Related papers: Shock-type singularity of the hyperbolic-parabolic…
This paper deals with the fully parabolic attraction-repulsion chemotaxis system \begin{align*} u_t=\Delta u-\chi\nabla \cdot (u\nabla v)+\xi \nabla\cdot(u \nabla w), \quad v_t=\Delta v-v+u, \quad w_t=\Delta w-w+u, \quad x \in \Omega,\ t>0…
In this work, we rigorously study chemotaxis effect versus haptotaxis effect on boundedness, blow-up and asymptotical behavior of solutions for a combined chemotaxis-haptotaxis model in 2D settings. It is well-known that the corresponding…
The exact structure of a shock is computed in a multiple-speed discrete-velocity gas, the nine-velocity gas, wherein the multiplicity of speeds ensures nontrivial thermodynamics. Obtained as a solution of the model Boltzmann equations, the…
We consider the fully parabolic, spatially heterogeneous chemotaxis-growth system \begin{align*} \begin{cases} u_t = \Delta u - \nabla\cdot(u\nabla v) + \kappa(x)u-\mu(x)u^2, \\ v_t = \Delta v - v + u \end{cases} \end{align*} in bounded…
It is shown in this paper that blow-up does not occur in the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain \(\Omega \subset \mathbb{R}^2\): \begin{equation*} \begin{cases} u_t =…
We study a class of non-linear parabolic systems relevant in turbulence theory. Those systems can be viewed as simplified versions of the Prandtl one-equation and Kolmogorov two-equation models of turbulence. We restrict our attention to…
In this paper we develop a field-theoretic description for run and tumble chemotaxis, based on a density functional description of crystalline materials modified to capture orientational ordering. We show that this framework, with its…
There has been considerable effort expended towards understanding high temperature superconductors (HTSC), and more specifically the cuprate phase diagram as a function of doping level. Yet, the only agreement seems to be that HTSC is an…
Cobalt carbonate hydroxide (CCH) is a pseudocapacitive material with remarkably high capacitance and cycle stability. Previously, it was reported that CCH pseudocapacitive materials are orthorhombic in nature. Recent structural…
We consider the energy super critical $d+1$ dimensional semilinear heat equation $$\partial_tu=\Delta u+u^{p}, \ \ x\in \Bbb R^{d+1}, \ \ p\geq 3, \ d\geq 14.$$ A fundamental open problem on this canonical nonlinear model is to understand…
In this paper, we study the singularity formation phenomenon of the 1D model of Electron Magnetohydrodynamics (EMHD). we will construct a solution whose $C^3$-norm blows up in finite time. In the end, we will show that the solution is in…
We employ granular hydrodynamics to investigate a paradigmatic problem of clustering of particles in a freely cooling dilute granular gas. We consider large-scale hydrodynamic motions where the viscosity and heat conduction can be…
This paper is devoted to the analysis of the classical Keller-Segel system over $\mathbb{R}^d$, $d\geq 3$. We describe as much as possible the dynamics of the system characterized by various criteria, both in the parabolic-elliptic case and…
We study driven 1d lattice gas models with two types of particles and nearest neighbor hopping. We find the most general case when there is a shock solution with a product measure which has a density-profile of a step function for both…
In recent years, a lot of attention has been drawn to the question of whether logistic kinetics is sufficient to enforce the global existence of classical solutions or to prevent finite-time blow-up in various chemotaxis models. The current…
Motivated by the work of Bryant on constant mean curvature (CMC) $1$-immersions of surfaces into the hyperbolic space H^3 and after the results of Tarantello (2023), we pursue a possible parametrization for the moduli space of (CMC)…
This paper is concerned with a parabolic-parabolic-parabolic chemotaxis system with indirect signal production, modelling the impact of phenotypic heterogeneity on population aggregation \begin{equation*} \begin{cases} u_t = \Delta u -…
In this paper we consider the one dimensional spring-block model describing earthquake faulting. By using geometric singular perturbation theory and the blow-up method we provide a detailed description of the periodicity of the earthquake…
A mathematical model of the metabolic process of formation of the hemostasis in a blood-carrying vessel is constructed. As distinct from the earlier developed model of the multienzyme prostacyclin-thromboxane system of blood, this model…
This review article offers a survey of the research program focused on a systematic computational search for extreme and potentially singular behavior in hydrodynamic models motivated by open questions concerning the possibility of a…