Papers

We examine a 4-dimensional generalization of predator-prey models describing interactions among four species. We show that these systems exhibit a rich variety of asymptotic behaviors, with solutions that are either integrable, leading to non-sustainable dynamics characterized by unbounded population growth or collapse, or chaotic, where all species coexist over time. Moreover, we review the integrable cases, which are classified into two-parametric families, and examine three-parametric families that behave similarly to the known two-parametric integrable cases. We also analyze chaotic Lotka-Volterra models in terms of their Lyapunov exponents and Poincare' surfaces of section, which turn out to reveal underlying structures that are compatible with what is expected from sustainable dynamics.

This paper deals with a certain family of series which is derived from generating functions of the Frobenius-Euler and Apostol type polynomials and numbers. When special cases of this series are examined, it is shown that it generates some well-known families of zeta functions, such as the Hurwitz zeta and the Lerch zeta, etc. We also give an open problem for these relation. We give analytic continuation of this function. By using the Cauchy Residue Theorem, We show that this series interpolates the Frobenius-Euler type polynomials at negative integers. Moreover, by applying the Laplace transform to the generating function for these polynomials, we get infinite series representations involving the Lerch zeta function and these polynomials. In addition, We show that this interpolation function can be representation in terms of the Lerch zeta function and the Hurwitz zeta function. Some special values for the results are also given.

In recent years, higher categorical variants of idempotents have become an object of interest. Notably, they appear when categorifying algebraic structures motivated by mathematical physics. Working in the setting of bicategories, we give a rigorous definition of the bicategorical idempotent completion as outlined by Gaiotto and Johnson-Freyd in arXiv:1905.09566. We show that this agrees with the Cauchy completion of a locally idempotent complete bicategory in the sense of Lawvere.

We investigate the structure of Poisson algebras of dimensions at most three over an arbitrary field. Our approach is based on the internal structure of the associated commutative associative and Lie algebras, with particular emphasis on the associative square $P^2$, the derived Lie algebra $[P,P]$, the Lie center, the associative annihilator, idempotents, ideals and decomposability. We obtain a complete classification in dimensions one and two and give a structural classification in dimension three. In dimension two, we prove that the associative and Lie multiplications cannot be simultaneously non-zero. In dimension three, the classification is organized according to the dimension and position of the derived Lie algebra and, in the case of trivial Lie multiplication, according to the dimension of $P^2$. The arbitrary-field setting leads to phenomena which do not occur over the complex field. In particular, quadratic and cubic field extensions appear naturally in the classification, some families depend on equivalence classes of symmetric bilinear forms and on the structure of three-dimensional Lie algebras over the ground field, and characteristic $2$ gives an additional family of Poisson algebras with both multiplications non-zero. Over the complex field, the resulting classification specializes, up to changes of basis and notation, to the known classifications in dimensions at most three.

We study ground-state solutions of the 4th order nonlinear Schrödinger equation with mixed dispersion via the elliptic profile equation $\Delta^2 Q + 2a \Delta Q + bQ - |Q|^\alpha Q=0$, focusing on symmetry breaking of least-action ground states in 2D in the regime $a=1$, $b=1+\epsilon$, $0<\epsilon \ll 1$, where nonradiality was proved by Lenzmann and Weth [26]. We establish a quotient-to-mass relation converting small-$\epsilon$ asymptotics of the Weinstein quotient into mass asymptotics, in any dimension and symmetry class. In 1D we prove the sharp small-$\epsilon$ rate for the quotient for every $\alpha>0$ and deduce the rates for the mass, action and potential norm. In two and higher dimensions the same relation, combined with the quotient asymptotics of Lenzmann-Weth [26] and Mandel-Oliveira e Silva [27], gives the mass rates for the Knapp-type nonradial and the radial ground states. The resulting threshold $\alpha_K(d)=8/(d+1)$ coincides for $d\leq3$ with the existence of normalized minimizers threshold of Fernández-Jeanjean-Mandel-Mariş [16], connecting the least-action and mass-constrained formulations. We complement this with a detailed numerical study. For the cubic nonlinearity we construct a nonradial branch from a Knapp-type example, compare it with radial and angular-mode branches, and verify the predicted rates. For the quartic nonlinearity, turning points and branching are present in the mass-energy diagrams in radial or two- and four-peak nonradial solutions. Here, although least-action ground states are nonradial for small $\epsilon$, the normalized ground states of the same mass appear radial, a consequence of branching. For $\alpha\geq4$ we find only radial ground states, consistent with the conjectured sharp form of the Stein-Tomas inequality on $\mathbb S^1$, equivalent to the radial and nonradial masses having the same leading coefficient.

