Papers

We prove that the largest Orlicz space into which the Hardy averaging operator maps coincides with the largest rearrangement-invariant (r.i.) space mapping into that space; in other words, the optimal Orlicz domain is automatically the optimal r.i.\ domain. More generally, we show that every Luxemburg–Orlicz norm is equivalent to a sublinear functional which makes more computable a certain expression for a dual norm arising in the $\K$-theory of interpolation. As applications we obtain Orlicz-space mapping properties for the Hardy–Littlewood maximal function, approximate identities and Calderón–Zygmund singular integral operators. These mapping properties are shown to be optimal for the Hardy-Littlewood maximal function and the approximate identities.

In every dimension three and higher we construct a nonzero function that satisfies the stationary Schrödinger equation with bounded, real-valued potential and decays like $\exp(-c|x|^{4/3})$. This gives a counterexample to the Landis conjecture in dimension three and higher.

This paper establishes an existence and uniqueness theorem for a class of functional ordinary differential equations under assumptions weaker than those found in the literature. We then consider the problem of designing a lens that refracts rays of two distinct frequencies emitted from a point source into a common collimated beam. We derive a system of functional ordinary differential equations whose solvability characterizes the existence of such lenses. Using the existence theorem, we prove that such a lens exists in two dimensions. We then extend the corresponding result to three dimensions.

In this paper, we investigate the low Mach number limit for the three-dimensional compressible Navier–Stokes–Korteweg equations in the whole space under a small stationary external force. We first construct a family of small stationary solutions uniformly with respect to the Mach number $\epsilon$ and prove that both the stationary density fluctuation and the compressible component of the stationary velocity are of order $\epsilon^2$. For ill-prepared non-stationary perturbations around these stationary solutions, we establish the existence and uniqueness of global strong solution by combining uniform high-order energy estimates with a low-frequency Besov estimate and a Kawashima-type compensating functional. The main difficulty is that Korteweg tensor not only changes the elliptic structure of the stationary problem, but also modifies the dispersive mechanism of the acoustic modes. In Korteweg-symmetric variables, the associated spectral projections are uniformly bounded zero-order Fourier multipliers, while the acoustic-capillary phase is wave-like at low frequencies and Schrödinger-like at high frequencies. Since the source terms generated by the stationary coefficients are generally not integrable in time, we decompose the Duhamel source according to its time-integrability and frequency behavior. Dyadic dispersive estimates, high-frequency damping estimates, and maximal regularity for the heat equation yield the global-in-time convergence rate $\epsilon^{\min\{1/r,\,1/2-1/p\}}$ in the mixed Besov norms $L^r(0,\infty;\dot B^s_{p,1})$. As a consequence, Besov embeddings also yield quantitative convergence in the mixed Lebesgue norms $L^r(0,\infty;L^p)$.

This paper is concerned with the analysis of a one-dimensional power-law model for compressible fluid dynamics on $\mathbb{R}$, in which the shear stress takes the form $\mu |\partial_{x}u|^{p-2}\partial_{x}u$, where $\mu$ is the viscosity coefficient and $u$ is the velocity. We prove that, in the singular limit $p\rightarrow\infty$, the solutions converge to functions $(\rho,u)$ satisfying $|\partial_{x}u|\leq 1$, $\tau = \pi \partial_{x}u$, $\pi \geq 0$, and $\pi (1 - |\partial_{x}u|) = 0$ a.e. on $\mathbb{R}$. Moreover, we rigorously justify the existence of weak solutions to the limiting equation. The convergence as $p \to \infty$ is obtained via domain truncation and compactness arguments, of which the key challenge is to show that the density remains bounded away from zero and infinity on any compact subset. This extends the recent result of Bresch, Burtea, and Szlenk [Nonlinearity 26 (2026), no. 5, Paper No. 055010.] from one-dimensional periodic domain to the whole real line.

We prove the existence of infinitely many smooth nonradial stationary solutions to the surface quasi-geostrophic (SQG) equation with finite kinetic energy. Our construction is based on a family of nonradial sign-changing solutions to the two-dimensional half-Yamabe equation, obtained via a Lyapunov–Schmidt reduction and concentrated at the vertices of a regular polygon. As the number of vertices tends to infinity, the associated stationary SQG solutions converge to a radial stationary profile centered at the origin, together with a lower-order vortex sheet correction.

