Papers

We study the one-dimensional Zakharov-Rubenchik system with periodic initial conditions. This system models the interaction of low-frequency waves with high-frequency waves in different physical phenomena. Using semigroup theory, we prove the global well-posedness of the associated linear equation in $H^q(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})$, $q,s \in \mathbb{R}$. Subsequently, we consider the modified Zakharov-Rubenchik system and, by means of a change of variables and the Banach Fixed Point Theorem, we obtain local well-posedness in the space $H^s(\mathbb{T})\times H^s(\mathbb{T})\times H^{s+1}(\mathbb{T})$, $s>3/2$.

We study a generalization of the classical optimal insulation problem by replacing the standard Laplacian with the \ti{p-Laplacian}, leading to a \ti{p-Poisson equation} with Robin boundary conditions. We also investigate the corresponding eigenvalue problem. This work extends and complements some results from \cite{bu1,pietra1,pietra2} to the nonlinear setting.

We study the behaviour of a given volume of liquid confined between two rough solid plates. When the separation between the plates is small relative to the liquid volume, capillary bridges are expected to form, which minimise Gauss's capillary energy locally. We derive a $\Gamma$-expansion for the energy as the plate separation approaches zero, yielding a dimensionally reduced problem in terms of the wetted regions on the plates. At leading order, the energy is determined by the area of the wetted regions, while the next order term is given by their perimeter, weighted by appropriate functions of the relative adhesion coefficients. This provides a framework for a subsequent phase-field approximation, which is employed in numerical simulations to study the evolution of the droplets under the normal movement of the plates. The theory also gives justification to models for capillarity that assume that liquids fill up the rough topography to the Kelvin radius.

We introduce defect measures for the heat flow of a class of harmonic maps with free boundary and initiate their analysis. Our flow is motivated by singular perturbations on the boundary of the domain, emphasizing structural differences between harmonic maps with free boundary and their classical counterpart. The harmonic maps we consider are a reformulation of maps introduced by Da Lio and Rivière and arise in several problems in the spectral geometry of Steklov eigenvalues. Contrary to the classical case, the flow we consider exhibits features of global nature and the blow-up analysis turns out to be much more challenging.

This paper studies the simultaneous recovery of point-source locations and temporal strengths in both the drift and diffusion terms of a stochastic parabolic equation with Neumann boundary conditions, using only boundary observations on a nonempty portion of the boundary after the sources have become inactive. By separating the stochastic equation into its expectation and centered parts, the drift channel reduces to a deterministic source problem with scalar Laplace moments, while the diffusion channel is controlled by Itô's isometry, which preserves the full $L^2$-energy of the integrand. Under suitable nondegeneracy and separation assumptions, we establish Lipschitz stability for the location of a single source in either channel, logarithmic stability for the drift strength, and Lipschitz stability for the complete diffusion strength including its sign. For multiple sources, we obtain Lipschitz stability up to permutation for drift locations and weighted temporal moments in dimensions two and three, and for diffusion locations and strengths in dimensions one through three. The proof employs explicit adjoint probes, including complex-isotropic polynomial annihilators in dimensions two and three and spectral synthesis in one dimension, combined with boundary-to-source estimates derived from Green's identities. These results extend and improve upon existing deterministic stability theories by revealing the fundamentally different temporal information carried by stochastic integrals, a distinction that is further illustrated by numerical experiments.

We introduce and study a Yamabe-type variational problem associated with the curl operator on middle-degree forms on a closed oriented Riemannian manifold $(M^n,g)$ with $n\equiv3\pmod4$. The corresponding curl–Yamabe constant is defined by minimizing a conformally invariant, gauge-invariant quotient built from $\|{\rm curl}\,\alpha\|_{\frac{2n}{n+1}}$ and $\int_M\langle{\rm curl}\,\alpha,\alpha\rangle$, and its Euler–Lagrange equation is the critical curl–Yamabe equation ${\rm curl}\,\alpha=\frac{n+1}{2}|\alpha|^{\frac{2}{n-1}}\alpha$. Using sharp curl–Sobolev inequalities on the sphere and an Aubin-type cut-off construction, we prove that $0<Y_{\rm curl}(M,[g])\le Y_{\rm curl}(\mathbb{S}^n)$ for every such manifold, and that the infimum is attained whenever $Y_{\rm curl}(M,[g])<Y_{\rm curl}(\mathbb{S}^n)$. Moreover, for $n>3$ we establish the strict inequality for manifolds that are not locally conformally flat, via a test-form expansion detecting the Weyl tensor, and deduce existence of a minimizer solving the curl–Yamabe equation.

