Related papers: Self-similar extinction for a fast diffusion equat…

We classify the blow up self-similar profiles for the following reaction-diffusion equation with weighted reaction $$ u_t=(u^m)_{xx} + |x|^{\sigma}u^m, $$ posed for $(x,t)\in\real\times(0,T)$, with $m>1$ and $\sigma>0$. In strong contrast…

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 construct solutions of the fast diffusion equation, which exist for all $t\in\mathbb{R}$ and are singular on the set $\Gamma(t):= \{ \xi(s) ; -\infty <s \leq ct \}$, $c>0$, where $\xi\in C^3(\mathbb{R};\mathbb{R}^n)$, $n\geq 2$. We also…

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…

In this paper, we present an approach to characterising self-similar fast-reaction limits of systems with nonlinear diffusion. For appropriate initial data, in the fast-reaction limit as k tends to infinithy,spatial segregation results in…

We consider the inhomogeneous Landau equation with $\gamma \in (\sqrt{3},2]$ and construct smooth, strictly positive initial data that develop a finite time singularity. The $C^{\alpha}$-norm of the distribution function blows up for every…

This paper is concerned with a type of nonlinear reaction-diffusion equation, which arises from the population dynamics. The equation includes a certain type reaction term $u^\alpha(1- \sigma \int_{\R^n}u^\beta dx)$ of dimension $n \ge 1$…

The aim of this paper is to study the singular solutions to fractional elliptic equations with absorption $$ \left\{\arraycolsep=1pt \begin{array}{lll} (-\Delta)^\alpha u+|u|^{p-1}u=0,\quad & \rm{in}\quad\Omega\setminus\{0\},\\[2mm]…

We establish symmetrization results for the solutions of the linear fractional diffusion equation $\partial_t u +(-\Delta)^{\sigma/2}u=f$ and itselliptic counterpart $h v +(-\Delta)^{\sigma/2}v=f$, $h>0$, using the concept of comparison of…

We study nonnegative solutions to the Cauchy problem for the Fractional Fast Diffusion Equation on a suitable class of connected, noncompact Riemannian manifolds. This parabolic equation is both singular and nonlocal: the diffusion is…

We study an anisotropic, possibly non-homogeneous version of the evolution $p$-Laplacian equation when fast diffusion holds in all directions. We develop the basic theory and prove symmetrization results from which we derive $L^1$ to…

We study the long time behaviour of solutions of semi-linear parabolic equation of the following type $\partial_t u-\Delta u+a_0(x)u^q=0$ where $a_0(x) \geq d_0 \exp(\frac{\omega(|x|)}{|x|^2})$, $d_0>0$, $1>q>0$ and $\omega$ a positive…

We seek for self-similar solutions describing the time-dependent evolution of self-gravity systems with either spherical symmetry or axisymmetric disk geometry. By assuming self-similar variable $x\equiv r/at$ where $a$ is isothermal sound…

Let $L$ be a positive definite self-adjoint operator on the $L^2$-space associated to a $\si$-finite measure space. Let $H$ be the dual space of the domain of $L^{1/2}$ w.r.t. $L^2(\mu)$. By using an It\^o type inequality for the $H$-norm…

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space $\mathbb{R}^n$, where $n \ge 2$, assuming that the diffusion matrix depends on the space variable $x$ and has a finite…

We study the diffusion equation with an appropriate change of variables. This equation is in general a partial differential equation (PDE). With the self-similar and related Ansat\"atze we transform the PDE of diffusion to an ordinary…

Bifurcations of self-similar solutions for reversing interfaces are studied in the slow diffusion equation with strong absorption. The self-similar solutions bifurcate from the time-independent solutions for standing interfaces. We show…

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 consider the semilinear heat equation $u_t=\Delta u+|u|^{p-1}u-|u|^{q-1}u$ in $\mathbb{R}^n\times(0,T)$, where $n=5$, $p=\frac{n+2}{n-2}$ and $q\in(0,1)$. By the presence of $-|u|^{q-1}u$, this equation has a finite time extinction…

In this paper, we investigate the extinction behavior of nonnegative solutions to the Sobolev critical fast diffusion equation in bounded smooth domains with the Dirichlet zero boundary condition. Under the two-bubble energy threshold…