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In this work, we consider a more general version of the nonlocal thermistor problem, which describes the temperature diffusion produced when an electric current passes through a material. We investigate the doubly nonlinear problem where the ...

Alopecia areata (AA) is an autoimmune disease whose clinical phenotype is characterized by the formation of distinct hairless patterns on the scalp or other parts of the body. In this paper, we study a three-component chemotaxis model for AA, ...${\mathrm{}}^{}$${\mathrm{}}^{}$$$

In this paper, we deal with the following quasilinear chemotaxis–consumption system with degenerate signal-dependent motility $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\nabla }·(v^{\alpha }\mathrm{\nabla }{u}^m)+\alpha \mathrm{\nabla }·(v^{\alpha -1}u\mathrm{\nabla }v)+ru-\mu u^2,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v-uv,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}\right)\end{array}$$under the homogeneous Neumann boundary condition in $\mathrm{\Omega }\subset {\mathbb{R}}^n(n\ge 1)$ ...$\mathrm{}$$\mathrm{}$$$\begin{array}{c}\hfill {}_{{}^{}\mathrm{}}{}_{{}^{\mathrm{}}\mathrm{}}\end{array}$$$\mathrm{}$$\mathrm{}$$\mathrm{}$

The aim of this paper is to investigate the initial-boundary value problem of a possibly degenerate reaction-diffusion system over $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with $n\ge 1$ of the following form $$\begin{array}{c}\hfill \left\{\begin{array}{cc}& {\partial }_t{m}_i-\kappa \mathrm{\Delta }{m}_i+{|m_i|}^{\gamma -2}{m}_i={({\partial }_{x_i}p)}^2,\hfill \\ \hfill & -\mathrm{\nabla }·[\mathbf{m}\mathrm{\nabla }p]=S,\hfill \end{array}\right)\end{array}$$with $\mathbf{m}=\text{diag}(m_1,\cdots ,m_n)$, the diffusivity $\kappa >0$,...$\mathrm{}$$\mathrm{}$$\mathrm{}$

This paper is concerned with the traveling waves for a modified Holling-Tanner predator–prey model with degenerate diffusion. Different from the established approaches of constructing upper-lower solutions, we construct a new and suitable pair of ...${}^{}$$\mathrm{}$${}^{}\mathrm{}\sqrt{}$$\mathrm{}$

In this study, we provide a nonconforming spectral approach for the Stokes equations with nonstandard boundary conditions on a single domain. The discrete spaces are defined in such a way that the discrete approximations for the velocity are ...

In this paper, the following cross-diffusion system is investigated $\{\begin{array}{cc}{u}_t=\mathrm{\nabla }\cdot \left({(u+1)}^m\mathrm{\nabla }u\right)-\mathrm{\nabla }\cdot \left(\frac{u{(u+1)}^{\beta -1}\mathrm{\nabla }v}{{(1+|\mathrm{\nabla }v{|}^2)}^{\alpha }}\right)+a-b{u}^r,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ 0=\mathrm{\Delta }v-v+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}$ in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$ ($n\ge 2$) with smooth boundary $\partial \mathrm{\Omega }$. Under the condition that $\alpha >\frac{2n-mn-2}{2(n-1)}$, $m\ge 1$, ...$\frac{\mathrm{}}{\mathrm{}}$${}_{\mathrm{}}$

In this paper, we consider a class of non-autonomous Kirchhoff equations with two nonnegative potentials in ${\mathbb{R}}^N$ ($N=1,2,3$). Under certain basic assumptions on the potentials, the existence and nonexistence of normalized ground state solutions are ...${}^{\mathrm{}}$${}^{\mathrm{}}$

We provide the well-posedness for a partially dissipative viscous system of balance laws in smooth Sobolev spaces under the same assumptions as in the case of inviscid balance laws. A priori estimates for coupled hyperbolic-parabolic linear ...

In this paper, we study the following two-species chemotaxis system with generalized volume-filling effect and general kinetic functions ...$\mathrm{}{\mathbb{}}^{}$$\mathrm{}$${}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}$${}_{}{}_{}{}_{}{}_{}$$\mathrm{}\mathrm{}$${}^{\mathrm{}}$$\mathrm{}$$\mathrm{}$${}^{}\mathrm{}$${}_{\mathrm{}}{}_{\mathrm{}}\mathrm{}\mathrm{}$${}_{\mathrm{}}\mathrm{}{}_{\mathrm{}}\mathrm{}$

We consider a system of two nonlocal and nonlinear partial differential equations that describe some aspects of yeast cell-cell communication. We study local and global existence and uniqueness of solutions. We consider mild solutions and we ...

