Papers

We prove Korn's first inequalities with and without boundary conditions in $GSBD^p$ ($1<p<\infty$) in $C^{1,1}$-regular thin domains with Lipschitz thickness boundaries with a constant $C h^{-p},$ that is optimal if the domain mid-surface contains a flat region, e.g., if the domain contains a piece of plate. For the one with boundary condition, we impose zero Dirichlet boundary conditions on the thin lateral faces of the shell, where we prove that the same estimate holds with no subtracted rigid motion like in the classical case for Sobolev vector fields.

We study a strongly damped viscoelastic wave equation with a logarithmic source whose measurable exponent satisfies $2<p_-\le p(x) \le p_+<2^*$. An optimal pointwise correction to the source–potential inequality defines a shifted energy. Under suitable kernel conditions, we prove finite-time blow-up for negative shifted energy and for a class of nonnegative shifted energies with no fixed upper energy threshold. Explicit upper lifespan bounds follow from an estimate for reaching negative shifted energy and a concavity argument. We also construct smooth blow-up data at every prescribed real energy level and show that our lifespan estimate covers additional data in the constant-exponent case. Explicit examples are given to illustrate the blow-up results.

For $N\ge2$, $0<\beta<N$ and $q>1$, we study maximizers of the critical singular anisotropic Moser–Trudinger integral \[ \int_{\mathbb{R}^N} \frac{\Phi_{N,q,\beta} (\lambda_N(1-\beta/N)|u|^{N/(N-1)})} {F^o(x)^\beta}\,\mathrm{d}x, \qquad \|F(\nabla u)\|_N^a+\|u\|_q^b\le 1. \] where $a>0$, $0<b\le N$ and $\Phi_{N,q,\beta}$ is the integrable Taylor remainder. The supremum is attained for every $a>0$ when $b<N$. For $b=N$ and $1<q<q_+$, the first retained Taylor term separates two regimes, with \[ q_-:=\frac{N^2(N-2)}{(N-1)(N-\beta)},\qquad q_+:=\frac{N^2}{N-\beta}. \] Attainment holds for every $a>0$ if $1<q<q_-$, while for $q>1$, $q_-\le q<q_+$, there is a finite threshold $a_c>N$ and attainment holds if and only if $0<a\le a_c$. An exact Euclidean reduction shows that the threshold is independent of the anisotropy. The proof combines critical–subcritical scaling with a nonlinear Green-function concentration bound. Radial flux and Pohozaev identities determine the first correction to the subcritical supremum, and refined Green tests give the strict comparison needed for attainment at the finite threshold, including in dimension two.

We consider the initial value problem for the Euler–Korteweg equations in the class of irrotational flows. In every space dimension $d \geqslant 2$, we establish a quantitative lower bound for the lifespan of solutions that explicitly exhibits its dependence on the Weber number, the Mach number, and the Sobolev norm of the initial perturbation. In particular, this lower bound can be taken arbitrarily large in a low-Weber-number regime under a suitable algebraic relation between the Mach and Weber numbers.

Seasonal succession and human intervention profoundly influence dengue fever transmission, and both humans and mosquito vectors can disperse over long distances. This paper proposes a nonlocal reaction-diffusion dengue model with moving boundaries that incorporates dry-wet seasonality and impulsive intervention. We first present the existence and uniqueness of the global solution to the model, then introduce the definition of the generalized principal eigenvalue, prove its uniqueness when the kernel functions coincide via the method of separation of variables, and further discuss its associated properties. Using this eigenvalue as an index, we obtain a spreading-vanishing dichotomy and derive two criteria that characterize the spreading and vanishing states, respectively. Our results extend both the nonlocal diffusion model without impulsive intervention and the model with impulsive intervention but without seasonal succession. Finally, we validate our theoretical findings and investigate how various factors affect dengue transmission through numerical simulations. The numerical results indicate that a shorter dry season makes disease containment more difficult, whereas more intensive impulsive intervention enhances control efficacy.

