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In the first part of the present paper, we show that strong convergence of $(v_{0 \varepsilon})_{\varepsilon \in (0, 1)}$ in $L^1(\Omega)$ and weak convergence of $(f_{\varepsilon})_{\varepsilon \in (0, 1)}$ in $L_{\textrm{loc}}^1(\overline…

We study the large-time asymptotic behavior of solutions to the one-dimensional damped pressureless Euler-Poisson system with variable background states, subject to a neutrality condition. In the case where the background density converges…

Hydrodynamic systems arising in swarming modelling include nonlocal forces in the form of attractive-repulsive potentials as well as pressure terms modelling strong local repulsion. We focus on the case where there is a balance between…

We study the leading order behaviour of positive solutions of the equation -\Delta u +\varepsilon u-|u|^{p-2}u+|u|^{q-2}u=0,\qquad x\in\R^N, where $N\ge 3$, $q>p>2$ and when $\varepsilon>0$ is a small parameter. We give a complete…

We study the global existence of a unique strong solution and its large-time behavior of a two-phase fluid system consisting of the compressible isothermal Euler equations coupled with compressible isentropic Navier-Stokes equations through…

We consider a parabolic equation of the form u_t=\Delta u +f(u)+h(x,t) in R^N\times (0,\infty), where f in C^1(R) is such that f(0)=0 and f'(0)<0 and h is a suitable function on R^N\times (0,\infty). We show that under certain conditions,…

The paper investigates the existence of global attractors and their upper semicontinuity for a structural damped wave equation on $\mathbb{R}^{N}: u_{tt}-\Delta u+(-\Delta)^\alpha u_{t}+u_{t}+u+g(u)=f(x)$, where $\alpha\in (1/2, 1)$ is…

In this work, we explore the global existence of strong solutions for a class of partially diffusive hyperbolic systems within the framework of critical homogeneous Besov spaces. Our objective is twofold: first, to extend our recent…

The paper studies the issue of stability of solutions to the Navier-Stokes and damped Euler systems in periodic boxes. We show that under action of fast oscillating-in- time external forces all two dimensional regular solutions converge to…

We consider a general hyperbolic model of chemotaxis in the multidimensional case. For this system we show the global existence of smooth solutions to the Cauchy problem and we determine their asymptotic behavior. Since this model does not…

In this paper we deal with the asymptotic behavior as $t$ tends to infinity of solutions for linear parabolic equations whose model is $$ \begin{cases} u_{t}-\Delta u = \mu & \text{in}\ (0,T)\times\Omega,\\[0.7 ex] u(0,x)=u_0 & \text{in}\…

We consider a second-order nonlocal parabolic MEMS equation with Dirichlet boundary conditions: \[ u_t-\Delta u=\frac{\lambda}{(1-u)^2\bigl(1+\int_\Omega\frac{1}{1-u}\,dx\bigr)^2},\quad x\in\Omega,\ t>0, \] where…

We study the asymptotic behavior of global solutions to hydrodynamical systems modeling the nematic liquid crystal flows under kinematic transports for molecules of different shapes. The coupling system consists of Navier-Stokes equations…

We consider a generalized degenerate diffusion equation with a reaction term $u_t=[A(u)]_{xx}+f(u)$, where $A$ is a smooth function satisfying $A(0)=A'(0)=0$ and $A(u),\ A'(u),\ A''(u)>0$ for $u>0$, $f$ is of monostable type in $[0,s_1]$…

This paper investigates the critical behavior of global solutions to a parabolic equation with a Hartree-type nonlinearity of the form $$\left\{\begin{array}{ll} u_{t}+(-\Delta)^{\frac{\beta}{2}} u= (\mathcal{K}\ast |u|^{p})|u|^{q},&\qquad…

In this article, we study a one-dimensional nonlocal quasilinear problem of the form $u_t=a(\Vert u_x\Vert^2)u_{xx}+\nu f(u)$, with Dirichlet boundary conditions on the interval $[0,\pi]$, where $0<m\leq a(s)\leq M$ for all $s\in…

This paper gives necessary and sufficient conditions for the convergence of the solution of a weakly damped second order linear differential equation that is subjected to outside forcing, for which solutions of the unforced equation are…

In this paper we study the existence and summability of the solutions to the following parabolic-elliptic system of partial differential equations with discontinuous coefficients: \begin{equation*} \begin{cases} u_t -…

We study the long time behavior of solutions to the nonlocal diffusion equation $\partial_t u=J*u-u$ in an exterior one-dimensional domain, with zero Dirichlet data on the complement. In the far field scale, $\xi_1\le|x|t^{-1/2}\le\xi_2$,…

We determine the large-time behavior of unbounded solutions for the so-called viscous Hamilton Jacobi equation, $u_t - \Delta u + |Du|^m = f(x)$, in the quadratic and subquadratic cases (i.e., for $1<m\leq 2$), with a particular focus on…