the fully parabolic coupling, combined with a lack of coercivity and global Lipschitz continuity in both the chemotactic drift and noise terms. To overcome this, we introduce a tailored truncated system to establish local existence via Banach's fixed point theorem. Using this local existence and pathwise uniqueness, we construct a maximal local strong solution. Finally, by introducing a specialized Lyapunov functional, we derive uniform estimates to extend this solution globally.

We develop heat-flow estimates in critical homogeneous Triebel–Lizorkin spaces and apply them to two-dimensional Keller–Segel type equations. For $q>2$, we show that the heat-flow from $\dot F^0_{1,q}(\mathbb R^2)$ to $L^2(0,T;\dot F^1_{1,q}(\mathbb R^2))$ is unbounded. This motivates the introduction of new spaces, time-refined Triebel–Lizorkin spaces, in which the time norm is taken before the dyadic summation. In this framework, we establish a family of heat smoothing estimates together with the corresponding maximal regularity. We further prove Banach-valued Peetre estimates, Banach-valued Jawerth-type estimate, homogeneous Poisson estimate, and an endpoint bilinear estimate for the Keller–Segel drift. As an application, we obtain local well-posedness for arbitrary initial data and global well-posedness for sufficiently small initial data in the critical space $\dot F^0_{1,q}(\mathbb R^2)$, $2\le q<\infty$, for the two-dimensional parabolic–elliptic Keller–Segel equation. The same analytic framework also yields analogous well-posedness results for a related two-component drift–diffusion system.

The main results of this paper are threefold. First, we prove an observability inequality for the Airy equation on the real line from Hausdorff-thick sets in a time-trace sense for every observation time $T>0$. The observation functional is a block supremum of $L^2(0,T)$ time traces over the Hausdorff-thick set. Second, we prove observability inequalities for the Airy equation on the real line with observations on some periodic sets, which in particular yields observability on a class of spatial point sequences. Third, we give a necessary and sufficient condition for finite point observability for the Airy equation on the torus with a bounded real-valued potential. Indeed, for a finite observation set $F$, a Kalman rank condition on a finite-dimensional invariant subspace is found. As a corollary, we obtain sharp point observability results for the Airy and linear KdV equations on the torus. Moreover, for potentials with finite order regularity assumptions we prove that every observation set with an accumulation point, in particular any set of positive Hausdorff dimension, gives an observability inequality for each observation time $T>0$. Since the Airy equation has neither high frequency exponential decay nor pointwise smoothing effects, which are essential in recent works on Hausdorff type observation results on heat equations, we introduce several new ideas adapted to the Airy case.

We study the semiclassical propagation of edge-states for a two-dimensional Dirac operator with a spatially varying mass. The zero set of the mass models the interface between topological phases and is allowed either to consist of two disjoint smooth curves or to possess an isolated singular point. For disjoint interfaces, we prove that a wave packet initially localized on one component remains confined near this interface over long semiclassical time scales and we quantify the influence of the distance between the two components on the accuracy of the approximation. For singular interfaces, we determine the regime in which the wave-packet approximation remains valid as the packet approaches the singularity and identify the scaling at which our estimates lose their uniformity.

This paper studies a free boundary problem for the Boltzmann equation that models the interaction of rarefied gas with a moving wall—a classical piston problem in kinetic theory. The piston motion is governed by Newton's law under the drag force exerted by the gas, with or without an additional Hookean restoring force. A central challenge for such a problem is the strong coupling between the kinetic equation and the free moving boundary, for which the classical Lagrangian formulation is unavailable due to low-regularity of solutions. Our approach relies on two new ingredients: a conformal transformation is introduced to reduce the moving domain to a fixed domain, and a coupled energy structure linking the kinetic distribution and free boundary variables is uncovered to reveal intrinsic dissipation and cancellation mechanisms at the interface. These structural observations lead to a global nonlinear theory for the fully coupled system. As a result, we establish the global existence, uniqueness, nonlinear stability, and exponential convergence to equilibrium of solutions near a global Maxwellian. This provides the first rigorous global well-posedness theory for a fully coupled Boltzmann free boundary problem of piston type.