We establish the global well-posedness of weak solutions to the Cauchy problem for a broad class of multi-dimensional $n\times n$ Keyfitz-Kranzer type systems with homogeneous flux. Our proof relies on a novel viscous approximation that yields the necessary compactness estimates. By exploiting the specific structure of this proposed approximation, we derive a priori uniform $L^{\infty}$ bounds and prove the $L^1_{loc}$ precompactness of the sequence of approximate solutions. This framework allows us to rigorously justify the vanishing viscosity limit. Furthermore, under suitable assumptions on the initial data, we employ the relative entropy method to analyze the $L^2$-time decay of large perturbations of the viscous shock, up to a dynamical shift.Evan

We prove that for every dimension $n \geq 2$ and parameter $p \in ]1,n[$, the hedgehog $u_0(x) = x/|x|$ is the unique minimiser of the $p$-energy $\mathcal{E}_p [u]:=\int_{\mathbf{B}^n}|\nabla u|^p\,{\rm d}x$ in the class of $W^{1,p}$-mappings from the $n$-dimensional unit ball $\mathbf{B}^n$ to the $(n-1)$-dimensional unit sphere $\mathbf{S}^{n-1}=\partial\mathbf{B}^n$ whose trace equal to the identity on $\mathbf{S}^{n-1}$. This answers in the affirmative an open question on p.29 in [Robert M. Hardt, Singularities of harmonic maps, Bull. Amer. Math. Soc. (N.S.) 34 (1997), 15–34].

Consider the heat equation: \begin{equation}\label{eq:00} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\,dt+b(u(t,x))\,dt, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} It is well known that superlinear growth of the coefficient $b$ may lead to finite-time blow-up. In this paper, we show that, for every locally Lipschitz reaction coefficient $b:\R\to\R$, there always exists a locally Lipschitz noise coefficient $\sigma:\R\to\R$ such that the stochastic heat equation \begin{equation}\label{eq:01} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\,dt+b(u(t,x))\,dt+\sigma(u(t,x))\,dB_t, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} has a unique global solution for every $u_0\in C_c(D)$. Here $(B_t)_{t\ge0}$ is a real-valued Brownian motion. In particular, if $b=0$, the equation has a global solution for every locally Lipschitz $\sigma$, without any growth restriction at infinity. The proof uses reflected scalar barriers, a Lyapunov estimate, truncation, and comparison techniques.

Shubin, Vakilian, and Wolff proved a Fourier uncertainty principle for sets of sufficiently small local density at the reciprocal scale and asked whether every density below one is admissible. We answer this question negatively in dimension one by constructing sequences of pairs of $1/2$-density thin sets and unit vectors whose total position and Fourier mass outside these sets tends to zero. This obstruction persists for every scaled reciprocal profile $\rho_\kappa(x)=\min\{1,\kappa/|x|\}$, $\kappa>0$. On the other hand, for $0<\kappa\le1$, the uncertainty estimate holds whenever the density $$ \varepsilon<\frac{1}{2(1+32\kappa)}. $$ Thus the critical density tends to $1/2$ as $\kappa\downarrow 0$. The obstruction uses odd Gaussian packets to transfer norm bounds from a free-group model. The positive estimate uses a Fourier-complementary anti-Wick operator and quadratic straightening of the reciprocal geometry.

It is well known that solutions to the compressible Euler equations can develop singularities in finite time. In this paper, we carry out a detailed analysis on behaviors of solutions up to the time of the first singularity for the one-dimensional compressible Euler equations with general smooth initial data. Our main results consist of three parts. First, for an open dense set of $C^3$ initial data, we show that the solution of Euler equations is twice continuously differentiable except at most finitely many points when the first singularity happens, using Thom's Transversality Theorem. Second, for any initial data in the open dense set of $C^3$ functions given in the first result, we provide the precise asymptotic description of the solution in a semi-neighborhood in the $(x,t)$-plane of each singular point at the time of the first singularity, and verify that the solution has a cusp-type singularity with Hölder exponent $1/3$ at each singular point. The proofs of the first two results are based on the representation of the solution in terms of a semilinear system. Third, for smooth initial data with small BV norm, we construct two Finsler type optimal transport metrics, then under these metrics show that the solution depends Lipschitz continuously on the initial data up to the time of the first singularity, with uniformly bounded Lipschitz constants. In particular, the $C^{1/3}$ generic singularity is stable in this sense. On the other hand, since our first two results hold for an open and dense set of initial data, any Hölder continuous cusp singularity with exponent other than $1/3$ is unstable under initial perturbations.

