Related papers: Large time behavior for a nonlocal nonlinear gradi…
This work is concerned with the large time behavior of solutions to the barotropic compressible Navier-Stokes equations in $\mathbb{R}^{d}(d\geq2)$. Precisely, it is shown that if the initial density and velocity additionally belong to some…
In this article we investigate the temporal regularity of strong solutions to the stochastic $p$-\com{L}aplace system in the degenerate setting, $p \in [2,\infty)$, driven by a multiplicative nonlinear stochastic forcing. We establish $1/2$…
In this paper, we mainly consider large time behavior for the classical free wave equation $u_{tt}-\Delta u=0$ in $\mathbb{R}^n$. We derive some large time optimal estimates for the quantity of solution $\|u(t,\cdot)\|_{L^2}$ with initial…
We study the existence of large solutions for nonlocal Dirichlet problems posed on a bounded, smooth domain, associated to fully nonlinear elliptic equations of order $2s$, with $s\in (1/2,1)$, and a coercive gradient term with subcritical…
We deal with a nonlocal interaction equation describing the evolution of a particle density under the effect of a general symmetric pairwise interaction potential, not necessarily in convolution form. We describe the case of a convex (or…
In this paper, we study the Navier-Stokes-Korteweg equations governed by the evolution of compressible fluids with capillarity effects. We first investigate the global well-posedness of solution in the critical Besov space for large initial…
We study both the local and global existence of a gradient flow of the Sinai-Ruelle-Bowen entropy functional on a Hilbert manifold of expanding maps of a circle equipped with a Sobolev norm in the tangent space of the manifold. We show…
In this paper, we investigate a three-dimensional fluid-particle coupled model. % in whole space $\mathbb{R}^3$. This model combines the full compressible Navier-Stokes equations with the Vlasov-Fokker-Planck equation via the momentum and…
We study the large time behaviour of the reaction-diffsuion equation $\partial_t u=\Delta u +f(u)$ in spatial dimension $N$, when the nonlinear term is bistable and the initial datum is compactly supported. We prove the existence of a…
In this paper, we are concerned with the H\"older regularity for solutions of the nonlocal evolutionary equation $$ \partial_t u+(-\Delta_p)^s u = 0. $$ Here, $(-\Delta_p)^s$ is the fractional $p$-Laplacian, $0<s<1$ and $1<p<2$. We…
The $L^{p}$ theory for non-isentropic Navier-Stokes equations governing compressible viscous and heat-conductive gases is not yet proved completely so far, because the critical regularity cannot control all non linear coupling terms. In…
We consider large time asymptotics for damped nonlinear Schr\"{o}dinger equations. It is known that the nonlinear solution asymptotically behaves like a linear solution when time $t$ tends to infinity in the energy space. We prove that its…
We consider a model of the compressible non-Newtonian fluids for power-law flow fulfilling a periodic domain in ${\mathbb R}^3,$ in which the extra stress tensor is induced by a potential with $p(t,x)$-structure. The local-in-time existence…
We consider the Schr\"odinger equation with nonlinear dissipation \begin{equation*} i \partial _t u +\Delta u=\lambda|u|^{\alpha}u \end{equation*} in ${\mathbb R}^N $, $N\geq1$, where $\lambda\in {\mathbb C} $ with $\Im\lambda<0$. Assuming…
We study the asymptotic behavior of the nonlinear parabolic flows $u_{t}=F(D^2 u^m)$ when $t\ra \infty$ for $m\geq 1$, and the geometric properties for solutions of the following elliptic nonlinear eigenvalue problems: F(D^2 \vp) &+…
Let $p(t,x)$ be the fundamental solution to the problem $$ \partial_{t}^{\alpha}u=-(-\Delta)^{\beta}u, \quad \alpha\in (0,2), \, \beta\in (0,\infty). $$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$…
In this paper we consider a nonlinear Fokker-Planck equation with asymptotically small parameters. It describes the diffusion of finite-size particles in the presence of a fixed distribution of obstacles in the limit of low-volume fraction.…
We study the large time behavior of nonnegative solutions of the Cauchy problem $u_t=\int J(x-y)(u(y,t)-u(x,t))\,dy-u^p$, $u(x,0)=u_0(x)\in L^\infty$, where $|x|^{\alpha}u_0(x)\to A>0$ as $|x|\to\infty$. One of our main goals is the study…
We investigate a system describing the flow of a compressible two-component mixture. The system is composed of the compressible Navier-Stokes equations coupled with non-symmetric reaction-diffusion equations describing the evolution of…
We study the large-time behaviour of the solutions $u$ of the evolution equation involving nonlinear diffusion and gradient absorption $\partial_t u - \Delta_p u + |\nabla u|^q=0$. We consider the problem posed for $x\in {\mathbb R}^N $ and…