We identify a physical-space balance-law mechanism underlying bilinear null forms for the free Dirac equation in three space dimensions. For each spatial direction, we decompose a spinor into the two eigenspaces of the directional Dirac symbol. The principal parts of the corresponding modes propagate in opposite directions, while the transverse derivatives and the mass term couple them. After integration over the transverse variables, their charge densities satisfy a pair of one-dimensional balance laws, and a div–curl interaction estimate controls the mixed product of these densities. The algebraic anticommutation condition defining the spinorial null form exchanges exactly the same two eigenspaces. This identifies the algebraic cancellation with the interaction selected by the balance laws. Combined with angular localization, the argument yields frequency-localized $L^2$ spacetime estimates while preserving the natural first-order formulation of the Dirac equation. In the massless case, the estimate has the same lower-frequency scaling as the three-dimensional wave null-form estimate and gains half a derivative over the direct product bound. In the massive case, we obtain a channel-dependent refinement. For the pseudoscalar interaction, anticommutation with the full massive Dirac Hamiltonian gives an additional frequency-to-mass factor $K/m$ for low-frequency interactions within the same energy branch, where $K$ is the frequency scale and $m$ is the mass. This gain is absent in the scalar channel. The bilinear estimates also yield factorized bounds for cubic spinorial null forms arising in nonlinear Dirac models.

We investigate a coupled thermoelastic plate system consisting of a fourth-order displacement equation and a heat evolution equation linked through a spatially varying coupling factor $\alpha(x)$. The model accounts for thermoelastic interactions through the operators $\operatorname{div}(\alpha(x)\nabla \theta)$ and $\operatorname{div}(\alpha(x)\nabla u_t)$. We establish the well-posedness of the direct problem under homogeneous Neumann conditions for $u$ and Dirichlet conditions for $\theta$, deriving optimal energy estimates and demonstrating continuous dependence of solutions on the given data. We further introduce an input-output operator corresponding to the considered inverse problem and show that it is compact and Lipschitz continuous, confirming the ill-posed nature of the associated inverse problem. Using these properties, the inverse problem is formulated as a minimization problem for the Tikhonov functional, and we establish the existence of a minimizer.

We study the Cauchy problem for the three-dimensional barotropic compressible Navier–Stokes equations with a time-independent potential force near a spatially nonconstant stationary state. The potential is controlled in unweighted homogeneous Besov spaces; in particular, no polynomial spatial-weight condition involving $(1+|x|)^j\nabla^j\phi$ is imposed. For initial data relative to the stationary state that are sufficiently small in $\dot H^{\frac12-\delta}\cap\dot H^3$, we establish the existence and uniqueness of a global strong solution in $H^3$, while allowing the initial $L^2$ norm to be arbitrarily large. If the initial data are bounded in $\dot B^s_{2,\infty}$ for $s\in[-\frac32,-1)$, then the solution and its first spatial derivative decay at the optimal rates $(1+t)^{-\frac{k-s}{2}}$ with $k=0$ and $1$, respectively. The analysis relies on refined homogeneous energy estimates and a frequency-localized description for the dissipative and asymptotic structures of the system.

We study the periodic homogenization of a class of viscous Hamilton–Jacobi equations with fast dependence on the unknown. This problem combines features of first-order Hamilton–Jacobi equations with \(u^\epsilon/\epsilon\)-periodic Hamiltonians and semilinear heat equations with rapidly oscillating positive potentials. In this paper, we prove qualitative homogenization results for $H(y,s,p)=F(s,p)+\eta W(y,s,p)$ when $|\eta|$ is sufficiently small, with the smallness threshold depending on the Lipschitz constant of the initial datum, and establish a large-time averaging result for a general class of evolutionary cell problems.

We construct a one-parameter family of small self-similar solutions to the three-dimensional one-phase Muskat problem with surface tension. The solutions have the form $\eta_\varepsilon(t,x) = t^{1/3}U_\varepsilon(t^{-1/3}x)$ and emanate from the conical initial data $\eta_\varepsilon(0,x)=\varepsilon|x|$. The profiles are perturbations of the linear capillary regularization of the cone, and we identify the leading quadratic correction. The proof combines a raywise inverse estimate for the linear similarity operator, a favorable high–high-to-low cancellation in the quadratic term, and finite-order tame estimates for the Dirichlet–Neumann operator on asymptotically conical graphs. These estimates yield the solutions by a contraction argument and show that the conical singularity is instantaneously rounded for positive time.

