Related papers: Refined existence theorems for doubly degenerate c…
In this paper we deal with positive solutions for singular quasilinear problems whose model is $$ \begin{cases} -\Delta u + \frac{|\nabla u|^2}{(1-u)^\gamma}=g & \mbox{in $\Omega$,}\newline \hfill u=0 \hfill & \mbox{on $\partial\Omega$,}…
Assuming that $0<\chi<\sqrt{\frac{2}n}$, $\kappa\ge 0$ and $\mu>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing \[ \begin{split} u_t &= \Delta u - \chi \nabla\cdot ( \frac{u}{v}…
In this paper, we obtain $C^{1,\alpha}$ estimates for weak solutions of certain quasilinear parabolic equations satisfying nonstandard growth conditions, the prototype examples being $$u_t - \text{div} (|\nabla u|^{p-2} \nabla u +…
This series of papers is concerned with the global solvability, boundedness, regularity, and uniqueness of weak solutions to the following parabolic-parabolic chemotaxis system with a logistic source and chemical consumption:…
We consider the following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 3$: \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot…
We study the boundary value problem $-{\rm div}(\log(1+ |\nabla u|^q)|\nabla u|^{p-2}\nabla u)=f(u)$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded domain in $\RR^N$ with smooth boundary. We distinguish the cases where…
In the present work we propose and study a time discrete scheme for the following chemotaxis-consumption model (for any $s\ge 1$), $$ \partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t v - \Delta v = - u^s v \quad…
This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=\Delta u-\chi_1 \nabla\cdot (\frac{u}{w} \nabla…
In this paper we first study partial regularity of weak solutions to the initial boundary value problem for the system $-\mbox{div}\left[(I+\mathbf{m}\otimes \mathbf{m})\nabla p\right]=S(x),\ \ \partial_t\mathbf{m}-D^2\Delta…
This paper addresses the questions of existence and uniqueness of strong solutions to the homogeneous Dirichlet problem for the double phase equation with operators of variable growth: \[ u_t - div \left(|\nabla u|^{p(z)-2} \nabla u+ a(z)…
We study a degenerate elliptic system with variable exponents. Using the variational approach and some recent theory on weighted Lebesgue and Sobolev spaces with variable exponents, we prove the existence of at least two distinct nontrivial…
We consider the Cauchy problem in $\mathbb{R}^{n}$ for wave and beam equations with frictional, viscoelastic damping, and a new power nonlinearity. In addition to the solution and its total energy, we define the following quantity:…
We consider the sublinear problem \begin {equation*} \left\{\begin{array}{r c l c} -\Delta u & = &|u|^{q-2}u & \textrm{in }\Omega, \\ u_n & = & 0 & \textrm{on }\partial\Omega,\end{array}\right. \end {equation*} where $\Omega \subset…
The flux-limited Keller--Segel system \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v), \\[] v_t = \Delta v - v + u^{\theta} \end{cases} \end{align*} is considered under homogeneous Neumann…
The aim of this paper is to prove global in time existence of weak solutions for a viscoelastic phase separation. We consider the case with singular potentials and degenerate mobilities. Our model couples the diffusive interface model with…
We prove the global-in-time existence of large-data finite-energy weak solutions to an incompressible hybrid Vlasov-magnetohydrodynamic model in three space dimensions. The model couples three essential ingredients of magnetized plasmas: a…
This paper deals with the two-species chemotaxis-competition system $u_t = d_1 \Delta u - \chi_1 \nabla \cdot (u \nabla w) + \mu_1 u(1 - u - a_1 v)$, $v_t = d_2 \Delta v - \chi_2 \nabla \cdot (v \nabla w) + \mu_2 v(1 - a_2 u - v)$, $0 = d_3…
We study the anisotropic, incompressible Cahn-Hilliard-Navier-Stokes system with variable density in a bounded smooth domain $\Omega \subset \mathbb{R}^d$. This work extends previous results on the isotropic case by incorporating…
We consider the Dirichlet problem for a compressible two-fluid model in three dimensions, and obtain the global existence of weak solution with large initial data and independent adiabatic constants \Gamma,\gamma>=9/5. The pressure…
This paper is concerned with the two-dimensional chemotaxis-fluid model \begin{equation*} \begin{cases} n_t+u\cdot\nabla n=\Delta (n\phi(v))+\mu n(1-n),\\ v_t+u\cdot\nabla v=\Delta v-nv,\\ u_t+ \kappa (u\cdot\nabla) u=\Delta…