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We study the quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source involving the exponents depending on the spatial variables: $u_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }·\left(u,{\left(u,+,1\right)}^{r-1},\mathrm{\nabla },\upsilon \right)+\xi \mathrm{\nabla }·\left(u,{\left(u,+,1\right)}^{r-1},\mathrm{\nabla },\omega \right)+au-b{u}^{m(x)}$, $0=\mathrm{\Delta }\upsilon -\beta \upsilon +\alpha u$, $0=\mathrm{\Delta }\omega -\delta \omega +\gamma u$, where $\alpha$, ...$$$$$$$$$$$\mathrm{}$$\mathrm{}$$\mathbb{}$$\mathrm{}\left(\mathrm{},,\right)$${\mathbb{}}^{}$$\left(,,\mathrm{}\right)$$$${}_{\mathrm{}}\left(\right)\left\{,,\frac{{\left(,,\mathrm{}\right)}_{}}{},,\mathrm{}\right\}$$$${}_{\mathrm{}}\left(\right)\left\{,,\mathrm{},,\mathrm{}\right\}$$$

This paper concerns the asymptotics of certain parabolic–elliptic chemotaxis-consumption systems with logistic growth and constant concentration of chemoattractant on the boundary. First we prove that in two dimensional bounded domains there ...

The transport of charged particles, which can be described by the Maxwell–Ampère Nernst–Planck (MANP) framework, is essential in various applications including ion channels and semiconductors. We propose a decoupled structure-preserving numerical ...

In recent years, the scalar auxiliary variable based (SAV-based) approach has become very popular and hot in the design of linear, high-order and unconditional energy stable schemes of dissipative systems. However, the nature of SAV-based ...

We prove linear convergence for a new family of modified Dirichlet–Neumann methods applied to quasilinear parabolic equations, as well as the convergence of the Robin–Robin method. Such nonoverlapping domain decomposition methods are commonly ...

This paper deals with an initial-boundary value problem in two-dimensional smoothly bounded domains for the system $$\begin{array}{c}\hfill \left\{\begin{array}{c}{n}_t+\mathbf{u}·\mathrm{\nabla }n=\mathrm{\Delta }n-\mathrm{\nabla }·(n\mathcal{S}(n)\mathrm{\nabla }v),x\in \mathrm{\Omega },t>0,\hfill \\ v_t+\mathbf{u}·\mathrm{\nabla }v=\mathrm{\Delta }v-v+w,x\in \mathrm{\Omega },t>0,\hfill \\ w_t+\mathbf{u}·\mathrm{\nabla }w=\mathrm{\Delta }w-w+n,x\in \mathrm{\Omega },t>0,\hfill \\ {\mathbf{u}}_t+\kappa (\mathbf{u}·\mathrm{\nabla })\mathbf{u}+\mathrm{\nabla }P=\mathrm{\Delta }\mathbf{u}+n\mathrm{\nabla }\varphi ,x\in \mathrm{\Omega },t>0,\hfill \\ \mathrm{\nabla }·\mathbf{u}=0,x\in \mathrm{\Omega },t>0,\hfill \end{array}\right)(\ast )\end{array}$$which ...$\mathbb{}$${}^{\mathrm{}}\mathrm{}$$\mathcal{}$$$\begin{array}{c}\hfill \mathcal{}{}_{\mathcal{}}{\mathrm{}}^{}\mathrm{}{}_{\mathcal{}}\mathrm{}\mathrm{}\end{array}$$$$\begin{array}{c}\hfill \mathrm{}\mathcal{}\mathrm{}{}_{}{}_{}\mathrm{}\mathbf{}\mathrm{}\mathrm{}\mathrm{}\end{array}$$$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathbb{}$

This article presents some controllability and stabilization results for a system of two coupled linear Schrödinger equations in the one-dimensional case where the state components are interacting through the Kirchhoff boundary conditions. ...${}^{\mathrm{}}$$\mathrm{}$

As in our previous work (SINUM 59(2):660–674, 2021) we consider space-time tracking optimal control problems for linear parabolic initial boundary value problems that are given in the space-time cylinder $Q=\mathrm{\Omega }\times (0,T)$, and that are controlled by the ...${}_{}$${}^{\mathrm{}}\mathrm{}{}^{\mathrm{}}\mathrm{}$${}^{\mathrm{}}$${}^{\mathrm{}}\mathrm{}{}^{\mathrm{}}\mathrm{}$${\stackrel{}{}}_{}\stackrel{}{}{}_{{}^{\mathrm{}}}$${\stackrel{}{}}_{}$$\stackrel{}{}$$$$\stackrel{}{}$${}^{\mathrm{}}$${\stackrel{}{}}_{}$

Starting from a mesoscopic description of cell migration and intraspecific interactions, we obtain by upscaling an effective reaction–diffusion–taxis equation for the cell population density involving spatial nonlocalities in the source term and ...