Bound states in the continuum (BICs) are localized eigenmodes with their frequencies in the radiation continuum of scattering states. The existence of a BIC implies the loss of uniqueness for scattering problems with given incident waves. Perturbed wave systems close to the ideal ones with a BIC exhibit strong resonance effects that are essential to numerous practical applications. A question of fundamental importance is whether a BIC is robust, i.e., whether it can continue its existence when the structure is slightly perturbed. In an earlier work [Yuan and Lu, Optics Letters, Vol.~42, pp.~4490-4493, 2017], for a class of BICs governed by the two-dimensional (2D) Helmholtz equation, which are not trivially protected by symmetry, we uncovered the conditions that ensure robustness and formally constructed the BIC in perturbed systems using a perturbation method. In this paper, we present a rigorous theory on the robustness of BICs in 2D dielectric structures with a single periodic direction. Specifically, we analyze the solvability and provide estimates for each order in the perturbation series, and prove the convergence of the series.

We develop a new phase-field (diffuse interface) approximation of multiphase curvature motion with triple junction drag - an important sharp interface model for the evolution of microstructure in polycrystalline materials during heat treatment. This sharp interface model arises as gradient flow for the total length of the interfacial network with respect to a certain metric. Accordingly, we derive our diffuse interface approximation - a coupled system of partial differential equations that is a variant of the Allen-Cahn system - from a variational perspective, in the style of minimizing movements: starting from a discrete in time approximation that entails a convex optimization problem to advance from one time step to the next. In the process, we propose a simple integral expression that counts the number of junctions using the order parameter that appears to be new even for the standard multiphase Allen-Cahn system. The convergence of the resulting flow to the desired sharp interface limit is then verified via the method of matched asymptotic expansions. Numerical convergence studies against both known exact solutions as well as highly accurate benchmark solutions obtained via front tracking provide clear further evidence for this convergence. Moreover, the method retains the most desirable feature of diffuse interface methods: Automatic handling of topological changes in the network of interfaces.

We present a robust physical-space approach to establish time decay and global asymptotics of solutions to variable-coefficient Schrödinger equations in $(3+1)$-dimensions. As an immediate nonlinear application, we obtain new small data global existence and asymptotics results for quasilinear Schrödinger equations with cubic, Hamiltonian nonlinearity, variable coefficients in their linear part, and possibly outside obstacles, even in the presence of trapped bicharacteristics (provided that they are suitably unstable). Our approach relies on three primary ingredients. First is the concept of a good commutator, which extends Klainerman's classical commuting vector field method, and develops upon earlier approaches of Cuccagna–Georgiev–Visciglia and Rodnianski–Tao in the variable-coefficient case, and of Ifrim–Tataru, Ifrim–Koch–Tataru in the nonlinear case. Second, we apply Ifrim and Tataru's testing-by-wave-packets method, which is key for obtaining global asymptotics and handling sharp decay assumptions on the coefficients. Finally, we introduce a systematic technique for analyzing the control provided by the good commutator, termed two-scale elliptic analysis. Together, these techniques significantly broaden the applicability of physical-space methods to a wider class of variable-coefficient nonlinear problems.

We quantify viscous energy loss for two-dimensional Delort vortex sheets. Let $u^\nu$ be Leray-Hopf solutions on $\mathbb{T}^2$ with uniformly bounded kinetic energy and total vorticity variation, and write $\omega_0^\nu=\mu_0^\nu+f_0^\nu$, where $\mu_0^\nu\geq0$ and $f_0^\nu$ is bounded in $L^p$, $p>1$. For every fixed $0<\delta<T$, $\nu\int_\delta^T\|\omega^\nu(t)\|_2^2\,\mathrm{d}t\lesssim_{\delta,T}\frac{1}{|\log\nu|}$. This improves the $O(|\log\nu|^{-1/2})$ bound of De Rosa and Marcotullio under the same assumptions and thereby disproves their Conjecture 1.6 (arXiv:2602.15670, v1). The proof uses a sharp $L^2$-$H^1$-$H^{-1}$ interpolation inequality for nonnegative densities, retaining the total interaction energy of the positive vorticity rather than only its largest local mass. The bound also remains effective when the observation time grows with the Reynolds number. If the initial velocities are relatively compact in $L^2$, the loss still vanishes whenever $\log T_\nu=o(|\log\nu|)$; in particular, any prescribed energy loss must wait at least until $\nu^{-a}$ for some $a>0$. Previous estimates covered only $T_\nu=o(\exp(|\log\nu|^\kappa))$, $\kappa<1/2$, so this gives a polynomial lower bound on the energetic lifetime of the inviscid vortex-sheet model. If instead $f_0^\nu$ is bounded in $L(\log L)^\alpha$, the rate is $O(|\log\nu|^{-q_\alpha})$, $q_\alpha=\min\{2\alpha,1\}$, and the loss vanishes when $\log T_\nu=o(|\log\nu|^{q_\alpha})$. On $\mathbb{R}^2$, exact radial solutions attain these exponents for $0<\alpha\leq1/2$. At the endpoint, a bounded-energy $L^p$ family attains the rate $1/|\log\nu|$, while every fixed radial datum dissipates $o(1/|\log\nu|)$ and can lose a fixed amount of energy only on the diffusive scale $1/\nu$.