Pulse interventions represent a highly effective measure for infection control, as they influence disease transmission through short-term actions. In addition, the habitat ranges of dengue vectors and hosts exhibit periodic variations driven by environmental and climatic factors. To investigate the effects of impulsive interventions and domain evolution on disease transmission, we propose a dengue fever reaction-diffusion model that incorporates impulsive perturbations in a periodically varying domain. By applying the Poincar$\acute{e}$ map and the Krein-Rutman theorem, we establish the existence of the principal eigenvalue for the periodic eigenvalue problem with impulses, thereby extending previous studies on reaction-diffusion equations in fixed domains without impulsive effects. Sufficient conditions governing the long-term dynamics of periodic solutions are derived using the comparison principle and monotone iteration theory. Numerical simulations corroborate the theoretical findings and elucidate the effects of impulsive-intervention intensity and periodic domain evolution on disease transmission patterns. Our results indicate that increasing the intensity of pulse interventions suppresses disease transmission, whereas a larger magnitude of domain variation impedes disease control.

We prove the existence of traveling wave solutions for a diffusive Nicholson model with a delayed linear harvesting term and two delayed-response channels. Our approach combines monotone heteroclinic connections of the associated scalar delay equation with the large-speed traveling-wave theorem of Faria, Huang and Wu. For \(0<\sigma<r\), we use a previous upper- and lower-solution construction under an explicit parameter condition. For \(0<r\leq\sigma\), we establish a new monotone heteroclinic connection. To fit the abstract framework, we embed the scalar equation into an auxiliary two-dimensional reaction–diffusion system and prove that every relevant traveling wave lies on the invariant diagonal. We then determine an explicit parameter region in which the required spectral hypotheses hold. Consequently, the original diffusive Nicholson equation admits traveling waves connecting the trivial and positive equilibria for all sufficiently large wave speeds.

Let $q_m=2m/(m+1)$ and put \[ \beta_0=\frac{3}{2}+\frac{1}{\log 2}=2.9426950408\ldots, \] where $\log$ is the natural logarithm. We give a proof scheme showing that, for every $\varepsilon>0$, there is $C_\varepsilon<\infty$ such that every complex-valued function $f:\{-1,1\}^n\to\C$ of Fourier degree at most $m$ satisfies \[ \left(\sum_{A\subseteq[n]}\abs{\wh f(A)}^{q_m}\right)^{1/q_m} \le C_\varepsilon m^{\beta_0+\varepsilon}\norm{f}_\infty. \] The improvement over the $m^9$ estimate of the earlier draft has two ingredients. The first three Fourier levels are estimated at the scales $m$, $m^{3/2}$, and $m^{5/3}$. These losses are encoded in the weight $M^{\mu_r/r}r^B\min\{r,L_M\}^8$, $L_M\asymp_B \log (M+1)$, with $\mu_r=\lceil3r/2\rceil$ for $r\ge2$. A parity-compatible central window handles $r<L_M$, while a shrinking balanced window handles $r\ge L_M\asymp_B\log(M+1)$, no parity is used in this regime. The leading high-degree loss–gain factor is $\ee\,2^{-B}$; careful uniform bounds close the bootstrap for every $B>1/\log2$. The formula for $\mu_r$ is explained below.

We prove the fixed-potential conjecture of Fefferman, Shapiro, and Weinstein for inversion-symmetric magnetic double wells. We construct a number $L_0>0$ and one smooth, compactly supported, nonpositive, nonradial single-well potential $v$, independent of both the coupling parameter $\lambda$ and the well displacement, such that for every fixed $L\ge L_0$ the associated magnetic double-well Hamiltonian has infinitely many exact degeneracies of its two lowest eigenvalues as $\lambda\to\infty$. The magnetic hopping coefficient also vanishes along an infinite sequence tending to infinity. The potential consists of a radial core and two small, reflection-related perturbations supported on one-sided quadratic cusps. Their log-flat profiles isolate two dominant hopping contributions, which we evaluate by steepest descent after inserting the radial ground-state asymptotics. The resulting asymptotic cosine expression, together with error bounds for the even–odd splitting, yields infinitely many exact eigenvalue crossings and changes of ground-state parity.

