Narrow Results

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

This paper proposes a review focused on exotic chemotaxis and cross-diffusion models in complex environments. The term exotic is used to denote the dynamics of models interacting with a time-evolving external system and, specifically, models derived with the aim of describing the dynamics of living systems. The presentation first, considers the derivation of phenomenological models of chemotaxis and cross-diffusion models with particular attention on nonlinear characteristics. Then, a variety of exotic models is presented with some hints toward the derivation of new models, by accounting for a critical analysis looking ahead to perspectives. The second part of the paper is devoted to a survey of analytical problems concerning the application of models to the study of real world dynamics. Finally, the focus shifts to research perspectives within the framework of a multiscale vision, where different paths are examined to move from the dynamics at the microscopic scale to collective behaviors at the macroscopic scale.

In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding Keller–Segel chemotaxis-only model possesses a striking feature of critical mass blowup phenomenon, namely, subcritical mass ensures boundedness, whereas, supercritical mass induces the existence of blow-ups. Herein, we show that this critical mass blow-up phenomenon stays almost the same in the full chemotaxis-haptotaxis model and that any global-in-time haptotaxis solution component vanishes exponentially and the other two solution components converge exponentially to that of chemotaxis-only model in a global sense for suitably large chemo-sensitivity and in the usual sense for suitably small chemo-sensitivity. Therefore, haptotaixs is neither good nor bad than chemotaxis, showing negligibility of haptotaxis effect in the underlying chemotaxis-haptotaxis model.

This paper deals with the analysis of qualitative properties involved in the dynamics of Keller–Segel type systems in which the diffusion mechanisms of the cells are driven by porous-media flux-saturated phenomena. We study the regularization inside the support of a solution with jump discontinuity at the boundary of the support. We analyze the behavior of the size of the support and blow-up of the solution, and the possible convergence in finite time toward a Dirac mass in terms of the three constants of the system: the mass, the flux-saturated characteristic speed, and the chemoattractant sensitivity constant. These constants of motion also characterize the dynamics of regular and singular traveling waves.

A novel class of implicit constitutive relations is studied, wherein the stress and the linearized strain appear linearly, that describe material response in elastic porous bodies like rocks, ceramics, concrete, cement, bones and metals. The constitutive relation is applied to a body with a crack subjected to non-penetration conditions between the opposite crack faces. To treat well-posedness of a corresponding variational inequality, we rely on a new approximation by thresholding dilatation and apply the Lions existence theorem on pseudo-monotone variational inequalities. An analytical solution to a specific example (without crack) under uniform triaxial loading is constructed, wherein blow-up can take place at a finite load, and this difficulty is overcome within a thresholding approximation so that blow-up does not occur.

In this paper, we consider a Kirchhoff-type parabolic equation with logarithmic nonlinearity. By making a more general assumption about the Kirchhoff function, we establish a new finite time blow-up criterion. In particular, the blow-up rate and the upper and lower bounds of the blow-up time are also derived. These results generalize some recent ones in which the blow-up results were obtained when the Kirchhoff function was assumed to be a very special form.

We investigate the formation and propagation of singularities for the system of one-dimensional Chaplygin gas. Under suitable assumptions we construct a physically meaningful solution containing a new type of singularities called "delta-like" solution for this kind of quasilinear hyperbolic system with linearly degenerate characteristics. By a careful analysis, we study the behavior of the solution in a neighborhood of a blow-up point. The formation of this new kind of singularities is related to the envelop of different characteristic families, instead of characteristics of the same family in the standard situation. This shows that the blow-up phenomenon for systems with linearly degenerate characteristics is quite different from the problem of shock formation for the system with genuinely nonlinear characteristic fields. Different initial data can lead to different delta-like singularities: the delta-like singularity with point-shape and the delta-like singularity with line-shape.

In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.

In this paper, we prove the existence of an extremal function for the Adams–Moser–Trudinger inequality on the Sobolev space $H_0^m(\mathrm{\Omega })$, where $\mathrm{\Omega }$ is any bounded, smooth, open subset of ${ℝ}^{2m}$, $m\ge 1$. Moreover, we extend this result to improved versions of Adams’ inequality of Adimurthi-Druet type. Our strategy is based on blow-up analysis for sequences of subcritical extremals and introduces several new techniques and constructions. The most important one is a new procedure for obtaining capacity-type estimates on annular regions.

We study the fractional Laplacian problem

In this paper, we consider a degenerate hyperbolic inequality in an exterior domain under three types of boundary conditions: Dirichlet-type, Neumann-type, and Robin-type boundary conditions. Using a unified approach, we show that all the considered problems have the same Fujita critical exponent. Moreover, we answer some open questions from the literature regarding the critical case.