In this paper, we investigate the analytic properties of the Hardy-type operator $-\Delta + \mu H_0$ on the $d$-dimensional integer lattice graph $\mathbb{Z}^d$, where $\mu \geq -1$ and $H_0$ is a critical Hardy potential at infinity–arising naturally from discrete Hardy inequalities. We derive sharp fixed-pole Green kernel estimates on $\mathbb{Z}^d$ and obtain a necessary and sufficient weighted summability condition characterizing the solvability of the nonnegative Poisson problem associated with this operator. Building upon this classification, we provide a complete characterization of the nonexistence of positive solutions to the semilinear elliptic inequality $$ -\Delta u + \mu H_0 u \geq W u^p \quad \text{in } \mathbb{Z}^d, $$ where $p > 0$, $d \geq 3$, $\mu \geq -1$, and $H_0, W \in C(\mathbb{Z}^d)$ are strictly positive potentials satisfying the asymptotic behaviors $H_0(x) \sim |x|^{-2}$ and $W(x) \sim |x|^{\theta}$ as $|x| \to \infty$, with $\theta \in \mathbb{R}$.

In the one-dimensional log-modified nonlinear Schrödinger equation, there are finitely many normalised standing-waves, up to space translation and phase multiplication. In the defocusing case, there is exactly one normalised standing-wave having a prescribed mass and least energy. Both results are obtained by exploiting properties of the mass function: the analyticity in the focusing case, and the monotonicity in the defocusing case.

We prove the Alexandrov–Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov–Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.

We establish two rigidity theorems for positive rupture solutions of the conformally invariant equation $\Delta^2 u=u^{-7}$ in $\mathbb R^3\setminus\{0\}$, which extend continuously to the origin with $u(0)=0$. First, we prove that if the associated conformal metric $g=u^{-4}|dx|^2$ has nonnegative scalar curvature, then every such solution is radially symmetric and has the sharp rupture profile $u(x)\sim(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$ as $x\to 0$; in particular, the metric is complete at the origin. The principal novelty is a global rigidity theorem requiring neither curvature nor symmetry: the single global condition $u(x)=o(|x|)$ at infinity forces $u(x)\equiv(\frac{4}{3})^{\frac{1}{4}}|x|^{\frac{1}{2}}$. This result provides a sharp answer to the uniqueness question posed by McKenna and Reichel in a class with no a priori symmetry assumption. Finally, we construct complementary examples demonstrating the essential roles of the curvature and growth hypotheses.

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 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).

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 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.

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].

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 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.

Dispersive and Strichartz estimates are fundamental tools for establishing the well-posedness and long-time behavior of solutions to nonlinear partial differential equations. While these estimates are well-understood in the boundaryless Euclidean setting, the presence of a geometric boundary introduces severe analytical complexities, such as the continuous formation of caustics. In this work, we establish sharp local-in-time dispersive estimates for the semiclassical Schrödinger equation inside a three-dimensional cylindrical domain $\Omega \subset \mathbb{R}^3$ subject to homogeneous Dirichlet boundary conditions. This paper provides the first comprehensive microlocal treatment for this anisotropic geometric setting, extending the optimal strictly convex boundary results of Ivanovici \cite{Ivanovici2023} to the parabolic setting. The primary analytical challenge in our cylindrical model stems from the fact that the boundary curvature is non-uniform, depending explicitly on the tracking angle of classical trajectories and vanishing identically along the flat longitudinal axis. Crucially, we demonstrate that the quadratic structure of the Schrödinger phase function ($\partial_\zeta^2 \Phi = 2t$) establishes global non-degeneracy, completely bypassing the arduous low-frequency trajectory ray-tracing mandatory in hyperbolic wave equations. By exploiting this structural advantage alongside a streamlined Littlewood-Paley dyadic block decomposition, we prove that the zero-frequency axial tail can be consistently integrated down to the flat limit $\eta=0$. This yields sharp global Strichartz estimates featuring an explicit derivative loss exponent of $\rho(q) = \frac{3}{2}\left(\frac{1}{2}-\frac{1}{q}\right)$.

We prove a Jacobian estimate underlying both the Heintze–Karcher comparison and the Alexandrov–Bakelman–Pucci method for submanifolds of arbitrary codimension. The estimate retains the contributions of ambient curvature along geodesics emanating from the submanifold and requires no curvature sign assumption. As applications, we obtain a quantitative Fenchel–Willmore inequality with explicit curvature remainders under nonnegative $n$-intermediate Ricci curvature, Michael–Simon Sobolev and isoperimetric inequalities under nonnegative $(n-1)$-intermediate Ricci curvature, and corresponding extensions under quadratic curvature decay. In each result, the required intermediate Ricci curvature condition depends only on the dimension of the submanifold and is independent of its codimension.

