Related papers: Large Time Behavior of a Nonlocal Diffusion Equati…
We investigate the long-time behavior of global solutions to the energy critical heat equation in $R^5$ \begin{equation*} \begin{cases} \pp_t u=\Delta u+|u|^{\frac{4}{3}} u ~&\mbox{ in }~ R^5 \times (t_0,\infty), u(\cdot,t_0)=u_0~&\mbox{ in…
In this paper, we study large-time asymptotics for heat and fractional heat equations in two discrete settings: the full lattice \(\mathbb Z^d\) and finite connected subgraphs with Dirichlet boundary condition. These results provide a…
We study the large time behavior of non-negative solutions to the singular diffusion equation with gradient absorption $$ \partial_t u-\Delta_{p}u+|\nabla u|^q=0 \quad \hbox{in} \ (0,\infty)\times\real^N, $$ for $p_c:=2N/(N+1)
We consider the following Cauchy problem for three dimensional energy critical heat equation \begin{equation*} \begin{cases} u_t=\Delta u+u^{5},~&\mbox{ in } \ {\mathbb R}^3 \times (0,T),\\ u(x,0)=u_0(x),~&\mbox{ in } \ {\mathbb R}^3.…
We consider the Cauchy problem for the generalized Kadomtsev--Petviashvili--Burgers equation in 2D. This is one of the nonlinear dispersive-dissipative type equations, which has a spatial anisotropic dissipative term. Under some suitable…
We study the large time behavior of solutions to a non-local diffusion equation, $u_t=J*u-u$ with $J$ smooth, radially symmetric and compactly supported, posed in $\mathbb{R}_+$ with zero Dirichlet boundary conditions. In sets of the form…
In this paper, we study the large time behavior of solutions of a class of parabolic fully nonlinear integro-differential equations in a periodic setting. In order to do so, we first solve the ergodic problem}(or cell problem), i.e. we…
The large time behavior of nonnegative solutions to the reaction-diffusion equation $\partial_t u=-(-\Delta)^{\alpha/2}u - u^p,$ $(\alpha\in(0,2], p>1)$ posed on $\mathbb{R}^N$ and supplemented with an integrable initial condition is…
We consider positive solutions for the fractional heat equation with critical exponent \begin{equation*} \begin{cases} u_t = -(-\Delta)^{s}u + u^{\frac{n+2s}{n-2s}}\text{ in } \Omega\times (0, \infty), u = 0\text{ on }…
We investigate the large-time asymptotics of solution for the Cauchy problem of the nonlocal focusing modified Kortweg-de Vries (MKdV) equation with step-like initial data, i.e., $u_0(x)\rightarrow 0$ as $x\rightarrow-\infty$,…
In the paper, we consider the large time behavior of solutions to the convection-diffusion equation u_t - Delta u + nabla cdot f(u) = 0 in R^n times [0,infinity), where f(u) ~ u^q as u --> 0. Under the assumption that q >= 1+1/(n+beta) and…
In this paper, we consider the Cauchy problem for a semilinear damped wave equation with the nonlinear term $|u|^{1+2/n} \mu(|u|)$, where $\mu$ is a modulus of continuity. In recent papers by Ebert,Girardi,Reissig (Math. Ann. 378 (2020))…
We consider the classical Cauchy problem for the linear heat equation and integrable initial data in the Euclidean space $\mathbb{R}^N$. In the case $N=1$ we show that given a weighted $L^p$-space $L_w^p(\mathbb{R})$ with $1 \leq p <…
This paper studies the global existence and uniqueness of strong solutions and its large-time behavior for the compressible isothermal Euler equations with a nonlocal dissipation. The system is rigorously derived from the kinetic…
We consider the Cauchy problem for linearly damped nonlinear Schr\"odinger equations \[ i\partial_t u + \Delta u + i a u= \pm |u|^\alpha u, \quad (t,x) \in [0,\infty) \times \mathbb{R}^N, \] where $a>0$ and $\alpha>0$. We prove the global…
We study the large-time behavior in all $L^p$ norms and in different space-time scales of solutions to a heat equation with a Caputo $\alpha$-time derivative posed in $\mathbb{R}^N$. The initial data are assumed to be integrable, and, when…
The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case…
We study a time-fractional semilinear heat equation $$\partial^{\alpha}_t u -\Delta u = u^{p},\ \ \mbox{in}\ (0,T)\times\mathbb{R}^N,\ \ u(0)=u_0\ge0$$ with $u_0\in L^{1}(\mathbb{R}^N)$ and $p=1+2/N$. Here $\partial_t^{\alpha}$ denotes the…
In this paper, we consider the global Cauchy problem for the $L^2$-critical semilinear heat equations $ \partial_t h=\Delta h\pm |h|^{\frac4d}h, $ with $h(0,x)=h_0$, where $h$ is an unknown real function defined on $ \R^+\times\R^d$. In…
Let $N\ge 1$ and let $f\in C[0,\infty)$ be a nonnegative nondecreasing function and $u_0$ be a possibly singular nonnegative initial function. We are concerned with existence and nonexistence of a local in time nonnegative solution in a…