We study global regularity of solutions to Dirichlet or Neumann elliptic problems in spherical sectors $S_{D,R}$ of radius $R>0$ in $\mathbb{R}^N, N\ge 2$, where $D$ is the bounded domain on the unit sphere $\mathbb{S}^{N-1}$ which spans the spherical sector. One of the main results shows that boundedness of the gradient of the solutions of Poisson equations holds whenever $\lambda_1(D)\ge N-1$, where $\lambda_1(D)$ is the first nontrivial eigenvalue of the Laplace Beltrami operator $-\Delta_{\mathbb{S}^{N-1}}$ on the domain $D$ with Dirichlet or Neumann boundary conditions on $\partial D$. As an example of Maz'ya shows, the condition on the eigenvalue is sharp. For general spherical sectors and for $p$-Laplacian equations, $p>1$ we prove weighted global lipschitzianity of the solutions, as well as second order regularity.

This paper provides an abstract framework for studying inertial manifolds associated with a class of nonlocal parabolic problems. In particular, by suitably modifying the nonlocal term outside the absorbing ball and changing the scale of time, we derive a corresponding spectral gap condition. As applications, we establish the existence of inertial manifolds for two classes of two-dimensional modified nonlocal parabolic equations on a square domain, whose diffusion coefficients depend on the $L^2$-norm of the solution and of its gradient, respectively.

The method of Convex Integration has revealed a number of rather disturbing facts concerning well-posedness of the Euler system of gas dynamics. In particular, there is a dense set of "wild" initial data, for which the problem admits infinitely many physically admissible (entropy) weak solutions. We identify the class of initial data enjoying the following properties: (a) they give rise to a family of weak solutions with increasing entropy profiles; (b) the solutions are "discrete", meaning they attain only a finite number of constant states; (c) the solutions reach a prescribed terminal entropy profile when time goes to infinity.

Let $(A_i,\Phi_i)$ be finite energy $\mathrm{SU}(2)$ monopoles of charge $k>0$ on an asymptotically conical $3$-manifold with one end, with masses $m_i\to\infty$. After passing to a subsequence, the mass-renormalized energy measures concentrate at finitely many points $x_a$ with concentration weights $4\pi K_a$, where $K_a$ is the total charge of the complete finite cluster of mass-one Euclidean monopoles lying over $x_a$. We prove that, on the complement $M$ of these points, the fields abelianize exponentially. After translating the Higgs fields by their masses along the unit Higgs directions and applying gauge transformations, the translated pairs converge smoothly locally to a reducible monopole $(A_\infty,\Phi_\infty)$ of the form \[ \Phi_\infty=-u\Psi_\infty, \qquad F_{A_\infty}=-*du\,\Psi_\infty, \qquad u=4\pi\sum_aK_aG(\,\cdot\,,x_a), \] where $\Psi_\infty$ is a parallel unit section and $G$ is the minimal positive Green function. Consequently, $x_a$ is a Dirac singularity of charge $K_a$. The singular part of the residual limit is determined by the weighted $0$-cycle of concentration points and total cluster charges, and does not retain the individual Euclidean profiles or their separation hierarchy. We also show that $k-\sum_aK_a$ is exactly the charge escaping through the asymptotically conical end, and describe the residual flat abelian ambiguity.

In dimension one, we introduce a one-parameter family of nonlocal energies, including subcritical Gagliardo seminorms and Riesz functionals, acting on scalar functions whose derivative is given by a periodic configuration of screened Dirac masses. We prove that the global minimizers are given by the equi-spaced configurations of particles. This result is achieved by representing the energy functionals as interaction potentials depending on the geodesic distances between the particles, exploiting the complete monotonicity properties of such potentials and invoking the celebrated Cohn-Kumar Universality Theorem.

