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The Jordan-Moore-Gibson-Thompson equation \[ \tau u_{ttt} + \alpha u_{tt} = \beta \Delta u_t + \gamma \Delta u + (f(u))_{tt} \] is considered in a smoothly bounded domain $\Omega \subset\mathbb{R}^n$ with $n\leq 3$, where $\tau>0,\beta>0,\gamma>0$, and $\alpha\in\mathbb{R}$. Firstly, it is seen that under the assumption that $f\in C^3(\mathbb{R})$ is such that $f(0)=0$, gradient blow-up phenomena cannot occur in the sense that for any appropriately regular initial data, within a suitable framework of strong solvability, an associated Dirichlet type initial-boundary value problem admits a unique solution $u$ on a maximal time interval $(0,T_{max})$ which is such that \[ \mbox{if $T_{max}<\infty$, then } \limsup_{t\nearrow T_{max}} \|u(\cdot,t)\|_{L^\infty(\Omega)}=\infty. \] This is used to, secondly, make sure that if additionally $f$ is convex and grows superlinearly in the sense that \[ f''\ge 0 \mbox{ on $\mathbb{R}$,} \qquad \frac{f(\xi)}{\xi} \to +\infty \mbox{ as $\xi\to +\infty$} \qquad \mbox{and} \qquad \int_{\xi_0}^\infty \frac{d\xi}{f(\xi)} < \infty \mbox{ for some $\xi_0>0$,} \] then for some initial data the above solution must undergo some finite-time $L^\infty$ blow-up in the style described above.

In a smoothly bounded convex domain $\Omega\subset R^n$ with $n\ge 1$, a no-flux initial-boundary value problem for \[ \left\{ \begin{array}{l} u_t=\Delta \big(u\phi(v)\big), v_t=\Delta v-uv, \end{array} \right. \] is considered under the assumption that near the origin, the function $\phi$ suitably generalizes the prototype given by \[ \phi(\xi)=\xi^\alpha, \qquad \xi\in [0,\xi_0]. \] By means of separate approaches, it is shown that in both cases $\alpha\in (0,1)$ and $\alpha\in [1,2]$ some global weak solutions exist which, inter alia, satisfy $C(T):= {\rm esssup} {}_{t\in (0,T)} \int_\Omega u(\cdot,t)\ln u(\cdot,t) < \infty$ for all $T>0$, with $\sup_{T>0} C(T)<\infty$ if $\alpha\in [1,2]$.

A standard tool for modelling real-world optimisation problems is mixed-integer programming (MIP). However, for many of these problems, information about the relationships between variables is either incomplete or highly complex, making it difficult or even impossible to model the problem directly. To overcome these hurdles, machine learning (ML) predictors are often used to represent these relationships and are then embedded in the MIP as surrogate models. Due to the large amount of available ML frameworks and the complexity of many ML predictors, formulating such predictors into MIPs is a highly non-trivial task. In this paper, we introduce PySCIPOpt-ML, an open-source tool for the automatic formulation and embedding of trained ML predictors into MIPs. By directly interfacing with a broad range of commonly used ML frameworks and an open-source MIP solver, PySCIPOpt-ML provides a way to easily integrate ML constraints into optimisation problems. Alongside PySCIPOpt-ML, we introduce, SurrogateLIB, a library of MIP instances with embedded ML constraints, and present computational results over SurrogateLIB, providing intuition on the scale of ML predictors that can be practically embedded. The project is available at https://github.com/Opt-Mucca/PySCIPOpt-ML.

