Related papers: Behaviour near extinction for the Fast Diffusion E…

For $n\ge 3$, $0<m<\frac{n-2}{n}$, $\beta<0$ and $\alpha=\frac{2\beta}{1-m}$, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in…

We present a complete description of the similarity solutions $u_{\alpha}(x,t)=t^{-\alpha/2}f(\Vert x \Vert/\sqrt{t};\alpha)$ for the following nonlinear diffusion equation $$ u_{t}+\gamma\vert u_{t} \vert =\Delta u\qquad(-1<\gamma<1) $$…

We consider, for $a,l\geq1,$ $b,s,\alpha>0,$ and $p>q\geq1,$ the homogeneous Dirichlet problem for the equation $-\Delta_{p}u=\lambda u^{q-1}+\beta u^{a-1}\left\vert \nabla u\right\vert ^{b}+mu^{l-1}e^{\alpha u^{s}}$ in a smooth bounded…

We prove sharp estimates for the decay in time of solutions to a rather general class of non-local in time subdiffusion equations on a bounded domain subject to a homogeneous Dirichlet boundary condition. Important special cases are the…

We study the propagation properties of nonnegative and bounded solutions of the class of reaction-diffusion equations with nonlinear fractional diffusion: $u_{t} + (-\Delta)^s (u^m)=f(u)$. For all $0<s<1$ and $m> m_c=(N-2s)_+/N $, we…

In this paper, we investigated a density-dependent reaction-diffusion equation, $u_t = (u^{m})_{xx} + u - u^{m}$. This equation is known as the extension of the Fisher or Kolmogoroff-Petrovsky-Piscounoff equation which is widely used in the…

We investigate quantitative properties of nonnegative solutions $u(t,x)\ge 0$ to the nonlinear fractional diffusion equation, $\partial_t u + \mathcal{L}F(u)=0$ posed in a bounded domain, $x\in\Omega\subset \mathbb{R}^N$, with appropriate…

In this paper we consider the problem $$(P)\qquad \{{array}{rclll} u_t-\D u^m&=&|\n u|^q +\,f(x,t),&\quad u\ge 0 \hbox{in} \Omega_T\equiv \Omega\times (0,T), u(x,t)&=&0 &\quad \hbox{on} \partial\Omega\times (0,T) u(x,0)&=&u_0(x),&\quad x\in…

Assume that $N_m(x)$ denotes the density of the population at a point $x$ at the beginning of the reproductive season in the $m$th year. We study the following impulsive reaction-diffusion model for any $m\in \mathbb Z^+$…

This paper is concerned with supersolutions to parabolic equations of the form \begin{equation} \partial_t U (x,t)-D(x)\Delta U(x,t)=0, \quad (x,t)\in \mathbb{R}^N \times (0,\infty), \end{equation} where $D\in C(\mathbb{R}^N)$ is positive.…

For any $-1<m<0$, $\mu>0$, $0\le u_0\in L^{\infty}(R)$ such that $u_0(x)\le (\mu_0 |m||x|)^{\frac{1}{m}}$ for any $|x|\ge R_0$ and some constants $R_0>1$ and $0<\mu_0\leq \mu$, and $f,\,g \in C([0,\infty))$ such that $f(t),\, g(t) \geq…

We study the asymptotic behaviour near extinction of positive solutions of the Cauchy problem for the fast diffusion equation with a critical exponent. After a suitable rescaling which yields a non--linear Fokker--Planck equation, we find a…

The behavior near the extinction time is identified for non-negative solutions to the diffusive Hamilton-Jacobi equation with critical gradient absorption $\partial_t u - \Delta_p u + |\nabla u|^{p-1} = 0$ in $(0, \infty) \times…

When $2N/(N+1)<p<2$ and $0<q<p/2$, non-negative solutions to the singular diffusion equation with gradient absorption $$\partial\_tu-\Delta\_p u + |\nabla u|^q=0 \ \text{ in }\ (0,\infty)\times\mathbb{R}^N$$ vanish after a finite time. This…

In this paper we study the following one-dimensional reaction-diffusion problem $$ u_t+(-\Delta)^s u=f(x-c t, u) \;\:\textrm{ in } \mathbb{R}\times (0,+\infty), $$ where $s>\frac{1}{2}$, $c \in \mathbb{R}$ is a prescribed velocity, and $f$…

In this paper we consider radially symmetric solutions of the following parabolic--elliptic cross-diffusion system \begin{equation*} \begin{cases} u_t = \Delta u - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), & \\[2mm] 0= \Delta v…

We study the homogeneous Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $\partial_t u =-\mathcal{L} u^m$ posed on a bounded Euclidean domain $\Omega\subset\mathbb{R}^N$ with…

We consider a solution $u(\cdot,t)$ to an initial boundary value problem for time-fractional diffusion-wave equation with the order $\alpha \in (0,2) \setminus \{ 1\}$ where $t$ is a time variable. We first prove that a suitable norm of…

The global boundedness and asymptotic behavior are investigate for the solution of time-space fractional non-local reaction-diffusion equation (TSFNRDE) $$ \frac{\partial^{\alpha }u}{\partial t^{\alpha }}=-(-\Delta)^{s} u+\mu…

Our aim is to study the limit of the solution of reaction-diffusion porous medium equation with linear drift $\displaystyle\partial_t u -\Delta u^m +\nabla \cdot (u \: V)=g(t,x,u) $, as $m\to\infty.$ We study the problem in bounded domain…