We establish a sharp bound on the Hausdorff dimension of the non-branch singular set of branched stable minimal immersed hypersurfaces whose singular sets have locally finite $\mathcal H^{n-2}$-measure: the non-branch singular set is empty when $n=2$, discrete when $n=3$, and has Hausdorff dimension at most $n-3$ when $n\geq4$. We also construct a non-flat stable minimal cone in $\mathbb R^4$ arising from a branched minimal immersion whose vertex is a non-branch singularity. Taking products with Euclidean factors yields examples whose non-branch singular sets have Hausdorff dimension exactly $n-3$, showing that our regularity bound is sharp in every dimension $n\geq3$. The main ingredients in our proof are a generalized Schoen inequality and a corresponding branched sheeting theorem near stationary classical cones and unions of hyperplanes.

We study nonlinear reactive transport in a two-dimensional bulk domain containing a thin branched fracture composed of three narrow branches of thickness epsilon connected through a junction node of diameter O(epsilon). The microscopic model couples nonlinear parabolic reaction-diffusion equations in the bulk with an advection-diffusion equation in the fracture, where the longitudinal Peclet number is of order epsilon^{-1}, leading to advection-dominated transport along the branches. Nonlinear side-dependent flux conditions describe the coupling between the bulk and the fracture. As epsilon tends to zero, the fracture collapses to a one-dimensional graph, and we derive a recurrent structure of effective limit problems: a first-order hyperbolic problem on the graph satisfying the classical Kirchhoff transmission condition at the node, and reaction-diffusion equations in the bulk with nonlinear Robin conditions on the graph edges involving the graph solution. The node boundary conditions of the microscopic model do not affect these leading-order limits. To capture the influence of the node geometry, we construct node-layer and corner-layer correctors and determine subsequent terms of the asymptotic expansion. Their coefficients solve auxiliary boundary-value problems in unbounded domains with outlets at infinity and in corner-type geometries. We assemble a complete multiscale approximation combining bulk, branch, node-layer, and corner-layer contributions, and establish quantitative error estimates in appropriate energy norms. These estimates demonstrate the accuracy of the approximation relative to epsilon and depend explicitly on the corner angles of the limiting graph, reflecting the geometric complexity of the fracture network.

We develop a two-dimensional Allen-Cahn theory for interfaces with boundary in the line-bundle framework introduced by Fröhlich and Struwe. The central new object is a model solution on the punctured plane whose nodal set is a half-line. Near the boundary of an interface, this solution plays the role of the heteroclinic profile in the classical interior theory. However, unlike the interior setting, where the analysis in directions normal to the interface reduces to the ODE satisfied by the heteroclinic, the boundary model has non-flat level sets and must be studied as a genuinely two-dimensional elliptic solution. In the first part of this paper, we construct this model solution and develop its stability and invertibility theory. In the second part of this paper, we present the main application: we construct Allen–Cahn sections whose nodal sets concentrate on any prescribed finite collection of disjoint line segments in the plane. The construction uses a Lyapunov-Schmidt reduction, but the boundary introduces a new difficulty: it creates large error terms along the interior of the interface. Controlling these terms requires a refined ansatz and new gluing arguments beyond the standard interior Allen-Cahn theory.

For a class of weighted degenerate or singular problems on a cylinder, Almgren-type monotonicity methods are employed to derive asymptotic estimates of solutions near the edge between the base and the lateral surface, where a Robin boundary condition is imposed. As a relevant application, local asymptotics and unique continuation from the boundary are obtained for the Robin and Neumann spectral fractional Laplacians. The local decay rates of the solutions are the same for both the Neumann and the Robin problems and are explicitly quantized by a weighted spherical spectral problem with some symmetry.

We survey a circle of recent results showing that a minimal submanifold obeys a dichotomy: either it fills up space, spreading out like a space-filling curve, or it is confined, and confinement forces quantitative restrictions. On the confined side these restrictions are both geometric and function-theoretic: Euclidean volume growth, an optimal rate of convergence of the density, complex-curve rigidity for stable surfaces in $\bf{R}^4$, and a Liouville theorem forcing slowly growing harmonic functions on minimal disks to be constant. A prototypical restriction is Euclidean volume growth, which in these results is forced by the geometry rather than assumed. The mechanism is a volume doubling theorem that converts geometric confinement into quantitative rigidity: a stationary integral varifold trapped in a thin slab at a given scale cannot double its volume by more than a universal factor. We explain how this principle, and the height-excess bounds behind it, produce Euclidean volume growth for submanifolds of sublinearly growing height in every dimension and codimension, the optimal density rate in a slab, the complex-curve and Liouville rigidity above, a higher-codimension Bernstein theorem for disks, one-sided volume bounds, and an optimal stable Bernstein theorem in all dimensions generalizing Moser, Bombieri-De Giorgi-Miranda, Caffarelli-Nirenberg-Spruck and Ecker-Huisken. We also explain how this entire picture emerges from the structure theory of embedded minimal disks in $\bf{R}^3$, built on the one-sided curvature estimate and reflected globally in the half-space theorem, and how it extends as a weak analogue of that theory to all dimensions.