The Cauchy problem in $\mathbb R^n$ is considered for \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v),\\ 0 = \Delta v + u. \end{array} \right. \end{eqnarray*} For each $n\ge 10$, a statement on stability and attractiveness of the singular steady state given by \[ u_\star(x):=\frac{2(n-2)}{|x|^2},\qquad x\in\mathbb R^n\setminus\{0\}, \] is derived within classes of nonnegative radial solutions emanating from initial data less concentrated than $u_\star$. In particular, for any such $n$ it is shown that infinite-time blow-up occurs for all radial initial data which are less concentrated than $u_\star$ and satisfy \[ u_0(x) \ge \frac{2(n-2)}{|x|^2} - \frac{C}{|x|^{2+\theta}}\qquad \mbox{for all } x\in \mathbb R^n\setminus B_1(0) \] with some $C>0$ and some $\theta>\frac{n-2+\sqrt{(n-2)(n-10)}}{2}$. This is complemented by a result which, in the case when $3\le n \le 9$, asserts instability of $u_\star$ as well as the existence of a bounded absorbing set for all radial trajectories initially less concentrated than $u_\star$. In particular, previous knowledge on stability properties of $u_\star$, as having been gained for $n\ge 11$ in [24], is thereby extended to any dimension $n\ge 3$., Comment: 31 pages, 1 figure

We give an overview of analytical results concerned with chemotaxis systems where the signal is absorbed. We recall results on existence and properties of solutions for the prototypical chemotaxis-consumption model and various variants and review more recent findings on its ability to support the emergence of spatial structures.

We consider a phenotype-switching chemotaxis model for aggregation, in which a chemotactic population is capable of switching back and forth between a chemotaxing state (performing chemotactic movement) and a secreting state (producing the attractant). We show that the switching rate provides a powerful mechanism for controlling the densities of spontaneously emerging aggregates. Specifically, in two- and three-dimensional settings it is shown that when both switching rates coincide and are suitably large, then the densities of both the chemotaxing and the secreting population will exceed any prescribed level at some points in the considered domain. This is complemented by two results asserting the absence of such aggregation phenomena in corresponding scenarios in which one of the switching rates remains within some bounded interval.

The taxis-type migration-consumption model accounting for signal-dependent motilities, as given by \[ u_t = \Delta \big(u\phi(v)\big), v_t = \Delta v-uv, \qquad (*) \] is considered for suitably smooth functions $\phi:[0,\infty)\to R$ which are such that $\phi>0$ on $(0,\infty)$, but that in addition $\phi(0)=0$ with $\phi'(0)>0$. In order to appropriately cope with the diffusion degeneracies thereby included, this study separately examines the Neumann problem for the linear equation \[ V_t = \Delta V + \nabla\cdot \big( a(x,t)V\big) + b(x,t)V \] and establishes a statement on how pointwise positive lower bounds for nonnegative solutions depend on the supremum and the mass of the initial data, and on integrability features of $a$ and $b$. This is thereafter used as a key tool in the derivation of a result on global existence of solutions to (*), smooth and classical for positive times, under the mere assumption that the suitably regular initial data be nonnegative in both components. Apart from that, these solutions are seen to stabilize toward some equilibrium, and as a qualitative effect genuinely due to degeneracy in diffusion, a criterion on initial smallness of the second component is identified as sufficient for this limit state to be spatially nonconstant.

We consider the Keller-Segel-type migration-consumption system involving signal-dependent motilities, $$\left\{ \begin{array}{l} u_t = \Delta \big(u\phi(v)\big), \\[1mm] v_t = \Delta v-uv, \end{array} \right. \qquad \qquad$$ in smoothly bounded domains $\Omega\subset\mathbb{R}^n$, $n\ge 1$. Under the assumption that $\phi\in C^1([0,\infty))$ is positive on $[0,\infty)$, and for nonnegative initial data from $(C^0(\overline{\Omega}))^\star \times L^\infty(\Omega)$, previous literature has provided results on global existence of certain very weak solutions with possibly quite poor regularity properties, and on large time stabilization toward semitrivial equilibria with respect to the topology in $(W^{1,2}(\Omega))^\star \times L^\infty(\Omega)$. The present study reveals that solutions in fact enjoy significantly stronger regularity features when $0<\phi\in C^3([0,\infty))$ and the initial data belong to $(W^{1,\infty}(\Omega))^2$: It is firstly shown, namely, that then in the case $n\le 2$ an associated no-flux initial-boundary value problem even admits a global classical solution, and that each of these solutions smoothly stabilizes in the sense that as $t\to\infty$ we have $$ \begin{align*} u(\cdot,t) \to \frac{1}{|\Omega|}\int_\Omega u_0 \qquad \text{ and } \qquad v(\cdot,t)\to 0 \qquad \qquad (\star) \end{align*}$$ even with respect to the norm in $L^\infty(\Omega)$ in both components. In the case when $n\ge 3$, secondly, some genuine weak solutions are found to exist globally, inter alia satisfying $\nabla u\in L^\frac{4}{3}_{loc}(\overline{\Omega}\times [0,\infty);\mathbb{R}^n)$. In the particular three-dimensional setting, any such solution is seen to become eventually smooth and to satisfy ($\star$)., Comment: 21 pages