We mechanized proof of Gödel's first and second incompleteness theorems, Solovay's arithmetical completeness theorem of \mathbf{GL}, and related results in the Lean 4 theorem prover.

Let $G=(V,E)$ be a graph. The eccentric connectivity index of $G$ is defined as $$\xi^C(G)=\sum_{u\in V(G)}d_uecc(u)$$ where $d_u$ and $ecc(u)$ are the degree and eccentricity of $u$, respectively. For a strongly connected digraph $D=(V,A)$, the eccentric connectivity index is defined as $$\xi^C(D)=\frac{1}{2}\sum_{u\in V(D)}(d_u^++d_u^-)mecc(u)$$ where $d_u^+$ and $d_u^-$ are the out-degree and in-degree of $u$, respectively, and $mecc(u)$ is its m-eccentricity with respect to the maximum distance $md(u,v)=\max\{\vec d(u,v),\vec d(v,u)\}$. In this article, give formulas and bounds for the eccentric connectivity index of Cartesian and strong products of strongly connected digraphs and discuss the corresponding equality cases. Also, an attempt is made to study the self-centeredness of these products and establish conditions under which the Cartesian and strong products are self-centered.The results are extended to products of several digraphs.

Integrable partial differential equations, such as the Korteweg–De Vries (KdV) equation, admit infinite hierarchies of commuting higher symmetries. Their symmetry reductions give rise to integrable finite-dimensional dynamical systems solvable in terms of Abelian functions. This underlies the finite-gap integration method for the KdV equation, introduced by S.P. Novikov and subsequently extended to many other integrable systems. In this paper, I introduce the concept of reduction ideals and a new class of reductions for integrable differential–difference equations, leading to integrable finite-dimensional systems in both commutative and noncommutative settings. This class includes periodic reductions, symmetry reductions, and a broad family of other reductions defined in terms of the associated Lax operator. The reduction constraints define integrable maps that enable solutions of the reduced systems to be extended to solutions of the corresponding differential-difference equations. The construction is illustrated using the Volterra hierarchy.

We study Poisson algebras in which every Poisson subalgebra is a Poisson ideal, called Dedekind Poisson algebras, over arbitrary fields and without any finite-dimensionality assumption. In characteristic different from 2, combining the two operations by $x*y=xy+[x,y]$ connects the problem with Outcalt's classical classification of power-associative H-algebras and yields the same structural type. We obtain a complete classification by a direct two-operation argument valid in every characteristic. This approach also shows that characteristic 2 retains additional Poisson information which cannot be recovered from the single product $*$.

