相关论文: Large-time behavior of solutions of parabolic equa…

We construct asymptotically self-similar global solutions to the Hardy-H\'enon parabolic equation $\partial_t u - \Delta u = \pm |x|^{\gamma} |u|^{\alpha-1} u$, $\alpha>1$, $\gamma \in \mathbb{R}$ for a large class of initial data belonging…

Well-posedness in time-weighted spaces for quasilinear (and semilinear) parabolic evolution equations $u'=A(u)u+f(u)$ is established in a certain critical case of strict inclusion $\mathrm{dom}(f)\subsetneq \mathrm{dom}(A)$ for the domains…

Let $u$ be a solution of the Cauchy problem for the nonlinear parabolic equation $$ \partial_t u=\Delta u+F(x,t,u,\nabla u) \quad in \quad{\bf R}^N\times(0,\infty), \quad u(x,0)=\varphi(x)\quad in \quad{\bf R}^N, $$ and assume that the…

We establish uniform a-priori bounds for solutions of the quasilinear problem $-\Delta_Nu=f(u)$ in $\Omega$, with $u=0$ on $\partial\Omega$, where $\Omega\subset\mathbb{R}^N$ is a bounded smooth and convex domain, and $f$ is a positive…

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $…

We study large time behaviour of solutions of the Cauchy problem for equations of the form $\partial_tu-L u+\lambda u=f(x,u)+g(x,u)\cdot\mu$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form…

This paper studies the asymptotic behavior of solutions of the parabolic-parabolic chemotaxis model with logistic-type sources in heterogeneous bounded domains: \begin{equation*} \label{u-v-eq00} \begin{cases} u_t=\Delta u-\chi\nabla\cdot…

We consider the Cauchy-problem for the following parabolic equation: \begin{equation*} \displaystyle u_t = \Delta u+ f(u,|x|), \end{equation*} where $x \in \mathbb{R}^n$, $n >2$, and $f=f(u,|x|)$ is either critical or supercritical with…

This paper is concerned with the global existence and stability of solution to the quasi linear hyperbolic-parabolic chemotaxis system on the half-line,which was proposed in[1] to primarily describe the formation of coherent vascular…

We consider semilinear evolution equations of the form $a(t)\partial_{tt}u + b(t) \partial_t u + Lu = f(x,u)$ and $b(t) \partial_t u + Lu = f(x,u),$ with possibly unbounded $a(t)$ and possibly sign-changing damping coefficient $b(t)$, and…

In this paper, we consider the following nonlinear parabolic equation \[ \partial_{t}u\,=\,\sum_{i=1}^{n}\partial_{x_{i}}\left[(\vert u_{x_{i}}\vert-\delta_{i})_{+}^{p-1}\frac{u_{x_{i}}}{\vert…

In this paper we study the existence of bound states of the following class of quasilinear problems, \begin{equation*} \left\{ \begin{aligned} &-\varepsilon ^p\Delta_pu+V(x)u^{p-1}=f(u)+u^{p^\ast -1},\ u>0,\ \text{in}\ \mathbb{R}^{N}, &\lim…

We are concerned with nonexistence results for a class of quasilinear parabolic differential problems with a potential in $\Omega\times(0,+\infty)$, where $\Omega$ is a bounded domain. In particular, we investigate how the behavior of the…

This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic type $ u_t=\nabla\cdot(\phi(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+g(u)$, $\tau v_t=\Delta v-v+u$ in…

This work is devoted to the study of the nonlinear second-order neutral difference equations with quasi-differences of the form $$ \Delta \left( r_{n} \Delta \left( x_{n}+q_{n}x_{n-\tau}\right)\right)= a_{n}f(x_{n-\sigma})+b_n%, \ n\geq n_0…

We consider the quasilinear parabolic-parabolic Keller-Segel system $$ u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \qquad x\in\Omega, \ t>0, v_t=\Delta v -v + u, x\in\Omega, \ t>0, $$ under homogeneous Neumann boundary…

If $\Omega$ is a bounded domain in $\mathbb R^N$ and $f$ a continuous increasing function satisfying a super linear growth condition at infinity, we study the existence and uniqueness of solutions for the problem (P): $\partial_tu-\Delta…

We consider periodic homogenization of boundary value problems for second-order semilinear elliptic systems in 2D of the type $$ \partial_{x_i}\left(a_{ij}^{\alpha…

We study the nonlinear fractional reaction diffusion equation $\partial_{t}u + (-\Delta)^{s} u= f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together with Dirichlet boundary conditions on $\R^N \setminus \Omega$. We prove asymptotic…

This is the first of a two-parts work on the qualitative properties and large time behavior for the following quasilinear equation involving a spatially inhomogeneous absorption $$ \partial_tu=\Delta u^m-|x|^{\sigma}u^p, $$ posed for…