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Xianfa Song

Publications and source records attributed to Xianfa Song.

15 recordsLinked to original sources
arXiv2024

Critical line of exponents, scattering theories for a weighted gradient system of semilinear wave equations

In this paper, we consider the following Cauchy problem of a weighted gradient system of semilinear wave equations \begin{equation*} \left\{ \begin{array}{lll} u_{tt}-\Delta u=\lambda |u|^{\alpha}|v|^{\beta+2}u,\quad v_{tt}-\Delta v=\mu |u|^{\alpha+2}|v|^{\beta}v,\quad x\in \mathbb{R}^d,\ t\in \mathbb{R},\\ u(x,0)=u_{10}(x),\ u_t(x,0)=u_{20}(x),\quad v(x,0)=v_{10}(x),\ v_t(x,0)=v_{20}(x),\quad x\in \mathbb{R}^d. \end{array}\right. \end{equation*} Here $d\geq 3$, $\lambda, \mu\in \mathbb{R}$, $\alpha, \beta\geq 0$, $(u_{10},u_{20})$ and $(v_{10},v_{20})$ belong to $H^1(\mathbb{R}^d)\oplus L^2(\mathbb{R}^d)$ or $\dot{H}^1(\mathbb{R}^d)\oplus L^2(\mathbb{R}^d)$ or $\dot{H}^{\gamma}(\mathbb{R}^d)\oplus H^{\gamma-1}(\mathbb{R}^d)$ for some $\gamma>1$. Under certain assumptions, we establish the local wellposedness of the $H^1\oplus H^1$-solution, $\dot{H}^1\oplus \dot{H}^1$-solution and $\dot{H}^{\gamma}\oplus \dot{H}^{\gamma}$-solution of the system with different types of initial data.

math-ph↗
arXiv2021

$H^s_x\times H^s_x$ scattering theory for a weighted gradient system of 3D radial defocusing NLS

In this paper, using $I$-method, we establish $H^s_x\times H^s_x$ scattering theories for the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t+Δu=λ|v|^2u,\quad iv_t+Δv=μ|u|^2v,\quad x\in \mathbb{R}^3,\ t>0,\\ u(x,0)=u_0(x),\quad v(x,0)=v_0(x),\quad x\in \mathbb{R}^3. \end{array}\right. \end{equation*} Here $λ>0$, $μ>0$, $(u_0,v_0)\in H^s_x(\mathbb{R}^3)\times H^s_x(\mathbb{R}^3)$ and $\frac{1}{2}<s<1$.

math.AP↗
arXiv2020

Quenching, global existence and blowup phenomena in heat transfer

Basing on the relations between a system of ODE and a system of parabolic equations, we establish some general theories in heat transfer about quenching, global existence and blowup phenomena, obtain the conditions(even watershed) on f(u,v), g(u,v), a(x) and b(x) which let the solution be global existence, quench or blow up, and estimate the bounds for blowup time and quenching time.

math.AP↗
arXiv2019

Spacetime estimates and scattering theory for quasilinear Schrödinger equations in arbitrary space dimension

In this paper, we consider the following Cauchy problem of \begin{equation*} \left\{ \begin{array}{lll} iu_t=Δu+2δ_huh'(|u|^2)Δh(|u|^2)+V(x)u+F(|u|^2)u+(W*|u|^2)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $δ_h$ is a constant, $N\geq 1$, $h(s)$, $F(s)$, $V(x)$ and $W(x)$ are some real functions, $W(x)$ is even. Besides obtaining some sufficient conditions on global existence of the solution, we establish pseudoconformal conservation law and give Morawetz type estimates, spacetime bounds and asymptotic behaviors for the global solution. We bring two ideas to establish scattering theory, one is that we take different admissible pairs in Strichartz estimates for different terms on the right side of Duhamel's formula in order to keep each term independent, another is that we factitiously let a continuous function be the sum of two piecewise functions and chose different admissible pairs in Strichartz estimates for the terms containing these functions. Basing on the two ideas, we provide the direct and simple proofs of classic scattering theories in $L^2(\mathbb{R}^N)$ and $Σ$ for any space dimension($N\geq 1$) under certain assumptions. Here $$ Σ=\{u\in H^1(\mathbb{R}^N),\quad |xu|\in L^2(\mathbb{R}^N)\}. $$

math-ph↗
arXiv2019

A quasilinear Schrödinger equation with Hartree type nonlinearity

In this paper, we deal with the Cauchy problem of the quasilinear Schödinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=Δu+2uh'(|u|^2)Δh(|u|^2)+(W(x)\ast|u|^2)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $W(x)$ are some real valued functions. Our focus is to investigate how the interplay between the potential $W(x)$ and the quasilinear presence $h(s)$ affects the blowup in finite time and global existence of the solution. In a special, we can obtain the watershed condition on $W(x)$ in the following sense: If $W(x)\in L^1(\mathbb{R}^N)\cap \{L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)\} $, then exist $q_c$ and $q_s$ such that the solution is global existence for any initial data in the energy space when $q>q_c$ and the solution maybe blow up in finite time for some initial data when $q_s<q<q_c$, and for $q=q_c$ whether the solution is global existence or not depend on the initial data.