Habiro's cyclotomic expansion of the colored Jones polynomial has a higher rank analogue conjectured by Chen–Liu–Zhu for colored $SU(n)$ invariants. We prove this conjecture for torus knots $T(2,2p+1)$ and give a Bailey theoretic realization of the cyclotomic coefficients. For $n\geq2$ and $K_p=T(2,2p+1)$, the colored $SU(n)$ invariants admit an expansion $$ J_N^{SU(n)}(K_p;q) = \sum_{m=0}^{N} \left(\prod_{j=0}^{m-1}\{N-j\}\{N+n+j\}\right) H_m^{(n,p)}(q), $$ where $H_m^{(n,p)}(q)\in\mathbb Z[q^{\pm1}]$ is independent of the color $N$. The proof identifies the cyclotomic basis with a Newton basis and rewrites the resulting Newton coefficients as an ordinary Bailey transform. The Lin–Zheng formula for $T(2,2p+1)$ then gives a well-poised Bailey kernel. A terminating very-well-poised ${}_6\phi_5$ summation diagonalizes the kernel and reduces the integrality problem to an ordinary Bailey transition. We prove a uniform integrality theorem for these transitions in a formal integral $q$-difference operator algebra. As a direct corollary, the expansion yields the corresponding congruence relations and proves part(i) of the Chen–Liu–Zhu $SU(n)$ volume conjecture for $T(2,2p+1)$.

The notion of s-composability was introduced to compose optimized stepsize schedules while preserving sharp guarantees. We show that the same joint potential has a second use: it can serve directly as a certificate of sharp last-iterate performance. We develop this viewpoint for constant and silver schedules. For the constant schedule, we prove the s-composability statement posed as an open question by Grimmer et al. (2025), at every finite horizon, through an explicit nonnegative smooth-convex interpolation certificate. The result identifies the unique constant stepsize minimizing the worst-case final gradient norm of smooth convex gradient descent under an initial-distance bound, thereby proving the optimal stepsize, value, and uniqueness predicted by Conjecture 3 of Taylor et al. (2017) without resolving its full worst-case curve. Through the Moreau envelope, the same constant is also the unique constant relaxation minimizing the worst-case final residual of relaxed proximal point. More generally, for every positive s-composable schedule, we determine the exact worst-case final proximal objective-gap constant and give a matching one-dimensional example. Applying this result to the already s-composable original silver schedule yields its exact last-iterate objective-gap constant, closing a question left open by Wang et al. (2025).

A boxed mesh pattern is a mesh pattern whose selected entries lie in an empty axis-parallel rectangle. We classify coincidences of boxed patterns with classical and vincular patterns, exhibit a genuinely bivincular coincidence, and prove that no boxed–bivincular coincidence occurs for patterns of length at least five. Together with known results, this shows that every boxed pattern of length at least five has factorial growth and hence fails the Stanley–Wilf property. At length four, one exceptional orbit is enumerated by the semi-Baxter numbers, while the remaining exceptional orbit, $\{2143,3412\}$, is unresolved; we conjecture that it has factorial growth. For Box(123), we derive an exact maximum-insertion identity and prove the subfactorial upper bound $2^{5n}n^{\beta n}$, where $\beta=\log_2(2\cos(\pi/7))<0.85$. A closed enumeration remains open. We also prove a general first-moment formula for boxed mesh patterns that depends only on the length of the underlying pattern; in particular, the expected number of Box(123) occurrences in a uniformly random permutation of length $n$ is asymptotic to $n\log n$. For Box(12), we identify the occurrence statistic with the up-degree in the strong Bruhat order, obtaining its maximum, its mean, and an exact insertion identity for the distribution polynomials. We conjecture that the coefficients of these polynomials are unimodal.

We prove that balls are the unique minimisers of the Bogovskii constant among bounded connected Lipschitz domains in $\mathbb{R}^n$, $n\geq 2$, resolving Open Problem 2 of Gazzola, Grunau and Sperone (arXiv:2608.25861).