Two relaxation features of the migration-consumption chemotaxis system involving signal-dependent motilities, $$ \left\{ \begin{array}{l} u_t = \Delta \big(u\phi(v)\big), \\[1mm] v_t = \Delta v-uv, \end{array} \right. \qquad \qquad (\star)$$ are studied in smoothly bounded domains $\Omega\subset\mathbb{R}^n$, $n\ge 1$: It is shown that if $\phi\in C^0([0,\infty))$ is positive on $[0,\infty)$, then for any initial data $(u_0,v_0)$ belonging to the space $(C^0(\overline{\Omega}))^\star\times L^\infty(\Omega)$ an associated no-flux type initial-boundary value problem admits a global very weak solution. Beyond this initial relaxation property, it is seen that under the additional hypotheses that $\phi\in C^1([0,\infty))$ and $n\le 3$, each of these solutions stabilizes toward a semi-trivial spatially homogeneous steady state in the large time limit. By thus applying to irregular and partially even measure-type initial data of arbitrary size, this firstly extends previous results on global solvability in ($\star$) which have been restricted to initial data not only considerably more regular but also suitably small. Secondly, this reveals a significant difference between the large time behavior in ($\star$) and that in related degenerate counterparts involving functions $\phi$ with $\phi(0)=0$, about which, namely, it is known that some solutions may asymptotically approach nonhomogeneous states., Comment: 26 pages

Cutting plane selection is a subroutine used in all modern mixed-integer linear programming solvers with the goal of selecting a subset of generated cuts that induce optimal solver performance. These solvers have millions of parameter combinations, and so are excellent candidates for parameter tuning. Cut selection scoring rules are usually weighted sums of different measurements, where the weights are parameters. We present a parametric family of mixed-integer linear programs together with infinitely many family-wide valid cuts. Some of these cuts can induce integer optimal solutions directly after being applied, while others fail to do so even if an infinite amount are applied. We show for a specific cut selection rule, that any finite grid search of the parameter space will always miss all parameter values, which select integer optimal inducing cuts in an infinite amount of our problems. We propose a variation on the design of existing graph convolutional neural networks, adapting them to learn cut selection rule parameters. We present a reinforcement learning framework for selecting cuts, and train our design using said framework over MIPLIB 2017 and a neural network verification data set. Our framework and design show that adaptive cut selection does substantially improve performance over a diverse set of instances, but that finding a single function describing such a rule is difficult. Code for reproducing all experiments is available at https://github.com/Opt-Mucca/Adaptive-Cutsel-MILP., Comment: Added new computational experiments. Expanded Appendix

New estimates and global existence results are provided for a class of systems of cross diffusion equations arising from the modeling of chemotaxis with local sensing, possibly featuring a growth term of logistic-type as well. For sublinear non-increasing motility functions, convergence to the spatially homogeneous steady state is shown, a dedicated Lyapunov functional being constructed for that purpose.

We introduce a generalized concept of solutions for reaction-diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.

The chemotaxis system \begin{align*} u_t &= \Delta u - \nabla \cdot (u\nabla v), \\ v_t &= \Delta v - uv, \end{align*} is considered under the boundary conditions $\frac{\partial u}{\partial\nu}- u\frac{\partial v}{\partial\nu}=0$ and $v=v_\star$ on $\partial\Omega$, where $\Omega\subset\mathbb{R}^n$ is a ball and $v_\star$ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case $n=2$, while global weak solutions are constructed when $n\in \{3,4,5\}$. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.