We prove sharp $L^{4}$ Strichartz estimates for two-dimensional periodic Schrödinger flows with integral diagonal symbols $ak_{1}^2+bk_2^2$ where $a,b \in \mathbb Z \setminus \{0\}$. The estimates exhibit a sharp arithmetic dichotomy according to whether $-ab$ is a square: for frequencies bounded by $N$, the optimal loss is $(\log N)^{1/4}$ when $-ab$ is not a square and $N^{1/4}$ when $-ab$ is a square. As a consequence, we show that arbitrarily small perturbations of the rectangular aspect ratio can lead to a transition between global well-posedness and norm inflation for cubic hyperbolic NLS.

In this article, we study unique continuation properties at infinity for solutions to generalized Schrödinger equations with potential functions that belong to the reverse Hölder class. For equations of the form $-div (A \nabla u) + V u = 0$ in $\mathbb{R}^n$, where $A$ is bounded and elliptic and $V \in RH_p$ for some $p \in [\frac n 2, \infty]$, we prove that if a solution doesn't grow too quickly at infinity, then it must be trivial. We use $d_V$, the Agmon distance function associated to $V$, to quantify the threshold growth rate. More precisely, there exists a constant $\gamma_0 > 0$ so that if $|u(x)| \lesssim \exp(\gamma d_V(x, 0))$ for some $\gamma < \gamma_0$ and every $x \in \mathbb{R}^n$, then $u$ must be trivial. The result may be interpreted as a Liouville-type theorem and is related to Landis' conjecture. Our proof techniques are inspired by Z. Shen's exponential decay estimates for fundamental solutions of Schrödinger operators and involve the application of a Fefferman-Phong inequality.

We develop a variational formulation of the Ornstein-Uhlenbeck evolution equation based on a Gaussian divergence structure, in which the drift term is absorbed into the first variation of a weighted Dirichlet energy rather than treated as a lower-order perturbation. For the linear equation, the distributional, energy weak, and variational notions of solution coincide and determine the same unique solution. The same framework yields a variational formulation of the associated parabolic obstacle problem, for which we prove existence and then uniqueness under a standard additional regularity assumption on the obstacle. The same mechanism applies to nonlinear Gaussian gradient flows associated with convex integrands of standard $p$-growth. Finally, we extend the Gaussian-divergence formulation to the force-free kinetic Fokker-Planck equation with periodic position variables, establishing the same equivalence for the linear equation.

We study a novel class of evolution variational inequalities (EVI) driven by bifunctions $\mathsf{b}:\mathrm{D} \times \mathrm{D} \to \mathbb{R}$ defined on a subset $\mathrm{D}$ of a metric space $(\mathrm{X},\mathrm{d})$ and satisfying a natural dissipativity condition $$\mathsf{b}(x,y)+\mathsf{b}(y,x)\le \eta \mathrm{d}^2(x,y).$$ We provide general conditions for existence, stability, regularity, approximation and asymptotic behavior of solutions, by proposing a metric framework that generalizes the classical structure of evolutions driven by monotone operators in Hilbert spaces. A motivating application is the setting of minimax and multispecies coupled gradient flows, in which each of $N$ species evolves in the direction of steepest descent of its own energy functional and the joint system is not a gradient flow of any single energy. To construct solutions, we introduce a variational movement scheme (VMS), a time discretization scheme in which each update is a saddle point of a metrically penalized bifunction, generalizing the classical minimizing movement/ JKO scheme for single species gradient descent. In the coupled multispecies case, the VMS reduces to a Nash equilibrium problem. We prove existence of discrete solutions to the VMS via a general approach which combines dissipativity of $\mathsf{b}$ with a new notion of convexity along suitable $\textit{barycentric interpolations}$ (which is inspired by convexity along generalized geodesics, a crucial notion for gradient flows). Together, these conditions provide a zeroth-order notion of game-theoretic monotonicity applicable to general metric spaces without linear structure.