In this paper, we investigate a parabolic–parabolic–elliptic system that describes the initial stage of tumor-related angiogenesis, given by $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t=\mathrm{\Delta }u-\mathrm{\nabla }·(u\mathrm{\nabla }v)+\xi \mathrm{\nabla }·(u^m\mathrm{\nabla }w)+\mu u(1-u^{\alpha }),\hfill \\ v_t=\mathrm{\Delta }v+\chi \mathrm{\nabla }·(v\mathrm{\nabla }w)-v+u,\hfill \\ 0=\mathrm{\Delta }w-w+u.\hfill \end{array}\right)\end{array}$$We demonstrate that the model possesses a ...

This work is concerned with the fractional-in-time parabolic–parabolic Keller–Segel system in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ ($d\ge 2$), for distinct fractional diffusions of the cells and chemoattractant. We prove results on existence, uniqueness, continuous ...

This paper deals with the following indirect chemotaxis-consumption model with signal-dependent degenerate diffusion and logistic source $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_t=\mathrm{\Delta }\left(u,v^{\alpha }\right)+au-b{u}^l,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=\mathrm{\Delta }v-vw,\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=-\delta w+u,\hfill & x\in \mathrm{\Omega },t>0,\hfill \end{array}\right)\end{array}$$under homogeneous Neumann boundary conditions in a ...$\mathrm{}{\mathbb{}}^{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$$\mathrm{}$

In this paper we advocate for Isaacs’ method for the solution of differential games to be applied to the solution of optimal control problems. To make the argument, the vehicle employed is Pontryagin’s canonical optimal control example, which ...

The present paper is devoted to the study of the following double-phase equation $$\begin{array}{c}\hfill -{\text{div}(|\mathrm{\nabla }u|}^{p-2}\mathrm{\nabla }u+{\mu }_{\epsilon }{(x)|\mathrm{\nabla }u|}^{q-2}\mathrm{\nabla }u)+V_{\epsilon }{(x)(|u|}^{p-2}u+{\mu }_{\epsilon }{(x)|u|}^{q-2}u)=f(u)\text{in}{\mathbb{R}}^N,\end{array}$$where $N\ge 2$, $1<p<q<N$, $q<p_{}^{\ast }$ with $p_{}^{\ast }=\frac{\mathit{Np}}{N-p}$, $\mu :{\mathbb{R}}^N\to \mathbb{R}$ is a continuous non-negative function, ${\mu }_{\epsilon }(x)=\mu (\epsilon x)$, ...${\mathbb{}}^{}\mathbb{}$${}_{}$$\mathbb{}\mathbb{}$$$

This article considers a two competitive biological species system involving signal-dependent motilities and sensitivities and nonlinear productions ...$\mathrm{}{\mathbb{}}^{\mathrm{}}$${}_{}{}_{}{}_{}{}_{}{}_{}$$\mathrm{}\mathrm{}$${}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}{}_{\mathrm{}}$$$${}_{}{}_{}{}^{\mathrm{}}\mathrm{}$${}_{}{}_{}\mathrm{}$$\mathrm{}$${}_{}^{}\mathrm{}$$\underset{}{}{}_{}\mathrm{}$$$$\underset{}{}\frac{{}_{}}{{}_{}}$$\underset{}{}\frac{{}_{}^{}}{{}_{}}$$\mathrm{}{}_{\mathrm{}}\mathrm{}{}_{\mathrm{}}$$\mathrm{}{}_{\mathrm{}}\mathrm{}{}_{\mathrm{}}$

In this paper, we investigate the Cauchy problem of Keller–Segel–Navier–Stokes system with fractional diffusion. Making use of Fourier localization technique and Littlewood-Paley theory, we establish the global well-posedness of mild solution for ...

This paper deals with the semilinear Rayleigh–Stokes equation with the fractional derivative in time of order $\alpha \in (0,1)$, which can be used to model anomalous diffusion in viscoelastic fluids. An operator family related to this problem is defined, ...

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to ...${}^{}$