We prove sharp dispersive $L^1 \to L^\infty$ estimates for the three-dimensional attractive Coulomb operator $H_Z=-\Delta-Z|x|^{-1}$, where $Z>0$. The absolutely continuous part of the Schrödinger evolution decays at the free rate for short times, whereas its leading contribution decays like $|t|^{-1}$ for long times, with amplitude proportional to $Z$. This slower decay is driven by the threshold and is sharp when $Z^2|t|\gg1$.

We present an analytical solution for 1D atmospheric pollutant dispersion in a bounded domain. Using coordinate transformation and spectral decomposition, we obtain a Fourier series solution for time-dependent wind velocity. For constant wind, Laplace inversion yields dual representations via residue calculus and Poisson summation. The solution extends to semi-infinite domains, recovering classical error-function profiles.

We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property $TMCP^+(K,N)$. The cost for this problem is the time separation function $\ell(x,y)$ (the Lorentz distance), which represents the maximum amount a particle can age when traveling from $x$ to $y$. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we show that the Lorentzian timelike measure contraction property $TMCP^+(K,N)$ is equivalent to the $TMCP^+_{rLip}(K,N)$ condition, where the curvature dimension conditions are defined on gradient flow curves of reverse 1-Lipschitz functions.

In the pioneer work of vectorial free boundary problems \cite[Adv. Math. 280 (2015)]{ASUW15}, Andersson, Shahgholian, Uraltseva, and Weiss investigated the regularity theory of solutions and the free boundary of a singular cooperative system \begin{equation*} \Delta\mathbf{u} = \frac{\mathbf{u}}{|\mathbf{u}|} \chi_{\{|\mathbf{u}|>0\}} \quad\text{in }\Omega\subset\mathbb{R}^n, \qquad \mathbf{u}:\Omega\longrightarrow \mathbb{R}^m, \qquad n,m\ge 2. \end{equation*} At the level of solutions, the boundedness of the right-hand side directly yields $W^{2,p}_{\mathrm{loc}}$ regularity for every $1<p<\infty$ by standard elliptic estimates, while the corresponding $W^{2,\infty}_{\mathrm{loc}}$ regularity question was explicitly left as an open problem on page~753 of that paper. In this paper, we give a positive answer by establishing optimal $W^{2,\infty}_{\mathrm{loc}}$ regularity through a scale-invariant interior Hessian estimate for every weak solution. Unlike the scalar case, the classical Alt–Caffarelli–Friedman monotonicity formula cannot be applied directly to the present system. Our proof instead combines a localized Newtonian potential decomposition with an explicit projection onto quadratic harmonic polynomials on the unit sphere, which isolates the symmetric trace-free quadratic part of the Hessian. A key new observation is that the resulting matrix coefficients satisfy an exact linear ordinary differential equations with constant coefficients on the logarithmic scale. The main novelty of this work lies in a so-called two-regime argument based on the competition between the affine and quadratic parts of the rescaled solution, together with a continuity argument that connects the two regimes and yields a uniform bound for the coefficient. The resulting a priori estimate provides an analytic basis for further study of the fine structure of the free boundary.

We study the one-dimensional viscous Burgers equation on $(0,L)$ with the conservative boundary conditions $u_x=-u^2$. On the half-line, solutions approach a nonlinear self-similar profile, whereas on a bounded interval they converge to a nonconstant equilibrium. We describe explicitly how the dynamics passes between these states at the critical diffusive scale $t=cL^2$. For compactly supported initial data of mass $M$, the rescaled solution converges as $L\to\infty$ to an explicit crossover profile $\Phi_c$. In similarity variables, $\Phi_c$ converges to the half-line profile $f_M$ as $c\downarrow0$; after rescaling to domain variables, it converges to the interval equilibrium as $c\to\infty$. We also determine the sharp onset of confinement: \[ \lim_{c\downarrow0}-c\log|\Phi_c(0)-f_M(0)|=1\qquad(M\ne0). \] Moreover, on compact sets in similarity variables, the interval and half-line solutions differ in $C^k$ by at most $C_{k,\varepsilon}\exp(-(1-\varepsilon)L^2/t)$, uniformly in $L$, and the exponential constant is optimal. Thus we identify not only the transition scale $L^2$, but also the profile governing the crossover and the sharp rate at which the remote boundary becomes visible. We finally discuss the conclusions that persist for space-dependent diffusivity and illustrate the three asymptotic regimes numerically.