We study growth of solid tumors in a partial differential equation model for the interaction between tumor cells (TCs) and cancer stem cells (CSCs). We find that invasion into the cancer-free state may be separated into two regimes, depending on the death rate of tumor cells. In the first, staged invasion regime , invasion into the cancer-free state is lead by tumor cells, which are then subsequently invaded at a slower speed by cancer stem cells. In the second, TC extinction regime , cancer stem cells directly invade the cancer-free state. Relying on recent results establishing front selection propagation under marginal stability assumptions, we use geometric singular perturbation theory to establish existence and selection properties of front solutions which describe both the primary and secondary invasion processes. With rigorous predictions for the invasion speeds, we are then able to heuristically predict how the total cancer mass as a function of time depends on the TC death rate, finding in some situations a tumor invasion paradox , in which increasing the TC death rate leads to an increase in the total cancer mass. Our methods give a general approach for verifying linear determinacy of spreading speeds of invasion fronts in systems with fast-slow structure.

This paper is motivated by the “transfer of regularity” phenomenon for the incompressible Navier–Stokes equations (NSE) in dimension n ≥ 3 n \geq 3 ; that is, the strong solutions of NSE on R n \mathbb {R}^n can be nicely approximated by those on sufficiently large domains Ω ⊂ R n \Omega \subset \mathbb {R}^n under the no-slip boundary condition. Based on the spacetime decay estimates of mild solutions of NSE established by Miyakawa [On space-time decay properties of nonstationary incompressible Navier-Stokes flows in R n \mathbf {R}^n , Funkcial. Ekvac. 43 (2000), no. 3, 541–557], Schonbek [ L 2 L^2 decay for weak solutions of the Navier-Stokes equations, Arch. Rational Mech. Anal. 88 (1985), no. 3, 209–222], and others, we obtain quantitative estimates on higher-order derivatives of velocity and pressure for the incompressible Navier–Stokes flow on large domains under certain additional smallness assumptions of the Stokes system and/or the initial velocity, thus complementing the results obtained by Robinson [Using periodic boundary conditions to approximate the Navier-Stokes equations on R 3 \mathbb {R}^3 and the transfer of regularity, Nonlinearity 34 (2021), no. 11, 7683–7704] and Ożánski [Quantitative transfer of regularity of the incompressible Navier-Stokes equations from R 3 \mathbb {R}^3 to the case of a bounded domain, J. Math. Fluid Mech. 23 (2021), no. 4, Paper No. 98, 14].

In this paper, we investigate the existence, uniqueness, and exponential stability of almost periodic mild solutions for the chemotaxis-Navier-Stokes system with a matrix-valued sensitivity on a real hyperbolic space H d ( R ) ( d ⩾ 2 ) \mathbb {H}^d(\mathbb {R})\, (d\geqslant 2) and on the whole-line time-axis R t \mathbb {R}_t . First, we prove the existence and uniqueness of almost periodic mild solutions for the corresponding linear systems by using the dispersive estimates of the scalar and vectorial heat semigroups and the boundedness of the matrix-valued sensitivity. Then, we use the results of linear systems and fixed-point arguments, and we establish the well-posedness of almost periodic mild solutions for the chemotaxis-Navier-Stokes systems. Finally, we employ the Gronwall inequality to prove the exponential stability of such solutions. Our obtained results are also valid for such systems on a bounded domain with smooth boundary in the Euclidean space R d ( d ⩾ 2 ) \mathbb {R}^d\, (d\geqslant 2) .

We study the emergent behaviors of the Lohe model on the unit hyperbolic sphere H d \mathbb {H}^d under more general interconnection topologies. In previous literature, the Lohe model on H d \mathbb {H}^d was studied under the complete graph. Using the LaSalle invariance principle, we show that the Lohe model on H d \mathbb {H}^d converges to a synchronized state for directed graphs containing spanning trees if all oscillator frequencies are equal. Moreover, we show that exponential convergence is achieved by analyzing the error dynamics of the Lohe model on H d \mathbb {H}^d . We also provide several numerical simulations and compare them with theoretical results .

This paper is concerned with the existence of global-in-time weak solutions to the multicomponent reactive flows inside a moving domain whose shape in time is prescribed. The flow is governed by the 3D compressible Navier-Stokes-Fourier system coupled with the equations of species mass fractions. The fluid velocity is supposed to fulfill the complete slip boundary condition, whereas the heat flux and species diffusion fluxes satisfy the conservative boundary conditions. The existence of weak solutions is obtained by means of suitable approximation techniques. To this end, we need to rigorously analyze the penalization of the boundary behavior, viscosity and the pressure in the weak formulation .

In this paper we show the persistence property for solutions of the derivative nonlinear Schrödinger equation with initial data in weighted Sobolev spaces H 2 ( R ) ∩ L 2 ( | x | 2 r d x ) H^{2}(\mathbb {R})\cap L^2(|x|^{2r}dx) , r ∈ ( 0 , 1 ] r\in (0,1] .

We consider the question of global existence of smooth solutions to a multi-species aggregation-diffusion equation for a class of singular interaction kernels. We establish a smallness condition on the initial data which yields global existence of smooth solutions. We also give conditions on the species interaction which ensure that pointwise inequalities comparing species densities are preserved by the evolution.

This paper asks if the following iterative procedure approximately orthogonalizes a set of n n linearly independent unit vectors while preserving their span: in each iteration, access a random pair of vectors and replace one with the component perpendicular to the other, renormalized to be a unit vector. We provide a positive answer: any given set of starting vectors converges almost surely to an orthonormal basis of their span. We specifically argue that the n n -volume of the parallelepiped generated by the vectors approaches 1 (i.e. the parallelepiped approaches a hypercube). If A A is the matrix formed by taking these vectors as columns, this volume is simply det ( | A | ) \det (\lvert A\rvert ) where | A | = ( A ∗ A ) 1 / 2 \lvert A\rvert =(A^*A)^{1/2} . We show that O ( n 2 log ⁡ ( 1 / ( det ( | A | ) ε ) ) ) O(n^2\log (1/\left (\det (\lvert A\rvert )\varepsilon \right ))) iterations suffice to bring det ( | A | ) {\det (\lvert A\rvert )} above 1 − ε 1-\varepsilon with constant probability.

In the present work, we revisit the topic of translational eigenmodes in discrete models. We focus on the prototypical example of the discrete nonlinear Schrödinger equation, although the methodology presented is quite general. We tackle the relevant discrete system based on exponential asymptotics and start by deducing the well-known (and fairly generic) feature of the existence of two types of fixed points, namely site-centered and inter-site-centered. Then, turning to the stability problem, we not only retrieve the exponential scaling (as e − π 2 / ( 2 ε ) e^{-\pi ^2/(2 \varepsilon )} , where ε \varepsilon denotes the spacing between nodes) and its corresponding prefactor power-law (as ε − 5 / 2 \varepsilon ^{-5/2} ), both of which had been previously obtained, but we also obtain a highly accurate leading-order prefactor and, importantly, the next-order correction, for the first time, to the best of our knowledge. This methodology paves the way for such an analysis in a wide range of lattice nonlinear dynamical equation models.