In this paper, we investigate the following quasi-linear weighted $N$-Laplacian Liouville equation \begin{equation*}\label{0} -\Delta_N u=|x|^{N\alpha}e^{u}, \qquad x\in \R^N, \end{equation*} where $N \geq 2$. For $\al>0$, by carefully studying the linearized problem and applying the approximation method and bifurcation theory, we prove that, when the parameter $\alpha$ equals to the critical values $\alpha(k):=\frac{\sqrt{k(N-1)(k+N-2)}}{N-1}-1$ for $k \geq 2$, there exist non-radial solutions $u$ (bifurcating from $U_{\alpha(k)}$) to the above quasi-linear Hénon type Liouville equation such that $u\sim \ln|x|$, $|\nabla u|= O(|x|^{-1})$ at $\infty$ and $\int_{\R^N}|x|^{N\alpha}e^{u}\md x=N\left(\frac{N^2}{N-1}\right)^{N-1}(\alpha+1)^{N-1}\omega_N$. One should note that, $\alpha(k)=k-1$ for $k\geq2$ when $N=2$. Our results successfully extend the existence result of J. Prajapat and G. Tarantello in \cite{PT} concerning the $2$-dimension and Laplacian case (i.e., $N=2$) to the more general $N$-dimension and $N$-Laplacian cases ($N\geq 2$), and extend the results of F. Gladiali, M. Grossi, and S. L. N. Neves in \cite{GGN} and the authors in \cite{DDGL} from $1<p<N$ to the much more complicated limiting case $p=N$. We introduced some new ideas and overcame a series of crucial difficulties, including the nonlinearity nature of the $N$-Laplacian $\Delta_N$, the lack of Green integral representation formula and critical weighted Sobolev embedding inequality, the absence of Kelvin type transforms for linearized/difference equations, the invariance of the total mass under scalings of $u$, and the signs-changing and divergence (to $-\infty$) at $\infty$ of the solutions, which makes the suitable choices of the approximate problems, the (normalized) approximate function sequences and the working space to be quite difficult.

We construct explicit wave packet frames adapted to Euclidean balls and use them to obtain quantitative eigenvalue estimates for spatio–spectral limiting operators. Let \(d\geq 2\), let \(B_d(R)\subset \R^d\) be the Euclidean ball of radius \(R\), and let \(S\subset \R^d\) be a measurable set such that $\partial S$ has finite $(d-\eta)$-upper Minkowski content for $0 < \eta \leq 1$. We construct a unit-norm frame for \(L^2(B_d(R))\), with frame bounds depending only on the dimension $d$, whose elements are adapted to the radial and angular geometry of the ball. We prove quantitative Fourier localization estimates for this frame: Relative to \(S\), the frame decomposes into packets concentrated in \(S\), packets concentrated in \(\R^d\setminus S\), and an exceptional family whose cardinality is bounded explicitly in terms of \(R\), and the Minkowski content of \(\partial S\). As an application, we derive an upper bound for the plunge region of the spatio-spectral limiting operator associated to the sets $B_d(R)$ and $S$.

We study the weak stability of the kinetic Cucker-Smale (in short, KCS) model in a phase-spatially extended setting, which can be formally derived from the infinite Cucker-Smale model in the mean-field limit. For a bounded Lipschitz communication weight function, we derive finite-time Osgood-type weak stability for measure-valued solutions with exponential velocity tails and finite spatial second moments. Unlike the phase-spatially confined setting, the solution operator to the KCS model is not Lipschitz continuous with respect to initial data. This is due to the unbounded velocity tail and the corresponding absence of a uniform Lipschitz bound for the alignment force. As an application of weak stability, we obtain an i.i.d. sampling consequence: empirical measures generated from independent initial samples converge to the measure-value solution for the corresponding kinetic model in any finite time interval, in expectation.

A maximum principle for certain strictly convex functions belonging to $W^{2,1}_{\rm loc}(\Omega)\cap C^1(\Omega)$ is established and then applied to obtain a $W^{2,1 +\eps}_{\rm loc}(\Omega)$-estimate.