We establish the equivalence between the stochastic optimal control and path space formulations of the Schrödinger bridge problem (SBP) for the kinematic equation on a compact connected Lie group. Using the geometric concepts of horizontal lift and stochastic anti-development, we derive a Girsanov-type change-of-measure result, and show that the expected control energy equals the relative entropy of the controlled path law with respect to the reference Wiener measure. Thus, the SBP is equivalently a path space relative entropy minimization problem subject to prescribed endpoint marginals.Our result has three useful implications. From an analytic viewpoint, the shown equivalence helps prove the existence and uniqueness of the SB. From a probabilistic viewpoint, it helps interpret the SB as the most probable deviation of the uncontrolled stochastic dynamics consistent with the endpoint constraints. From a computational viewpoint, it allows using static Sinkhorn recursions to directly solve the relative entropy minimization problem and compute the optimal path measure. We illustrate the equivalence numerically on the torus $\mathbb{T}^2$. The code is publicly available at: https://github.com/gradslab/LargeDeviationSBP

The Hamming graph H(d,3)=K_3^{box d} (n=3^d vertices) is the graph of single-symbol errors of ternary codes, yet we ask a geometric question of it: can it be drawn in the plane with every edge exactly one unit long and no two non-adjacent vertices a unit apart? It can, for every d, joining two problems into one. Where does it live? As a Minkowski sum of unit triangles, each H(d,3) has a hidden flexibility carrying its coordinates up the constructibility ladder (compass, origami, and beyond), so one graph is realizable over many number fields at once, in the plane and in R^3 (edim(H(d,q))=q-1). Yet almost every faithful realization is transcendental: the Galois picture is a measure-zero shadow of a vast transcendental continuum, matching the exists-R hardness of unit-distance recognition. How crowded must a unit drawing be? We separate the ordinary crossing number from a unit-distance crossing number (absent from Schaefer's survey), and one recursion H(d,3)=H(d-1,3) box K_3 controls both. We prove cr(H)=Theta(n^2), with sharp one-page constant 7/6; and Omega(n^2) <= udcr(H) <= O(n^2 log n) by concentration, plus a closed-form majorant (3/2)n^2(L^2-L+1), L=log_3 n. The least-crossing drawing we find is constructible (origami): field and crossings are two readings of one geometry. All claims are verified computationally; the lower bound udcr=Omega(n^2 log n) is the central open problem.

LP-Newton solves a linear program through a sequence of nearest-point problems. Given an interior feasible point for an inequality-form LP, we construct the compact hull formed by the origin and its normalized constraint rows. Polar LP-Newton (P-LPN) follows the objective ray to the boundary of this row-polar hull, where it recovers a primal-dual optimum or a recession direction proving unboundedness. For rational row-polar data, we prove an outer-iteration bound quadratic in dimension and linear in binary input length, excluding inner Wolfe work. Restarting Wolfe at every outer iteration discards its terminal corral even though the hull is unchanged and the next target lies on the same ray. Corral-repair P-LPN (CR-P-LPN) instead repairs that corral and uses it to start the next projection. Verification over the full hull preserves P-LPN's projections, outer targets, and LP conclusion in exact arithmetic. Experiments in Julia and MATLAB show where repair helps. After common initialization, median P-LPN times across 180 single-LP tests are 1.62 times the corresponding CR-P-LPN times in Julia and 1.55 times in MATLAB. On three problems with 5,000 to 100,000 rows, the corresponding geometric-mean factors are 2.03 and 1.46. A separate end-to-end study includes initialization on 19 application-derived LPs. CR-P-LPN has lower median total time than the faster cold HiGHS mode, either dual simplex or the interior-point method with crossover, on 18 of the 19 workloads in each language. The geometric means of the ratios of HiGHS time to CR-P-LPN time are 4.93 and 3.06. Component comparisons identify terminal-corral reuse as the useful component, whereas candidate-first ordering alone shows no consistent gain. Repair is not always beneficial: it is slower on Klee-Minty tests, and an author-implemented simplex is faster on the retained Netlib models.

