Related papers: Vanishing time behavior of solutions to the fast d…
In this paper we study the asymptotic behavior as time goes to infinity of the solution to a nonlocal diffusion equation with absorption modeled by a powerlike reaction $-u^p$, $p>1$ and set in $\R^N$. We consider a bounded, nonnegative…
We study the large time behavior of non-negative solutions to the nonlinear diffusion equation with critical gradient absorption $$\partial\_t u - \Delta\_{p}u + |\nabla u|^{q\_*} = 0 \quad \hbox{in} (0,\infty)\times\mathbb{R}^N\ ,$$ for…
We consider the large time asymptotic behavior of the global solutions to the initial value problem for the nonlinear damped wave equation with slowly decaying initial data. When the initial data decay fast enough, it is known that the…
We consider the initial value problem for the thermal-diffusive combustion systems of the form: $u_{1,t}= Delta_{x}u_1 - u_1 u_2^m$, $u_{2,t}= d Delta_{x} u_2 + u_1 u_2^m$, $x in R^{n}$, $n geq 1$, $m geq 1$, $d > 1$, with bounded uniformly…
We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a subcritical exponent. We show that separable solutions are stable in some suitable sense by finding a class…
We classify the smooth self-similar solutions of the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u$ in $\mathbb{R}^n\times (0,T)$ satisfying an integral condition for all $p>1$ with positive speed. As a corollary, we prove that finite…
Self-diffusion in a two-dimensional simple fluid is investigated by both analytical and numerical means. We investigate the anomalous aspects of self-diffusion in two-dimensional fluids with regards to the mean square displacement, the…
We investigate the long time behavior of solutions to semilinear hyperbolic equation (E$_{\alpha}$): $ u^{\prime\prime}(t)+\gamma(t)u^{\prime}(t)+Au(t)+f(u(t))=g(t),~t\geq0, $ where $A$ is a self-adjoint nonnegative operator, $f$ a function…
The large time behavior of non-negative solutions to the viscous Hamilton-Jacobi equation $u_t - \Delta u + |\nabla u|^q = 0$ in the whole space $R^N$ is investigated for the critical exponent $q = (N+2)/(N+1)$. Convergence towards a…
We investigate qualitative properties of local solutions $u(t,x)\ge 0$ to the fast diffusion equation, $\partial_t u =\Delta (u^m)/m$ with $m<1$, corresponding to general nonnegative initial data. Our main results are quantitative…
We study the asymptotic behavior of solutions to the heat equation in nonhomogeneous media with critical singular density $$ |x|^{-2}\partial_{t}u=\Delta u, \quad \hbox{in} \ \real^N\times(0,\infty). $$ The asymptotic behavior proves to…
Let $u$ be the solution of $u_t=\Delta\log u$ in $\R^N\times (0,T)$, N=3 or $N\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\le u_0\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t)…
In this paper, we investigate the asymptotic behavior, as $\beta \to 0$, of positive solutions to the semilinear elliptic Robin problem \begin{equation*} \begin{cases} -\Delta u = u^p, & \text{in } \Omega,\\ u > 0, & \text{in } \Omega,\\…
We consider the semilinear diffusion equation $\partial$ t u = Au + |u| $\alpha$ u in the half-space R N + := R N --1 x (0, +$\infty$), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional…
This article is on the simultaneous diffusion approximation and homogenization of the linear Boltzmann equation when both the mean free path $\varepsilon$ and the heterogeneity length scale $\eta$ vanish. No periodicity assumption is made…
We prove existence and uniqueness of a global in time self-similar solution growing up as $t\to\infty$ for the following reaction-diffusion equation with a singular potential $$ u_t=\Delta u^m+|x|^{\sigma}u^p, $$ posed in dimension…
We establish the existence of self-similar solutions presenting finite time blow-up to the quasilinear reaction-diffusion equation $$ u_t=\Delta u^m + u^p, $$ posed in dimension $N\geq3$, $m>1$. More precisely, we show that there is always…
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…
We consider the following stochastic space-time fractional diffusion equation with vanishing initial condition:$$ \partial^{\beta} u(t, x)=- \left(-\Delta\right)^{\alpha / 2} u(t, x)+ I_{0+}^{\gamma}\left[\dot{W}(t, x)\right],\quad…
The Fast Diffusion Equation (FDE) $u_t= \Delta u^m$, with $m\in (0,1)$, is an important model for singular nonlinear (density dependent) diffusive phenomena. Here, we focus on the Cauchy-Dirichlet problem posed on smooth bounded Euclidean…