Related papers: Global solutions of a doubly tactic resource consu…

The global boundedness and the hair trigger effect of solutions for the nonlinear nonlocal reaction-diffusion equation \begin{align*} u_t=\Delta u+\mu u^\alpha(1-\kappa J*u^\beta),\quad\hbox{in} \;\mathbb R^N\times(0,\infty),\; N\geq 1…

We study the periodic and the Neumann boundary value problems associated with the second order nonlinear differential equation \begin{equation*} u'' + c u' + \lambda a(t) g(u) = 0, \end{equation*} where $g \colon…

The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} \displaystyle n_t=\Delta n-\nabla\cdot(nS(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in \Omega\times (0,T),\\ c_t=\Delta c-nc-u\cdot\nabla c ,…

In this paper we consider a doubly critical nonlinear elliptic problem with Neumann boundary conditions. The existence of blow-up solutions for this problem is related to the blow-up analysis of the classical geometric problem of…

We study the energy-critical half-wave maps equation: \[ \partial_t \mathbf{u} = \mathbf{u} \times |D| \mathbf{u} \] for $\mathbf{u} : [0, T) \times \mathbb{R} \to \mathbb{S}^2$. Our main result establishes the global existence and…

This paper deals with a problem which describes tuberculosis granuloma formation \begin{align*} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) - uv - u + \beta, &x \in \Omega,\ t>0, \\ v_t = \Delta v + v -uv + \mu w, &x \in…

A boundary control problem for the following generalized Korteweg-de Vries-Burgers-Huxley equation: $$u_t=\nu u_{xx}-\mu u_{xxx}-\alpha u^{\delta}u_x+\beta u(1-u^{\delta})(u^{\delta}-\gamma), \ x\in[0,1], \ t>0,$$ where…

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} \Delta u=0 \qquad &\text{in $\Omega$,}\\ u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u…

In this work, we study the Neumann initial boundary value problem for a three-component chemotaxis model in any dimensional bounded and smooth domains; this model is used to describe the branching of capillary sprouts during angiogenesis.…

We consider the following slightly supercritical problem for the Lane-Emden system with Neumann boundary conditions: \begin{equation*} \begin{cases} -\Delta u_1=|u_2|^{p_\epsilon-1}u_2,\ &in\ \Omega,\\ -\Delta u_2=|u_1|^{q_\epsilon-1}u_1, \…

In this paper, we investigate the global boundedness, asymptotic stability and pattern formation of predator-prey systems with density-dependent preytaxis in a two-dimensional bounded domain with Neumann boundary conditions, where the…

In this paper, we investigate the Cauchy problem of the parabolic-parabolic Keller-Segel system with the logistic-type term $au-bu^\gamma$ on $\mathbb{R}^N, N\geq2$. We discuss the global boundedness of classical solutions with nonnegative…

We investigate sufficient conditions for the presence of coexistence states for different genotypes in a diploid diallelic population with dominance distributed on a heterogeneous habitat, considering also the interaction between genes at…

This work considers the doubly degenerate nutrient model \begin{equation*}\label{AH1} \left\{ \begin{split} &u_t=\nabla\cdot\left(u^{m-1}v\nabla u\right)-\nabla\cdot\left(f(u)v\nabla v\right)+\ell uv,&&x\in\Omega,\,t>0, &v_t=\Delta v-uv,…

We prove global in time well-posedness for perturbations of the 2D stochastic Navier-Stokes equations \begin{equation*} \partial_t u + u \cdot \nabla u = \Delta u - \nabla p + \zeta + \xi \;, \quad u (0, \cdot) = u_{0}(\cdot) \;, \quad…

A doubly nonlinear parabolic equation of the form $\alpha(u_t)-\Delta u+W'(u)= f$, complemented with initial and either Dirichlet or Neumann homogeneous boundary conditions, is addressed. The two nonlinearities are given by the maximal…

The coupled chemotaxis fluid system \begin{equation} \left\{ \begin{array}{llc} n_t=\Delta n-\nabla\cdot(n S(x,n,c)\cdot\nabla c)-u\cdot\nabla n, &(x,t)\in \Omega\times (0,T), \displaystyle c_t=\Delta c-nc-u\cdot\nabla c,…

We consider the Hamiltonian system with Neumann boundary conditions: \[ -\Delta u + \mu u=v^{q }, \quad -\Delta v+ \mu v=u^{p} \quad \text{ in $\Omega$}, \qquad u, v >0 \quad \text{ in $\Omega$,} \qquad \partial_\nu u= \partial_\nu v=0…

The parabolic-elliptic cross-diffusion system \[ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm] 0 = \Delta v - \mu + u, \qquad \int_\Omega v=0, \qquad \mu:=\frac{1}{|\Omega|} \int_\Omega…

In this paper, we study the following the coupled chemotaxis--haptotaxis model with remodeling of non-diffusible attractant $$ \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+\mu u(1- u-w),…