This paper develops a hyperbolic-majorization preconditioned three-term nonlinear conjugate-gradient framework for nonconvex finite minimax optimization. An analytic symmetric positive definite metric is derived from a global quadratic majorization of the hyperbolic smoothing model and is used simultaneously as a curvature absorber, a preconditioner, and the line-search energy metric. With the displacement \(s_{k-1}\) fixed and a variable curvature response \(b_k\), an enhanced three-term direction is introduced together with an adaptive parameter \(\mu_k^\star\) that is maximal under the requirement that the baseline worst-case metric-energy constant be preserved. The resulting framework yields a Dai–Liao-type conjugacy relation, enhanced sufficient descent, a smoothing-parameter-uniform Armijo lower bound, fixed-smoothing global first-order convergence and complexity, and Clarke-stationary accumulation points under continuation.

Continuum angiogenesis systems often use local endothelial or vessel density as an oxygen-delivery proxy without distinguishing structurally formed vessels from pressure-supported vascular function. We formulate a two-dimensional model in which lumenized density determines a normalized nonlocal conductivity, a globally solved pressure field, and a flow-functional density that gates oxygen delivery and vessel regression. For fixed positive regularization parameters and signal relaxation times, we prove Lipschitz stability of the pressure–perfusion map and global bounded weak solvability; strong vessel compactness follows from a nonlocal ordinary differential equation stability estimate rather than spatial smoothing. As the oxygen and vascular endothelial growth factor timescales vanish simultaneously, weak solutions converge along a subsequence to a parabolic–elliptic–ordinary differential equation system, with both fast fields converging strongly in \(L^2(0,T;H^1)\). A locally frozen one-dimensional reduction admits monotone fronts precisely at or above the threshold \(B+2\sqrt{DR}\) for \(B\geq-\sqrt{DR}\) and \(-DR/B\) otherwise. Finite-volume experiments verify the analytical mechanisms and show that equal-mass vessel fields can generate distinct functional masses and oxygenation levels.

We say that $v\in V(G)$ is a good vertex if the Kirchhoff index remains unchanged when $v$ is removed, i.e. $Kf(G)=Kf(G-v)$. In 1991, Šoltés studied the Wiener index of a graph and posed the problem of identifying graphs for which the removal of an arbitrary vertex preserves the Wiener index. In this paper, we explore a similar concept: identifying Kirchhoff Šoltés graphs, i.e. graphs in which all vertices are good vertices. We show that the cycle $C_5$ is a Kirchhoff Šoltés graph. We consider several relaxed versions of the Kirchhoff Šoltés problem, where the primary objective is to identify graphs containing at least one good vertex. One of them is the \textit{$\beta$-Kirchhoff Šoltés problem}, which seeks to find an infinite family of graphs in which the proportion of good vertices is at least $\beta$, with $\beta \in (0,1]$ being a specified rational number. Another one involves constructing infinite families of graphs where the proportion of good vertices increases and asymptotically approaches a given real number $\gamma\in (0,1]$ as the order of the graph grows. We demonstrate that both relaxed versions have infinitely many solutions. In particular, we prove the existence of infinitely many graphs for which the proportion $\beta$ of good vertices, $1/7\leq \beta<1/5$ tends to a certain irrational number. Furthermore, we prove the existence of infinitely many graphs with half good vertices, and for each $s\in\mathbb{N}$, we construct an infinite family of graphs whose proportion of good vertices tends to $\frac{s+1}{2s+1}$. These findings could be pivotal in addressing the original problem of determining whether there are additional solutions beyond $C_5$.

We consider the focusing cubic nonlinear Schrodinger equation with a repulsive inverse-power potential $V(x)=a|x|^{-\mu}$, where $a>0$ and $1<\mu\leq 2$. At the mass-energy threshold, Miao, Murphy, and Zheng, as well as Ardila, Hamano, and Ikeda, established sharp scattering results in the positive virial region. In this paper, we continue to study the dynamics in the negative virial region and show that, in each time direction, solutions either blow up in finite time or grow up.

