Related papers: A Continuum of Extinction Rates for the Fast Diffu…

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term related to an index $s\in \R$ over the whole space $\R^n$ for any spatial dimension $n\geq 1$. Here, the diffusion-type source term…

We study the speed of propagation of fronts for the scalar reaction-diffusion equation $u_t = u_{xx} + f(u)$\, with $f(0) = f(1) = 0$. We give a new integral variational principle for the speed of the fronts joining the state $u=1$ to…

In this paper, we study the global well-posedness and optimal time decay rates of strong solutions to the diffusion approximation model in radiation hydrodynamics in $\mathbb{R}^3$. This model consists of the full compressible Navier-Stokes…

We consider the equation $\Delta u=Vu$ in exterior domains in $\mathbb{R}^2$ and $\mathbb{R}^3$, where $V$ has certain periodicity properties. In particular we show that such equations cannot have non-trivial superexponentially decaying…

We study the wave equation with potential $u_{tt}-\Delta u+Vu=0$ in two spatial dimensions, with $V$ a real-valued, decaying potential. With $H=-\Delta+V$, we study a variety of mapping estimates of the solution operators, $\cos(t\sqrt{H})$…

The initial value problem for the conservation law $\partial_t u+(-\Delta)^{\alpha/2}u+\nabla \cdot f(u)=0$ is studied for $\alpha\in (1,2)$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic…

Let $n\ge 3$, $0<m<\frac{n-2}{n}$, $\alpha=\frac{2\beta-1}{1-m}$ and $\frac{2}{1-m}<\frac{\alpha}{\beta}<\frac{n-2}{m}$. We give a new direct proof using fixed point method on the existence of singular radially symmetric forward…

We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $\sigma >…

On a smooth bounded Euclidean domain, Sobolev-subcritical fast diffusion with vanishing boundary trace is known to lead to finite-time extinction, with a vanishing profile selected by the initial datum. In rescaled variables, we quantify…

We consider solutions to linear parabolic equations with initial data decaying at spatial infinity. For a class of advection-diffusion equations with a spatially dependent velocity field, we study the behavior of solutions as time tends to…

In this paper we address the decay of solutions to the four-dimen\-sional energy-critical nonlinear heat equation in the critical space $\dot{H}^1$. Recently, it was proven that the $\dot{H}^1$ norm of solutions goes to zero when time goes…

We study the asymptotic large time behavior of singular solutions of the fast diffusion equation $u_t=\Delta u^m$ in $({\mathbb R}^n\setminus\{0\})\times(0,\infty)$ in the subcritical case $0<m<\frac{n-2}{n}$, $n\ge3$. Firstly, we prove the…

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the…

We study the asymptotic speed of a random front for solutions $u_t(x)$ to stochastic reaction-diffusion equations of the form \[ \partial_tu=\farc{1}{2}\partial_x^2u+f(u)+\sigma\sqrt{u(1-u)}\dot{W}(t,x),~t\ge 0,~x\in\Rm, \] arising in…

Solutions in self-similar form, either global in time or presenting finite time blow-up, to the supercritical fast diffusion equation with spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, \quad…

We prove the \textit{finite time extinction property} $(u(t)\equiv 0$ on $\Omega$ for any $t\ge T_\star,$ for some $T_\star>0)$ for solutions of the nonlinear Schr\"{o}dinger problem ${\rm i} u_t+\Delta u+a|u|^{-(1-m)}u=f(t,x),$ on a…

In this work, we construct a transformation between the solutions to the following reaction-convection-diffusion equation $$ \partial_t u=(u^m)_{xx}+a(x)(u^m)_x+b(x)u^m, $$ posed for $x\in\real$, $t\geq0$ and $m>1$, where $a$, $b$ are two…

This article discusses the analyticity and the long-time asymptotic behavior of solutions to space-time fractional diffusion equations in $\mathbb{R}^d$. By a Laplace transform argument, we prove that the decay rate of the solution as…

Ballistic annihilation with continuous initial velocity distributions is investigated in the framework of Boltzmann equation. The particle density and the rms velocity decay as $c=t^{-\alpha}$ and $<v>=t^{-\beta}$, with the exponents…

In this paper, we consider the Cauchy problem for the fractional Schr\"odinger equation $i D_t^\alpha u + (-\Delta)^{\frac{\beta}{2}} u =0$ with $0<\alpha<1$, $\beta>0$. We establish the dispersive estimates for the solutions. In…