We analyse the Cartesian perfectly matched layer (PML) approximation of the two-dimensional Helmholtz equation $-c^2\Delta u-\omega^2 u=f$, in which the propagation speed $c$ is variable inside the region of interest and equal to a constant $c_\infty$ outside a disc $B_R$. Existing analyses of the Cartesian PML assume a homogeneous, or piecewise homogeneous, background. We assume of $c$ only a two-sided bound, so that discontinuous speeds – layered media and compact inclusions – are covered. This is possible because the speed enters the PML sesquilinear form only through its zeroth order coefficient, the principal part being built from the stretching profiles alone. Well-posedness of the physical problem at this regularity follows from a Lippmann-Schwinger equation, uniqueness from Rellich's lemma and unique continuation. A Morawetz multiplier yields a sharp non-trapping condition on $c$, namely $\nabla c(x)\cdot(x-x_0)<c(x)$ for some $x_0$, which reduces in the radial case to the Herglotz condition $\frac{d}{d\rho}(\rho/c(\rho))>0$; its failure produces a trapped bicharacteristic. The variable speed is invisible to the essential spectrum: after factoring out $c^2$, the two operators differ by a compactly supported multiplication, so the spectral analysis of the homogeneous Cartesian layer applies verbatim. This yields well-posedness of the truncated problem for every $\omega>0$, with a stability constant independent of the layer width, and convergence to the physical solution in $H^1$ at a rate exponential in $\omega\gamma_M\delta_M$, where $\gamma_M$ and $\delta_M$ are the absorption strength and the layer thickness. Numerical experiments with a low-velocity inclusion confirm the exponential decay in both $\delta$ and $\omega$.

The aim of this work is to develop a self-contained existence and uniqueness theory for a doubly nonlinear parabolic problem with variable exponents and homogeneous Neumann boundary conditions. Our approach is based on Rothe's time-discretization method and builds on the results established by the author in a previous paper. We also investigate the asymptotic behavior of the solution. All arguments are presented in full detail, so that the paper is self-contained.

We prove linear stability of Kasner spacetimes in the direction of the Big Bang, up to an explicit finite dimensional space of non-decaying self-similar solutions. The fastest growing solution is the linearization of Taub's explicit Bianchi II solution, describing a Kasner transition. All other non-decaying solutions are inhomogeneous analogues of this, and the linearized Kasner metric. The key novelty is a notion of quasinormal modes on Kasner spacetimes, similar to quasinormal modes on stationary black holes spacetimes. All quiescent (as opposed to oscillatory) vacuum Big Bang spacetimes of dimension 4 are asymptotic to a Kasner spacetime from the perspective of a single observer at the Big Bang. They are expected to be highly unstable due to the oscillations conjectured by Belinski, Khalatnikov and Lifschitz. This paper provides a complete description of this instability on a linear level without symmetry assumptions.

OpenAI~\cite{OpenAIManuscript} has recently announced a proof of finite time singularity formation for the 3D Navier-Stokes equations, in the presence of a $C^\infty$-smooth body force. The construction in~\cite{OpenAIManuscript} has a few key features, among which we single out: (i) the angular mean of the solution satisfies specific Type II bounds which are anisotropic; (ii) the solution is exactly axisymmetric in a collapsing core region. In this paper we consider solutions of the 3D Navier-Stokes equations in the presence of a real-analytic body force, and assume that these solutions satisfy properties (i) and (ii) above. We prove that such solutions are in fact regular at the putative singular point. As a consequence, in the construction of~\cite{OpenAIManuscript}, and in any construction with properties (i) and (ii) whose force remains bounded in $C^2$ up to the singular time, the force can neither vanish identically near the singular point, nor be real analytic in the space variables, locally uniformly in time. The main idea of the proof is inspired by our earlier work~\cite{CIV26} and the companion paper~\cite{CIVEulerLength}: zooming-in at the putative singularity using the anisotropic length scales provided by the a priori bounds, we arrive at ancient limits whose PDE evolution imposes additional rigidity.