Given an embedded shrinker $\Sigma$ in $\mathbb{R}^{n+1}$ that is either closed, asymptotically conical, or a Cartesian product of such a shrinker with $\mathbb{R}^k$, we construct a closed embedded mean curvature flow whose tangent flow at the first singularity is modeled on $\Sigma$. We also prescribe the first-order asymptotics of the tangent flow. This result is a consequence of a more general theorem that allows us to construct mean curvature flows with an additional force whose tangent flow and first-order asymptotics at the first singularity are prescribed.

We consider a three-dimensional fluid-structure interaction problem coupling the incompressible Navier-Stokes equations in a time-dependent domain with a square-root damped plate equation, which is posed on the moving upper boundary of the fluid. We prove, a priori, the exponential decay of strong solutions for initial data that are sufficiently small in a suitable Sobolev space. The proof combines higher-order energy estimates, Stokes-type regularity bounds, and a nonlinear bootstrap scheme that closes under the smallness assumption.

We develop a pluripotential approach to complex Hessian equations on compact Hermitian manifolds. In this setting, the lack of closedness of the background metric introduces torsion terms that prevent a direct extension of the Kähler theory. Our main result is a uniform $L^\infty$ estimate for bounded $\omega$-$m$-subharmonic solutions of the equation \[ (\omega + dd^c u)^m \wedge \omega^{n-m} = cf\,\omega^n, \] under the assumption that $f \in L^p$, $f \ge 0$ for some $p>1$. The proof combines a weak comparison principle with torsion error, a capacity theory adapted to the Hermitian setting, and a nonlinear iteration scheme controlling the decay of sublevel sets. As applications, we obtain existence, stability and compactness results for weak solutions with $L^p$ densities. These results extend several aspects of the pluripotential theory. of complex Hessian equations beyond the Kähler framework.

We study continuous data assimilation for the two-dimensional Navier–Stokes equations on a smooth, bounded, connected domain with Navier-slip boundary conditions, using no interior observations. The available data consist only of finite-dimensional measurements of the tangential velocity on a non-empty relatively open subset $\Gamma\subset\partial\Omega$. We prove that sufficiently strong boundary feedback, constructed from sufficiently fine observations, generates a coercive spectral gap for the assimilation error. The limiting gap is identified with that of a mixed-boundary problem obtained by imposing a homogeneous Dirichlet condition on $\Gamma$, and is shown to be of order $\nu$. Combining this feedback-induced coercivity with an estimate of the non-linear error production in terms of the long-time averaged symmetric-gradient energy of the reference solution, we obtain a sufficient criterion for exponential synchronisation. We verify this criterion in the unforced case, for sufficiently small forcing when the unnudged Navier-slip form has an $\mathrm{L}^2$ spectral gap, for sufficiently large viscosity on domains without tangential rigid motions, and for sufficiently large viscosity in the presence of positive boundary friction. We also treat perfect slip on domains admitting tangential rigid motions, where synchronisation follows under a smallness condition on the non-rigid solenoidal component of the forcing.

In 1995, Brakke proposed a formulation of the Plateau problem for a given boundary $\Gamma$ that allows for triple junctions and tetrahedral singularities. Let $\Gamma$ be a smooth closed curve and let $G=\pi_1(\mathbb{R}^3\setminus \Gamma)$. For each proper finite index subgroup $N$ of $G$, Brakke constructs a $(\mathrm{\mathbf{M}}, 0, \infty)$-minimal surface $\Sigma_N$, which is obtained as the projection of the boundary of a perimeter-minimising fundamental domain in the covering space associated to $N$. We advance the theory in two ways. Firstly, we extend Brakke's construction to include all normal subgroups $N\triangleleft G$, i.e. possibly with infinite index. Secondly, we prove a compactness result which implies that there exists a proper normal subgroup $N_0\triangleleft G$ such that \[ \mathrm{Area}(\Sigma_{N_0})=\inf_{\{N\triangleleft G, N \neq G\}}\ \mathrm{Area}(\Sigma_N). \] A similar result holds when $\Gamma$ has many connected components. Furthermore, we study the spanning and minimising properties of the $\Sigma_N$s and $\Sigma_{N_0}$.