In this paper, we provide a thorough investigation of the blowing up behavior induced via diffusion of the solution of the following non-local problem:

we construct a solution which blows up in finite time and only at an interior point $x_0$ of $\mathrm{\Omega },$ i.e.

where

The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in [F. Merle and H. Zaag, Reconnection of vortex with the boundary and finite time quenching, Nonlinearity10 (1997) 1497–1550] and [G. K. Duong and H. Zaag, Profile of a touch-down solution to a nonlocal MEMS model, Math. Models Methods Appl. Sci.29 (2019) 1279–1348].

Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat Friedmann–Lemaître–Robertson–Walker spacetimes. For the case of accelerated expansion, we show that the blow-up in a finite time occurs for the equation with arbitrary power nonlinearity as well as upper bounds of the lifespan of blow-up solutions. Comparing to the case of the Minkowski spacetime, we discuss how the scale factor affects the lifespan of blow-up solutions of the equation.

The center problem for degenerate singular points of planar systems (the degenerate-center problem) is a poorly-understood problem in the qualitative theory of ordinary differential equations. It may be broken down into two problems: the monodromy problem, to decide if the singular point is of focus-center type, and the stability problem, to decide whether it is a focus or a center.

We present an outline on the status of the center problem for degenerate singular points, explaining the main techniques and obstructions arising in the study of the problem. We also present some new results. Our new results are the characterization of a family of vector fields having a degenerate monodromic singular point at the origin, and the computation of the generalized first focal value for this family V₁. This gives the solution of the stability problem in the monodromic case, except when V₁ = 1. Our approach relies on the use of the blow-up technique and the study of the blow-up geometry of singular points. The knowledge of the blow-up geometry is used to generate a bifurcation of a limit cycle.

The ℂP²-genus of a knot K is the minimal genus over all isotopy classes of smooth, compact, connected and oriented surfaces properly embedded in ℂP² - B⁴ with boundary K. We compute the ℂP²-genus and realizable degrees of (-2,q)-torus knots for 3 ≤ q ≤ 11 and (2,q)-torus knots for 3 ≤ q ≤ 17. The proofs use gauge theory and twisting operations on knots.

We investigate the well-posedness in the generalized Hartree equation $i{u}_t+\mathrm{\Delta }u+(|x{|}^{-(N-\gamma )}\ast |u{|}^p)|u{|}^{p-2}u=0$, $x\in {ℝ}^N$, $0<\gamma <N$, for low powers of nonlinearity, $p<2$. We establish the local well-posedness for a class of data in weighted Sobolev spaces, following ideas of Cazenave and Naumkin, Local existence, global existence, and scattering for the nonlinear Schrödinger equation, Comm. Contemp. Math. 19(2) (2017) 1650038. This crucially relies on the boundedness of the Riesz transform in weighted Lebesgue spaces. As a consequence, we obtain a class of data that exists globally, moreover, scatters in positive time. Furthermore, in the focusing case in the $L^2$-supercritical setting we obtain a subset of locally well-posed data with positive energy, which blows up in finite time.

This paper is concerned with the blow-up phenomenon of stochastic nonlocal heat equations. We first establish the sufficient condition to ensure that the stochastic nonlocal heat equations have a unique non-negative solution. Then the problem of blow-up solutions in finite time is considered.

In the present paper the small quantum cohomology ring of some Fano threefolds which are obtained as one- or two-curve blow-ups from ℙ³ or the quadric Q³ is explicitely computed. Because of systematic usage of the associativity property of quantum product only a very small and enumerative subset of Gromov–Witten invariants is needed. Then, for these threefolds the Dubrovin conjecture on the semisimplicity of quantum cohomology is proven by checking the computed quantum cohomology rings and by showing that a smooth Fano threefold X with b₃(X) = 0 admits a complete exceptional set of the appropriate length.

In this paper, we investigate the initial boundary value problem of the nonlinear fourth-order dispersive-dissipative wave equation. By using the concavity method, we establish a blow-up result for certain solutions with arbitrary positive initial energy.

By using the Lyapunov–Schmidt reduction method without perturbation, we consider existence results for the conformal scalar curvature on $S^n$ ($n\ge 3$) when the prescribed function (after being projected to ${\mathrm{\text{I}}\mathrm{\text{R}}}^n$) has two close critical points, which have the same value (positive), equal “flatness” (“twin”; flatness $<n-2$), and exhibit maximal behavior in certain directions (“pseudo-peaks”). The proof relies on a balance between the two main contributions to the reduced functional — one from the critical points and the other from the interaction of the two bubbles.