We study a singular Dirichlet problem driven by a regularized $p(x)$-Laplacian under a prescribed modular constraint. Using growth and differentiability estimates for the regularized energy, together with coercivity and modular compactness, we construct nonnegative constrained minimizers for a family of nonsingular approximating functionals. Under the admissibility of signed constrained variations, each minimizer satisfies an approximate Euler–Lagrange equation with a uniquely determined constraint multiplier. An explicit uniform boundary lower barrier and a compatible weight condition, combined with a variable exponent Hardy inequality, provide global dual control of the singular sources and convergence against every zero-trace Sobolev test function. For these approximate minimizing pairs, the singular-limit result gives subsequential strong Sobolev convergence of the states and convergence of their multipliers, while preserving positivity and the prescribed modular. The limiting pair is therefore a positive normalized weak solution of the singular problem, obtained by removing the reaction regularization while keeping the gradient regularization fixed.

We prove local regularity for axisymmetric suitable weak solutions of the 3D Navier-Stokes equations, which are smooth before the terminal time $t=0$, and satisfy Type~II pointwise bounds at a vanishing length scale $\ell(t)$. We say $\ell(t)$ is an Euler length if it is non-increasing, satisfies a doubling condition, and if $\ell(t)\to0$ and $(-t)/\ell(t)^2\to0$ as $t\to0^-$. This includes power laws $\ell(t)=(-t)^\gamma$ with $0<\gamma<1/2$, and logarithmic enlargements of the parabolic length. Our main result shows that local bounds of the type $|u(\cdot,t)| \leq C\ell(t)/(-t)$ and $|\nabla^2 u(\cdot,t)|\leq C/((-t)\ell(t))$ for all $t\in (-1,0)$ imply regularity. The proof adapts the circulation and potential-vorticity argument of our earlier paper~\cite{CIV26} to ancient limits obtained from the rescaled vorticity system by zooming in. We show that the second derivative a priori assumption may be replaced by a Hölder bound on the azimuthal vorticity and a corresponding bound on its potential vorticity.

We prove existence and uniqueness of a strong initial trace for every bounded kinetic solution of a stochastic scalar conservation law, although no initial value is prescribed and no nondegeneracy condition is imposed on the flux. The core of the proof is pathwise. After fixing a realization, the rescaled stochastic terms and kinetic measure vanish in the blow-up limit, so Panov's compactness argument applies; degenerate flux intervals are handled by recursive dimension reduction. The stochastic setting creates several additional difficulties. The martingale identities must remain valid on one common full-probability set throughout the reductions, which requires a parameterized stochastic-Fubini construction. Moreover, the pathwise trace is not automatically measurable because its exceptional sets may depend on the realization. Deterministic time averages and right-continuity of the filtration yield a jointly measurable initial trace, with local strong convergence along essential times, both almost surely and in mean.

We study the free minimization problem for a nonlinear, nonlocal functional associated with Robin-type nonlocal exterior conditions depending on a positive parameter $\alpha$. We prove that, for every $\alpha>0$, the minimizer exists, is unique, and satisfies suitable decay properties at infinity. We also investigate the regularity of the maps $\alpha \mapsto u_\alpha$ and $\alpha \mapsto E_\alpha$, where~$u_\alpha$ denotes the minimizer and $E_\alpha$ the corresponding energy. Finally, we derive asymptotic expansions of the energy as $\alpha\to+\infty$ and as $\alpha\to0^+$. The latter regime requires distinguishing between two cases, depending on whether the forcing term has zero mass. In one case, the limiting energy possesses a minimizer and $E_\alpha$ converges to its energy, but in the other case $E_\alpha$ diverges to $-\infty$.