Derived from a biophysical model for the motion of a crawling cell, the system \[(*)~\begin{cases}u_t=\Delta u-\nabla\cdot(u\nabla v)\\0=\Delta v-kv+u\end{cases}\] is investigated in a finite domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, with $k\geq0$. While a comprehensive literature is available for cases with $(*)$ describing chemotaxis systems and hence being accompanied by homogeneous Neumann-type boundary conditions, the presently considered modeling context, besides yet requiring the flux $\partial_\nu u-u\partial_\nu v$ to vanish on $\partial\Omega$, inherently involves homogeneous Dirichlet conditions for the attractant $v$, which in the current setting corresponds to the cell's cytoskeleton being free of pressure at the boundary. This modification in the boundary setting is shown to go along with a substantial change with respect to the potential to support the emergence of singular structures: It is, inter alia, revealed that in contexts of radial solutions in balls there exist two critical mass levels, distinct from each other whenever $k>0$ or $n\geq3$, that separate ranges within which (i) all solutions are global in time and remain bounded, (ii) both global bounded and exploding solutions exist, or (iii) all nontrivial solutions blow up in finite time. While critical mass phenomena distinguishing between regimes of type (i) and (ii) belong to the well-understood characteristics of $(*)$ when posed under classical no-flux boundary conditions in planar domains, the discovery of a distinct secondary critical mass level related to the occurrence of (iii) seems to have no nearby precedent. In the planar case with the domain being a disk, the analytical results are supplemented with some numerical illustrations, and it is discussed how the findings can be interpreted biophysically for the situation of a cell on a flat substrate.

The parabolic-elliptic cross-diffusion system \[ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm] 0 = \Delta v - \mu + u, \qquad \int_\Omega v=0, \qquad \mu:=\frac{1}{|\Omega|} \int_\Omega u dx, \end{array} \right. \] is considered along with homogeneous Neumann-type boundary conditions in a smoothly bounded domain $\Omega\subset R^n$, $n\ge 1$, where $f$ generalizes the prototype given by \[ f(\xi) = (1+\xi)^{-\alpha}, \qquad \xi\ge 0, \qquad \mbox{for all } \xi\ge 0, \] with $\alpha\in R$. In this framework, the main results assert that if $n\ge 2$, $\Omega$ is a ball and \[ \alpha<\frac{n-2}{2(n-1)}, \] then throughout a considerably large set of radially symmetric initial data, an associated initial value problem admits solutions blowing up in finite time with respect to the $L^\infty$ norm of their first components. This is complemented by a second statement which ensures that in general and not necessarily symmetric settings, if either $n=1$ and $\alpha\in R$ is arbitrary, or $n\ge 2$ and $\alpha>\frac{n-2}{2(n-1)}$, then any explosion is ruled out in the sense that for arbitrary nonnegative and continuous initial data, a global bounded classical solution exists.

This manuscript deals with the three-dimensional version of a flux-limited Keller-Segel system coupled to the incompressible Stokes equations through transport and buoyancy. The main goal consists in verifying that within a certain parameter regime, known as being optimal therefor in some fluid-free simplification, a feature of blow-up prevention by suitably strong flux limitation persists also in the framework of the considered full chemotaxis-fluid system. To achieve this, as a secondary objective of possibly independent interest the manuscript separately establishes some conditional bounds for fluid fields and taxis gradients in a fairly general setting. The application thereof to the specific problem under consideration thereafter facilitates the derivation of a result on global existence of bounded classical solutions for widely arbitrary initial data actually, indeed within the entire and essentially optimal parameter range, through an argument which appears to be signficantly condensed when compared to reasonings pursued in previous related works.