Let $A$ be a zero-dimensional monomial algebra over an algebraically closed field of characteristic zero, that is, a finite-dimensional quotient of a polynomial ring by a monomial ideal. Its automorphism group $G$ is a linear algebraic group, described through the homogeneous nilpotent derivations of $A$. We analyze the structure of $G$ in detail. Its identity component $G^0$ is a semidirect product of its unipotent radical and a reductive subgroup isomorphic to a product of general linear groups, and for each root degree we characterize when the associated derivations give rise to an additive root subgroup, and determine its dimension. Using the Lie brackets of these derivations, we then give an explicit algorithm that produces, out of the minimal monomial generators of the ideal, a family of root subgroups generating $G^0$ together with a maximal torus. Such a family is minimal in the generic case. We also show that the component group $G/G^0$ can be arbitrary: every finite group arises as the component group of the automorphism group of some zero-dimensional monomial algebra. Finally, we apply these results to the algebras $\mathbf{k}[\mathbf{x}]/\mathfrak{m}^d$, showing that the subgroup generated by a maximal torus and the outer root subgroups is exactly the subgroup of automorphisms with constant Jacobian determinant, and we deduce from this a new proof of Anick's theorem on the density of the tame automorphisms of $\mathbf{k}[\mathbf{x}]$.

We study local and global properties of positive solutions to the equation $-\Delta u=u^p+M|\nabla u|^q$ in a domain $\Omega$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting appropriate auxiliary functions over various regions and employing the maximum principle, we derive the local gradient estimates for all $(p,q)\in \mathbb R^2$, and further establish Liouville-type theorems. As an application, we acquire universal estimates for local solutions of elliptic equations with general nonlinearities. Our results extend partial conclusions established in Bidaut-Véron, Garcia-Huidobro and Véron [Math. Ann. 378 (1-2) (2020) 13-56].

In a very recent, beautiful paper, Junyi Xie and Ziquan Yang gave a new proof of Faltings' isogeny theorem over number fields by reducing to Tate's theorem over finite fields. We show that the same ideas can be adapted to prove finiteness of an isogeny class over number fields from the finiteness of an isogeny class over finite fields. Hence we obtain a new proof of the Shafarevich conjecture over number fields. The proof in this paper was obtained through the use of GPT6 Astra Ultra.

We study the sampling problem for non-decaying signals defined on a locally compact group. The signals are modeled as elements of suitable subspaces of weighted Orlicz and weighted mixed-norm Orlicz spaces, thereby allowing controlled growth at infinity. Our principal focus is on image spaces of idempotent integral operators. Under suitable oscillation estimates on the associated integral kernels, we establish deterministic sampling theorems that guarantee stable reconstruction from pointwise samples. We further prove an average sampling result in the same setting. In addition, we prove random sampling theorems for functions whose norm is essentially concentrated on a compact subset of the group. In particular, we show that, with high probability, sampling is possible from $\mathcal{O}\!\left(\mu(K)\log \mu(K)\right)$ randomly chosen sample points, where $K$ denotes the compact set on which the signal, belonging to Orlicz space, is essentially norm concentrated. In the same spirit, a random sampling result is proved for signals in mixed-norm Orlicz space. As applications of the abstract theory, we obtain sampling theorems for weighted shift-invariant spaces and Orlicz modulation spaces.

In this note, we prove that every primitive KMS-symmetric quantum Markov semigroup on a finite-dimensional matrix algebra satisfies a modified logarithmic Sobolev inequality (MLSI). We also construct a primitive quantum Markov semigroup without KMS symmetry that fails MLSI, and a primitive KMS-symmetric quantum Markov semigroup that fails complete MLSI (CMLSI). The latter construction also provides a non-primitive KMS-symmetric quantum Markov semigroup without MLSI. For the graph-based KMS-symmetric quantum Markov semigroups studied here, we prove MLSI and CMLSI when the underlying graph is connected and has at least three vertices. Finally, we establish CMLSI for a class of primitive bimodule KMS-symmetric quantum Markov semigroups arising from fermionic systems.