We prove that solution maps of the Vlasov–Poisson equation are discontinuous at the zero initial datum in $X_p=L_x^pL_v^\infty$ for every $1\leq p<\infty$, in $\mathbb{R}^{d} \times \mathbb{R}^{d}$ with $d \le 3$, and for both attractive and repulsive interactions. For every $T>0$, the trajectory of the solution in $L^\infty([0,T];X_p)$ is discontinuous at zero, and for every sufficiently small fixed $t>0$, the fixed-time map with values in $X_p$ is also discontinuous at zero. The counterexamples are smooth, nonnegative, bounded by one, and supported in a common compact subset of phase space. Their mass and initial $X_p$ norm tend to zero, whereas their solution norm at the observation time is at least one. The construction places narrow spatial packets on a lattice. Free transport allows a different velocity to select a packet at each spatial point, while a smallness estimate on the field ensures that the nonlinear characteristics are close to that of the free transport. For the same families, the phase-space $L^q$ norms converge to zero uniformly in time for every $1\leq q<\infty$.

Let $m \ge 4$. We prove that there is a forward orbit of the 2D Euler system for scale-invariant $m$-fold symmetric vorticities, namely \begin{equation*} \partial_tg + 2G\partial_\theta g = 0, \quad 4G + \partial_{\theta\theta}G = g, \end{equation*} which is dense in $S :=\{g \in L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1): g\sin(m\theta) \ge 0 \text{ and }\|g\|_{L^\infty} \le 1\}$ equipped with the topology of weak$^*$ convergence on $L^\infty(\mathbb{S}^1)$, where $L_{m,\mathrm{odd}}^\infty(\mathbb{S}^1)$ denotes the space of odd and $m$-fold symmetric functions in $L^\infty(\mathbb{S}^1)$. The system above was first derived by Elgindi and Jeong, who showed well-posedness in the $m$-fold symmetric class. In subsequent work by Elgindi, Murray and Said, it was shown that solutions with regulated vorticity relax to piece-wise constant steady states with finitely many jumps; our construction shows that this regularity assumption cannot be relaxed. The proof starts from a simple linear mechanism: by evolving prescribed step functions backwards from chosen times and superimposing the resulting data, one can arrange for a single linear orbit to approximate a countable dense family. We show that this mechanism persists for the nonlinear Euler coupling. The key stability estimates imply that the error introduced at each stage becomes arbitrarily small when the next approximation time is taken sufficiently far in the future, allowing the construction to be iterated indefinitely.

In this paper, we investigate the stochastic two-dimensional anisotropic magnetohydrodynamics system on $\mathbb T^2$ with velocity dissipation acting only in the horizontal direction. Under multiplicative noise, we prove the existence of global martingale solutions and global probabilistically weak solutions, together with pathwise uniqueness.

For every dimension $m\geq 3$, we show the existence of an open set $U\subset \mathbb R^{m+1}$, a smooth Riemannian metric $g$ on it, and an integral $m$-dimensional stationary varifold $V$ in $(U,g)$ with the following properties. $V$ has a flat tangent plane of multiplicity $2$ at an interior point $p$, it is a smooth immersed minimal surface in $U\setminus \{p\}$, and it has infinite topology in any neighborhood of $p$. The construction starts from an idea of three of the authors, who in \cite{DHS} tried to build a similar example with $C^{m-1, \alpha}$ regularity for $g$. That previous attempt contained an important error, which has been overcome using a crucial suggestion of gpt-astra-6.

