相关论文: Initial trace of positive solutions to fractional …

This paper considers the fractional Schr\"{o}dinger equation \begin{equation}\label{abstract} (-\Delta)^s u + V(|x|)u-u^p=0, \quad u>0, \quad u\in H^{2s}(\R^N) \end{equation} where $0<s<1$, $1<p<\frac{N+2s}{N-2s}$, $V(|x|)$ is a positive…

This paper continues the investigation of Du and Lou (J. European Math Soc, to appear), where the long-time behavior of positive solutions to a nonlinear diffusion equation of the form $u_t=u_{xx}+f(u)$ for $x$ over a varying interval…

Let n>2, $0<m\le (n-2)/n$, p>\max(1,(1-m)n/2), and $0\le u_0\in L_{loc}^p(R^n)$ satisfy $\liminf_{R\to\infty}R^{-n+\frac{2}{1-m}}\int_{|x|\le R}u_0\,dx=\infty$. We prove the existence of unique global classical solution of…

For $\alpha \in (1,2)$ we consider the equation $\partial_t u = \Delta^{\alpha/2} u - r b \cdot \nabla u$, where $b$ is a divergence free singular vector field not necessarily belonging to the Kato class. We show that for sufficiently small…

We consider the following equations: \begin{equation*} \left\{\begin{array}{ll} (-\triangle)^{\alpha/2}u(x)=f(v(x)), \\ (-\triangle)^{\beta/2}v(x)=g(u(x)), &x \in R^{n},\\ u,v\geq 0, &x \in R^{n}, \end{array} \right. \end{equation*} for…

Let $(u,v)$ be a nonnegative solution to the semilinear parabolic system \[ \mbox{(P)} \qquad \cases{ \partial_t u=D_1\Delta u+v^p, & $x\in{\bf R}^N,\,\,\,t>0,$\\ \partial_t v=D_2\Delta v+u^q, & $x\in{\bf R}^N,\,\,\,t>0,$\\…

In this paper we study the inverse problem of identifying a source or an initial state in a time-fractional diffusion equation from the knowledge of a single boundary measurement. We derive logarithmic stability estimates for both…

Let $0<\alpha,\beta<2$ be any real number. In this paper, we investigate the following semilinear system involving the fractional Laplacian \begin{equation*} \left\{\begin{array}{lll} (-\lap)^{\alpha/2} u(x)=f(v(x)), & (-\lap)^{\beta/2}…

In the case of any bounded open set $\Omega$ $\subset$ R d with boundary $\partial$$\Omega$, we first construct a directional trace in any direction $\theta$ of the unit sphere, for any u $\in$ L 2 ($\Omega$) whose the directional…

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…

We study the Cauchy problem for the fractional Schr\"{o}dinger equation $$ i\partial_tu = (m^2-\Delta)^\frac\alpha2 u + F(u) in \mathbb{R}^{1+n}, $$ where $ n \ge 1$, $m \ge 0$, $1 < \alpha < 2$, and $F$ stands for the nonlinearity of…

We consider the following class of fractional parametric problems \begin{equation*} \left\{ \begin{array}{ll} (-\Delta_{Dir})^{s} u= f(x, u)+t\varphi_{1}+h &\mbox{ in } \Omega\\ u=0 &\mbox{ on } \partial \Omega, \end{array} \right.…

Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^N$. We study positive solutions of equation (E) $-L_\mu u+ u^q = 0$ in $\Omega$ where $L_\mu=\Delta + \frac{\mu}{\delta^2}$, $0<\mu$, $q>1$ and…

We consider reaction-diffusion equations $\partial_tu=\Delta u+f(u)$ in the whole space $\mathbb{R}^N$ and we are interested in the large-time dynamics of solutions ranging in the interval $[0,1]$, with general unbounded initial support.…

In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective…

In the present work, we discuss the existence of a unique positive solution of a boundary value problem for nonlinear fractional order equation with singularity. Precisely, order of equation $D_{0+}^\alpha u(t)=f(t,u(t))$ belongs to $(3,4]$…

In this work we prove the existence of solution for a class of perturbed fractional Hamiltonian systems given by \begin{eqnarray}\label{eq00} -{_{t}}D_{\infty}^{\alpha}(_{-\infty}D_{t}^{\alpha}u(t)) - L(t)u(t) + \nabla W(t,u(t)) = f(t),…

This paper studies the linear stochastic partial differential equation of fractional orders both in time and space variables $\left(\partial^\beta + \frac{\nu}{2} (-\Delta)^{\alpha/2} \right) u(t,x)= \lambda u(t,x) \dot{W}(t,x)$, where…

In this paper, the existence of least energy solution and infinitely many solutions is proved for the equation $(1-\Delta)^\alpha u = f(u)$ in $\mathbf{R}^N$ where $0<\alpha<1$, $N \geq 2$ and $f(s)$ is a Berestycki-Lions type nonlinearity.…

In this paper we study the problem -\Delta u =\left(\frac{2+\alpha}{2}\right)^2\abs{x}^{\alpha}f(\lambda,u), & \hbox{in}B_1 \\ u > 0, & \hbox{in}B_1 u = 0, & \hbox{on} \partial B_1 where $B_1$ is the unit ball of $\R^2$, $f$ is a smooth…