We introduce a new method for studying gradient higher integrability for mixed local and nonlocal parabolic equations. More precisely, for $p>2d/(d+2)$ and $s \in (0,1)$, we prove that if the inhomogeneity $F \in L^{p(1+\sigma)}_{\mathrm{loc}}$ for some $\sigma>0$, then every weak solution satisfies $\nabla u \in L^{p(1+\eps)}_{\mathrm{loc}}$ for some $\eps>0$, together with quantitative local estimates. The proof relies on stopping time arguments and an intrinsic Calderón-Zygmund-type covering decomposition in which the inhomogeneity and nonlocal energy contributions are treated by classical and fractional maximal estimates.

We consider two canonical market-microstructure regularities: the long-memory of trade signs and the square-root law of meta-order impact. The point is not to propose new empirical laws, but to separate the clocks on which existing laws are defined. The sign-memory law is an event-time statement about the ordering and fragmentation of hidden orders. The square-root impact law is an operational-time statement about front motion in a locally linear latent order book. Starting from a discrete-time random-walk bid/ask reaction–diffusion order book with a separate event clock, we identify an event-time imbalance reduction, derive the operational-time front and impact equations in the locally linear regime, and then subordinate both sign and impact observables to calendar time. Fractional or tempered clock effects enter through this event-to-calendar projection, not through a different operational-time impact mechanism. This gives a compact clock-aware framework in which event-time sign persistence, operational-time square-root impact, and anomalous calendar-time effects appear as distinct but compatible consequences of a common order-book representation.

Let $\Omega\subset\mathbb R^n$ be a bounded convex domain that is thin around a chosen diameter segment. We compare its Neumann spectrum with the spectrum of that segment weighted by the $(n-1)$-dimensional volumes of its perpendicular sections. We prove an $O(\varepsilon^2)$ comparison of the mean-zero inverse operators and, consequently, an $O(\varepsilon^2)$ eigenvalue comparison for every fixed index in every dimension $n\ge2$. The constants depend only on the dimension and the eigenvalue index. Thin rectangles show that the quadratic exponent is optimal.

We study a conservative semilinear wave equation coupled through a zero-order interaction to a dissipative Moore–Gibson–Thompson equation on a bounded domain. The wave component carries no direct damping and is driven by a general focusing source $f(u)$. The augmented variable $w=v+\tau v_t$ reveals an exact coupled energy and a coercive potential-well geometry. The source assumptions are formulated through $H_\theta(s)=\frac1\theta sf(s)-F(s)$, $F(s)=\int_0^s f(r)\,d r$, $\theta>2$. Under $L^2$-subcritical $C^1$ growth, smallness at the origin, nonnegativity and radial monotonicity of $H_\theta$, and a nontrivial focusing condition, we establish local well-posedness, the exact energy identity, and a continuation alternative for arbitrary finite-energy data. For nonzero coupling, the linearized semigroup is strongly stable, whereas a wave-branch expansion precludes uniform exponential stability and positive-time compactness. Below the coupled well depth, the stable set is positively invariant and generates global solutions, while data with negative Nehari functional blow up in finite time without a sign condition on the initial velocities. At the critical level $E(0)=d$, nonzero coupling yields a complete trichotomy into stable entry, finite-time blow-up, or a stationary Nehari ground state. Under the same coupling condition, every stable trajectory converges weakly to zero without any compactness hypothesis. A renormalized high-frequency identity shows that vanishing of the accumulated nonlinear high–low flux is equivalent to relative compactness of the orbit and to strong convergence in the natural energy space. We give several sufficient criteria, including finite total variation of the nonlinear force in $L^2(\Omega)$.

In this paper, we study parametrization of solutions to the Liouville equation near their isolated singular points. We demonstrate that prescribing asymptotic behaviors forms a new type of problem leading to all such solutions. The harmonic functions determining the asymptotic behaviors play the role of “free data" as the boundary values in the boundary value problems.