In this paper, we use optimal mass transportation to derive an isoperimetric inequality and an interior Michael-Simon-Sobolev inequality for weakly twice differentiable integral varifolds of dimension and codimension at least $2$, with locally bounded first variation and $L^2_{loc}$ mean curvature. This generalizes a result of Brendle and Eichmair. In both inequalities the corresponding constants are optimal in codimension 2. As part of this process we also generalize Alexandrov's theorem for convex functions to this setting. This necessitates the use of measure and distribution-valued solutions to certain geometric differential operators, in addition to the theory of $L^p$ Taylor expansions started by Calderón and Zygmund.

In this paper, we study a heat-wave fluid-structure interaction system posed on a periodic strip geometry and investigate whether higher-order $H^s$-estimates for arbitrarily large values of $s$ can be established for the coupled system despite the inherent mismatch of parabolic and hyperbolic regularity. We establish global existence of weak solutions and Sobolev regularity up to the $H^3$-level, which, to the best of our knowledge, is the highest regularity currently established for this system. We further show that, even for arbitrarily regular initial data, neither the semigroup approach through higher-order generator domains $D(A^k)$ with $k\in\mathbb{N}$ nor the PDE-based tangential-normal recovery procedure yields closed estimates beyond $H^3$ due to the complexities of the coupled PDE system.

We investigate localization in a time-dependent contrained obstacle problem for the density of an active protein on a cell membrane. This free boundary problem has been derived in Logioti et al. [17] as a model for cell polarization under the influence of an external signal. We show that in the limit of small mass there is instantaneous localization, that is the mass of the active protein concentrates in the set where the signal is maximal for any positive time $t>0$. We give a precise a characterization of this concentration process on the appropriate slow time scale and illustrate this for a few examples of the external signal.

We establish the existence and uniqueness of solutions to a $(1+1)$-dimensional fractional stochastic Burgers-type equation featuring a Caputo fractional time derivative, a fractional Laplacian, and space-time white noise forcing modified by a Riemann-Liouville fractional integral. From a physical perspective, equations of this type describe non-Markovian dynamics with long-range temporal and spatial dependencies. The main difficulty is that the solution to the linearized equation has low spatial Hölder regularity, rendering the singular nonlinearity $G(u)\partial_x u$ classically ill-defined. The proof relies on two main ingredients: first, combining the Da Prato-Debussche decomposition with rough path theory to rigorously interpret the nonlinear product; and second, deriving Schauder-type estimates for fractional heat kernels given in terms of slowly decaying Mittag-Leffler functions. The nonlocality of the fractional time derivative requires a strong assumption on the initial condition.

In this paper, we study the modified Korteweg-de Vries-Burgers (mKdVB) equation \[u_t + (u^3)_x = \mu u_{xx} - \kappa u_{xxx}\] with $\mu>0$ and $\kappa>0$. Since the flux is non-convex, the (inviscid) Riemann problem may be solved by a composite wave of a degenerate Oleinik shock and a rarefaction wave. We prove the global existence of solutions and the time-asymptotic stability of the corresponding viscous-dispersive composite wave, up to a time-dependent shift, under small $H^1$ perturbations. The shock strength $\delta_S$ and the rarefaction strength $\delta_R$ need not be small and are restricted only by the explicit ratio $\delta_R\le\frac{5}{18}\delta_S$ and the quantity $\kappa\delta_S^2/\mu^2$ governing the monotonicity of the shock. The stability estimate is uniform with respect to the dispersion strength $\kappa$. As a key ingredient of the stability analysis, we establish structural properties of the degenerate viscous-dispersive shock profile in the monotone regime, including two-sided pointwise bounds on its derivative and the rates at which it converges to its two end states. These decay rates coincide with those of the corresponding purely viscous profile, with constants independent of the dispersion coefficient. In the absence of dispersion, namely, when $\kappa=0$, the time-asymptotic stability of the corresponding viscous composite wave was established by Huang, Wang and Zhang [26]. The present work extends their result to the viscous-dispersive setting and relaxes the restriction on the rarefaction strength. The proof relies on the method of $a$-contraction with shifts (for viscous conservation laws) developed by Kang and Vasseur [32,33,35].