This work considers a model for oncolytic virotherapy, as given by the reaction-diffusion-taxis system $$ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot (u\nabla v)-\rho uz, \\[1mm] v_t = - (u+w)v, \\[1mm] w_t = D_w \Delta w - w + uz, \\[1mm] z_t = D_z \Delta z - z - uz + \beta w, \end{array} \right. $$ in a smoothly bounded domain $\Omega\subset\mathbb{R}^2$, with parameters $D_w>0, D_z>0, \beta>0$ and $\rho\ge 0$.\\ % Previous analysis has asserted that for all reasonably regular initial data, an associated no-flux type initial-boundary value problem admits a global classical solution, and that this solution is bounded if $\beta<1$, whereas whenever $\beta>1$ and $\frac{1}{|\Omega|}\int_{\Omega} u(\cdot,0)>\frac{1}{\beta-1}$, infinite-time blow-up occurs at least in the particular case when $\rho=0$.\abs % In order to provide an appropriate complement to this, the present work reveals that for any $\rho\ge 0$ and arbitrary $\beta>0$, at each prescribed level $\gamma\in (0,\frac{1}{(\beta-1)_+})$ one can identify an $L^\infty$-neighborhood of the homogeneous distribution $(u,v,w,z)\equiv (\gamma,0,0,0)$ within which all initial data lead to globally bounded solutions that stabilize toward the constant equilibrium $(u_\infty,0,0,0)$ with some $u_\infty>0$., Comment: 21 pages

We consider a class of macroscopic models for the spatio-temporal evolution of urban crime, as originally going back to Short et al. (Math. Mod. Meth. Appl. Sci. 18, 2008). The focus here is on the question how far a certain nonlinear enhancement in the random diffusion of criminal agents may exert visible relaxation effects. Specifically, in the context of the system \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (u^{m-1} \nabla u) - \chi \nabla \cdot \Big(\frac{u}{v} \nabla v \Big) - uv + B_1(x,t), \\[1mm] v_t = \Delta v +uv - v + B_2(x,t), \end{array} \right. \end{eqnarray*} it is shown that whenever $\chi>0$ and the given nonnegative source terms $B_1$ and $B_2$ are sufficiently regular, the assumption \begin{eqnarray*} m>\frac{3}{2} \end{eqnarray*} is sufficient to ensure that a corresponding Neumann-type initial-boundary value problem, posed in a smoothly bounded planar convex domain, admits locally bounded solutions for a wide class of arbitrary initial data. Furthermore, this solution is seen to be globally bounded if both $B_1$ and $B_2$ are bounded and $\liminf_{t\to\infty} \int_\Omega B_2(\cdot,t)$ is positive. This is supplemented by numerical evidence which, besides illustrating associated smoothing effects in particular situations of sharply structured initial data in the presence of such porous medium type diffusion mechanisms, indicates a significant tendency toward support of singular structures in the linear diffusion case $m=1$., Comment: 29 pages, 12 figures

This manuscript considers a Neumann initial-boundary value problem for the predator-prey system $$ \left\{ \begin{array}{l} u_t = D_1 u_{xx} - \chi_1 (uv_x)_x + u(\lambda_1-u+a_1 v), \\[1mm] v_t = D_2 v_{xx} + \chi_2 (vu_x)_x + v(\lambda_2-v-a_2 u), \end{array} \right. \qquad \qquad (\star) $$ in an open bounded interval $\Omega$ as the spatial domain, where for $i\in\{1,2\}$ the parameters $D_i, a_i, \lambda_i$ and $\chi_i$ are positive. Due to the simultaneous appearance of two mutually interacting taxis-type cross-diffusive mechanisms, one of which even being attractive, it seems unclear how far a solution theory can be built upon classical results on parabolic evolution problems. In order to nevertheless create an analytical setup capable of providing global existence results as well as detailed information on qualitative behavior, this work pursues a strategy via parabolic regularization, in the course of which ($\star$) is approximated by means of certain fourth-order problems involving degenerate diffusion operators of thin film type. During the design thereof, a major challenge is related to the ambition to retain consistency with some fundamental entropy-like structures formally associated with ($\star$); in particular, this will motivate the construction of an approximation scheme including two free parameters which will finally be fixed in different ways, depending on the size of $\lambda_2$ relative to $a_2 \lambda_1$., Comment: 60 pages