In this paper we study qualitative properties of the eigenvalues and eigenfunctions of $-\Delta$ with Dirichlet boundary condition in a smooth bounded domain $\Omega$ with a small circular hole. In the literature, this is known as a "singular perturbation", in contrast with the "regular perturbation" case. Denoting by $\Omega_\epsilon:=\Omega\setminus B(P,\epsilon)$ where $B(P,\epsilon)$ is the ball centered at $P$ and radius $\epsilon$, for $P\in\Omega$ and $\epsilon$ small enough we investigate 1) quantitative estimates for the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$; 2) the simplicity of the eigenvalues of $-\Delta$ in $\Omega_\epsilon$; 3) the behavior of nodal sets of the eigenfunctions of $-\Delta$ in $\Omega_\epsilon$. A key ingredient in our analysis consists of pointwise estimates on the so-called $u$-capacitary potential firstly introduced in \cite{afhl}.

In this paper, we study the uniqueness of positive solutions to the semilinear elliptic Robin problem $$ \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\ \frac{\partial u}{\partial \nu} + \beta u = 0, & \text{on } \partial \Omega, \end{cases} $$ where $\beta > 0$, $p$ is subcritical, and $\Omega$ is a bounded smooth domain. It is known that the uniqueness of the solution depends on the shape of the domain. Even if $\Omega$ is a ball, the problem is open for arbitrary $\beta>0$, since the method of moving planes does not work for Robin boundary conditions . By scaling arguments and a careful analysis of the linearized problem, we prove uniqueness for any $\beta>0$ provided that $p$ and $\Omega$ satisfy suitable conditions. Finally, we study the effects of concave and convex nonlinearities.

We study extremizers for a trilinear Stein-Weiss inequality on $\mathbb{R}^n$. Within the known boundedness region, we prove attainment under two additional assumptions: all six weight exponents are nonnegative, and at least one pair of Lebesgue exponents is admissible. The proof combines symmetric decreasing rearrangement with a logarithmic radial reduction to a translation-invariant bilinear operator on $\mathbb{R}$ whose kernel belongs to $L^1\left(\mathbb{R}^2\right)$. A common-scale compactness argument rules out relative separation of the two arguments and yields norm attainment. We then derive the Euler-Lagrange system. In the fully symmetric case, every normalized nonnegative extremizing triple is diagonal. Finally, we establish the origin-centered Kelvin invariance of the resulting scalar equation at the scaling exponent and record the unweighted conformal example.

We consider a class of anisotropic energy integrals including fast and slow growth under a unified approach, proving that the local minimizers are locally Lipschitz continuous.

We consider the initial–boundary value problem for the semilinear wave equation with fractional structural damping $$ u_{tt}+(-\Delta)^{\theta}u_{t}-\Delta u=|u|^{p-1}u $$ in a bounded domain, where the exponent $\theta\in[0,1]$ interpolates between external frictional damping ($\theta=0$) and internal Kelvin–Voigt viscoelastic damping ($\theta=1$), and therefore parametrises the frequency dependence of the dissipation mechanism. For initial data with energy below the depth of the potential well and negative Nehari functional we prove finite-time blow-up together with an explicit upper bound for the blow-up time. The bound comes from a single concavity functional in which the fractional dissipation cancels identically; it therefore has the same form for every $\theta\in[0,1]$, and it admits a variant that is uniform in $\theta$. Conversely, for $1<p\le\frac{n+2\theta}{n-2}$ when $n\ge3$, we establish an explicit lower bound for the blow-up time. Mechanically, the two bounds delimit a guaranteed interval of existence and a guaranteed failure time for the model. The admissible range of exponents in the lower bound widens linearly with $\theta$, which quantifies how the strength of the internal damping enlarges the class of nonlinear loads for which such a guarantee can be computed; we do not claim monotonicity in $\theta$ of the numerical value of the bound. Both theorems are proved for a class of energy solutions specified by a short list of requirements, so that they are independent of any particular local existence theorem, and they carry over unchanged to the elasticity and plate operators used in structural models. The two endpoint cases recover known results for frictional and strong damping.