This paper is devoted to study the population dynamics of a single species in a one-dimensional environment which is modeled by a reaction–diffusion–advection equation with free boundary condition. We find three critical values $c_0$, 2 and ${\beta }^{\ast }$ for ...$$${}^{}\mathrm{}{}_{\mathrm{}}\mathrm{}$$\mathrm{}{}_{\mathrm{}}$${}_{\mathrm{}}\mathrm{}$$\mathrm{}{}^{}$${}^{}$$\mathrm{}$

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, ...

We discuss the control of a human crowd whose dynamics is governed by a regularized version of Hughes’ model, cf. Hughes (Transp Res Part B: Methodol 36(6):507–535, 2002. https://doi.org/10.1016/s0191-2615(01)00015-7). We assume that a finite ...

We study the linear heat equation on a half-space with a linear dynamical boundary condition. We are interested in an appropriate choice of the function space of initial functions such that the problem possesses a solution. It was known before ...

This paper is concerned with the numerical solution of compressible fluid flow in a fractured porous medium. The fracture represents a fast pathway (i.e., with high permeability) and is modeled as a hypersurface embedded in the porous medium. We ...

In this paper, we study a free boundary problem for a ratio-dependent predator–prey system in one space dimension, in which the free boundary is only caused by prey, representing the spreading fronts of prey. We discuss the long time behaviors of ...$$$$$$

In this paper, we introduce three series of Jacobi rational basis functions on the half line by using the matrix decomposition technique. The new basis functions are simultaneously orthogonal in both $L^2$- and $H^1$-inner products, and lead to diagonal ...

This paper reconsiders the two-species cancer invasion haptotaxis model without cell proliferation $$\begin{array}{c}\hfill \left\{\begin{array}{c}{c}_{1t}=\mathrm{\Delta }{c}_1-{\chi }_1\mathrm{\nabla }·(c_1\mathrm{\nabla }v)-f(v)m{c}_1,\hfill \\ c_{2t}=\mathrm{\Delta }{c}_2-{\chi }_2\mathrm{\nabla }·(c_2\mathrm{\nabla }v)+f(v)m{c}_1,\hfill \\ \tau m_t=\mathrm{\Delta }m+c_1+c_2-m,\hfill \\ v_t=-mv+\eta v(1-{\alpha }_1{c}_1-{\alpha }_2{c}_2-v)\hfill \end{array}\right)(\star )\end{array}$$in a bounded and smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with homogeneous ...${}_{\mathrm{}}{}_{\mathrm{}}\mathrm{}$$\mathrm{}\mathrm{}$${}^{\mathrm{}}\left(,\mathrm{},,,,,,\mathrm{},,,\right)$$\mathrm{}\mathrm{}$${}_{\mathrm{}}{}_{\mathrm{}}\mathrm{}$$$$\mathrm{}$$$$$$\mathrm{}$${}_{\mathrm{}}{}_{\mathrm{}}\mathrm{}$$$$\mathrm{}\mathrm{}$$$

The Swift–Hohenberg model is a very important phase field crystal model which can describe many crystal phenomena. This model with quadratic–cubic nonlinearity based on the H − 1-gradient flow approach is a sixth-order system which ...

This work studies a class of chemotaxis systems generalizing the prototype $$\begin{array}{c}\hfill \left\{\begin{array}{c}{\displaystyle u_t=d\mathrm{\nabla }·({(1+u)}^{m-1}\mathrm{\nabla }u)-\chi \mathrm{\nabla }·(u{(1+u)}^{\sigma -2}\mathrm{\nabla }v)+\mu u^{\alpha }(1-{\int }_{\mathrm{\Omega }}{u}^{\beta }),}\hfill \\ 0=\mathrm{\Delta }v-v+u,\hfill \end{array}\right)\end{array}$$with nonnegative initial data under zero-flux boundary conditions in a smooth bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^N$$(N\ge 1)$, where d, m, ...$$$\mathrm{}$$\mathrm{}$$$$\mathrm{}$$$\begin{array}{c}\hfill {\displaystyle \frac{}{\mathrm{}}\frac{\mathrm{}}{}}\end{array}$$${\displaystyle \frac{}{\mathrm{}}}$$$

We study weak solutions and its approximation of hyperbolic linear symmetric Friedrichs systems describing acoustic, elastic, or electro-magnetic waves. For the corresponding first-order systems we construct discontinuous Galerkin discretizations ...${}_{\mathrm{}}$