math.AP↗
arXiv2019

A free boundary problem for spreading under shifting climate

In this paper we consider a free boundary problem which models the spreading of an invasive species whose spreading is enhanced by the changing climate. We assume that the climate is shifting with speed c and obtain a complete classification of the long-time dynamical behaviour of the species. The model is similar to that in [9] with a slight refinement in the free boundary condition. While [9], like many works in the literature, investigates the case that unfavourable environment is shifting into the favourable habitat of the concerned species, here we examine the situation that the unfavourable habitat of an invasive species is replaced by a favourable environment with a shifting speed c. We show that a spreading-vanishing dichotomy holds, and there exists a critical speed$c_0$ such that when spreading happens in the case $c < c_0$, the spreading profile is determined by a semi-wave with forced speed c, but when $c \geq c_0$, the spreading profile is determined by the usual semi-wave with speed $c_0$.

math.AP↗
arXiv2019

Morawetz estimates and spacetime bounds for quasilinear Schrödinger equations with critical Sobolev exponent

In this paper, we study the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t=Δu + 2uh'(|u|^2)Δh(|u|^2) + F(|u|^2)u\mp A[h(|u|^2]^{2^*-1} h'(|u|^2)u,\ x\in \mathbb{R}^N, \ t>0\\ u(x,0)=u_0(x), \quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $F(s)$ are some real-valued functions, $h(s)\geq 0$ and $h'(s)\geq 0$ for $s\geq 0$, $N\geq 3$, $A>0$. Besides obtaining sufficient conditions on the blowup in finite time and global existence of the solution, we establish Morawetz estimates and spacetime bounds for the global solution based on pseudoconformal conservation law, which is an important tool to construct scattering operator on the energy space.

math-ph↗
arXiv2019

The role of potential, Morawetz estimate and spacetime bound for quasilinear Schrödinger equations

In this paper, we deal with the following Cauchy problem \begin{equation*} \left\{ \begin{array}{lll} iu_t = Δu + 2uh'(|u|^2)Δh(|u|^2) + V(x)u,\ x\in \mathbb{R}^N,\ t>0\\ u(x,0) = u_0(x), \quad x \in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $V(x)$ are some real functions. We take the potential $V(x)\in L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)$ as criterion of the blowup and global existence of the solution to (1.1). In some cases, we can classify it in the following sense: If $V(x)\in S(I)$, then the solution of (1.1) is always global existence for any $u_0$ satisfying $0 q_c}[L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)],\quad S(II)=\left\{\cup_{q<q_c}[L^q(\mathbb{R}^N)+L^{\infty}(\mathbb{R}^N)]\right\}\setminus S(I).$$ Under certain assumptions, we also establish Morawetz estimates and spacetime bounds for the global solution, for example, \begin{align*} &\int_0^{+\infty} \int_{\mathbb{R}^N }\frac{[|\nabla h(|u|^2)|^2 + |V(x)||u|^2]}{(|x|+t)^λ}dxdt\leq C,\\ & \|u\|_{L^{\bar{q}}_t (\mathbb{R}) L^{\bar{r}}_x(\mathbb{R}^N)} = \left(\int_0^{+\infty} \left(\int_{\mathbb{R}^N}|u|^{\bar{r}} dx\right)^{\frac{\bar{q}}{\bar{r}}} dt\right)^{\frac{1}{\bar{q}}} \leq C. \end{align*}

math-ph↗
arXiv2018

Global existence, blowup phenomena, and asymptotic behavior for quasilinear Schr\"{o}dinger equations

In this paper, we study the Cauchy problem of the quasilinear Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{lll} iu_t=\Delta u+2uh'(|u|^2)\Delta h(|u|^2)+F(|u|^2)u \quad {\rm for} \ x\in \mathbb{R}^N, \ t>0\\ u(x,0)=u_0(x),\quad x\in \mathbb{R}^N. \end{array}\right. \end{equation*} Here $h(s)$ and $F(s)$ are some real-valued functions, with various choices for models from mathematical physics. We examine the interplay between the quasilinear effect of $h$ and nonlinear effect of $F$ for the global existence and blowup phenomena. We provide sufficient conditions on the blowup in finite time and global existence of the solution. In some cases, we can deduce the watershed from these conditions. In the focusing case, we construct the sharp threshold for the blowup in finite time and global existence of the solution and lower bound for blowup rate of the blowup solution.