We construct a family of bounded, smooth, simply connected, nonconvex planar domains $\Omega_{a,\epsilon}$. Let $u_{a,\epsilon}$ be the corresponding torsion function satisfying \[ -\Delta u_{a,\epsilon}=1\quad\text{in }\Omega_{a,\epsilon},\qquad u_{a,\epsilon}=0\quad\text{on }\partial\Omega_{a,\epsilon}. \] There exists \(a^*\in(0,1)\) such that, for every \(a\in(a^*,1)\) and all sufficiently small \(\epsilon>0\), the origin is a unique global maximizer of $u_{a,\epsilon}$. Moreover, \[ \lim_{a\downarrow a^*}\lim_{\epsilon\to0} \lambda_{\max}\bigl(D^2u_{a,\epsilon}(0)\bigr)=0. \] In addition, the ratios $\text{diam}(\Omega_{a,\epsilon}) / \text{inrad}(\Omega_{a,\epsilon})$ are uniformly bounded. Hence the Hessian estimate proved by Steinerberger (J. Funct. Anal. 274, 1611–1630, 2018) for convex planar domains cannot be extended to smooth, simply connected, nonconvex planar domains. \vskip0.2cm Our domains are star-shaped, symmetric with respect to both coordinate axes and convex in the horizontal direction. When the limiting slit is sufficiently long, we further show that certain superlevel sets of the torsion function are not star-shaped. This strengthens the counterexample to the star-shapedness question raised by Gladiali and Grossi (Amer. J. Math. 144, 1221–1240, 2022) under stronger geometric assumptions.

We construct three counterexamples for the special Lagrangian curvature equation (SLCE). First, in dimension two, we use a post-focal branch of a parallel surface with constant positive Gauss curvature to construct an explicit Lipschitz viscosity solution which is not $C^1$. Second, still in dimension two, we construct a sequence of smooth admissible solutions on a fixed rectangle with uniformly bounded $C^1$-norm but unbounded curvature at one point; furthermore, we show that any uniform $C^{1,\beta}$ estimate fails for $\beta > 1/3$. Third, in dimension three and in the subcritical phase, we construct a Mooney-Savin type Lipschitz viscosity solution whose gradient has a jump discontinuity across an analytic surface. These examples demonstrate the sharpness of the recent a priori estimates by Qiu and Zhou, revealing that their structural assumptions-convexity and the critical phase-are strictly necessary.

Let $\mathcal{L}_a=-\Delta_x-\Delta_y+\frac{a}{2}|x|^{-2}$ with $a>0$ denote the Schrödinger operator on $L^2(\mathbb{R}^2_x\times \mathbb{R}^n_y)$, which involves a singular partial inverse-square potential. The purpose of this manuscript is twofold. First, relying on the explicit representation for the spectral measure associated with the operator $\mathcal{L}_a$ established by Zhang-Zhang [J. Geom. Anal. 35(3), Paper No. 71, 27pp (2025)], we investigate the decay estimate for a class of dispersive semigroups of the form $e^{it\phi(\sqrt{\mathcal{L}_a})}$, where $\phi: \mathbb{R}^+ \to \mathbb{R}$ is a smooth function. To handle the technical difficulty arising from the inhomogeneity of the phase function $\phi$, we adopt the frequency localization and the stationary phase method. In the second part of the paper, we first derive boundary Strichartz estimates for the fractional Schrödinger operator $e^{it\mathcal{L}_a^\nu}$, $0<\nu\neq\frac{1}{2}$. As applications of the established decay estimates, we further obtain Strichartz estimates for some concrete wave equations associated with the operator $\mathcal{L}_a$, which corresponds to $\phi(r)=r, r^2, r^2+r^4, \sqrt{1+r^2}, \sqrt{1+r^4}$, and $r^\mu,0<\mu\neq 1$. Most notably, our results unify and simplify existing dispersive estimates for the operator $\mathcal{L}_a$, while extending the relevant theory to more general scenarios.