math.AP↗
arXiv2015

Linking Theorems of Local Semiflows on Complete Metric Spaces

In this paper we prove some linking theorems and mountain pass type results for dynamical systems in terms of local semiflows on complete metric spaces. Our results provide an alternative approach to detect the existence of compact invariant sets without using the Conley index theory. They can also be applied to variational problems of elliptic equations without verifying the classical P.S. Condition. As an example, we study the resonant problem of the nonautonomous parabolic equation $ u_t-Δu-μu=f(u)+g(x,t) $ on a bounded domain. The existence of a recurrent solution is proved under some Landesman-Laser type conditions by using an appropriate linking theorem of semiflows. Another example is the elliptic equation $-Δu+a(x)u=f(x,u)$ on $R^n$. We prove the existence of positive solutions by applying a mountain pass lemma of semiflows to the parabolic flow of the problem.

math.DS↗
arXiv2013

Remarks on Scattering Properties of the Solution to a Nonlinear Schrödinger Equation with Combined Power-Type Nonlinearities

In this paper, we consider the Cauchy problem of Nonlinear Schrödinger equation \begin{align*} \left\{\begin{array}{ll}&i u_t+Δu=λ_1|u|^{p_1}u+λ_2|u|^{p_2}u, \quad t\in\mathbb{R}, \quad x\in\mathbb{R}^N &u(0,x)=φ(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*} where $N\geq 3$, $0<p_1<p_2<\frac{4}{N-2}$, $λ_1$ and $λ_2$ are real constants. Using the methods in \cite{Cazenave2} and analyzing the interaction between the nonlinearity $λ_1|u|^{p_1}u$ and $λ_2|u|^{p_2}u$, we not only partly solve the open problems of Terence Tao, Monica Visan and Xiaoyi Zhang's \cite{Tao} but also obtain other scattering properties of the solutions.

math.AP↗
arXiv2010

On the Global Existence and Blowup Phenomena of Schrödinger Equations with Multiple Nonlinearities

In this paper, we consider the global existence and blowup phenomena of the following Cauchy problem \begin{align*} \left\{\begin{array}{ll}&-i u_t=Δu-V(x)u+f(x,|u|^2)u+(W\star|u|^2)u, \quad x\in\mathbb{R}^N, \quad t>0, &u(x,0)=u_0(x), \quad x\in\mathbb{R}^N, \end{array} \right. \end{align*} where $V(x)$ and $W(x)$ are real-valued potentials with $V(x)\geq 0$ and $W$ is even, $f(x,|u|^2)$ is measurable in $x$ and continuous in $|u|^2$, and $u_0(x)$ is a complex-valued function of $x$. We obtain some sufficient conditions and establish two sharp thresholds for the blowup and global existence of the solution to the problem. These results can be looked as the supplement to Chapter 6 of \cite{Cazenave2}. In addition, our results extend those of \cite{Zhang} and improve some of \cite{Tao2}.

math.AP↗
arXiv2005

New Monotonicity Formulae for Semi-linear Elliptic and Parabolic Systems

In this paper, we establish a general monotonicity formula of the following elliptic system $$ Δu_i+f_i(u_1,...,u_m)=0 \quad {\rm in} Ω, \label{0.1} $$ where $Ω\subset\subset \mathbb{R}^n$ is a bounded domain, $(f_i(u_1,...,u_m))=\nabla F(\vec{u})$, and $F(\vec{u})$ is a given smooth function of $\vec{u}=(u_1,...,u_m)$, $m,n$ are two positive integers. We also set up a new monotonicity formula for the following parabolic system $$ \partial_t u_i-Δu_i-f_i(u_1,...,u_m)=0, in (t_1, t_2)\times \mathbb{R}^n, $$ where $t_1<t_2$ are two constants, $(f_i(\vec{u}))$ is given as above. Our new monotonicity formulae are focused on more attention to the monotonicity of non-linear terms. Our point of view is that we introduce an index called $β$ to measure the monotonicity of the non-linear terms in the problems. The index in the study of monotonicity formulae is very useful in understanding the behavior of blow up sequences of solutions. Corresponding monotonicity results for free boundary problems are also presented.

math.AP↗