We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed in $\real^N$ with $N\geq3$, where $$ 0<m<m_c=\frac{N-2}{N}, \qquad p>1, $$ and the critical value for the weight $$ σ=\frac{2(p-1)}{1-m}. $$ The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a \emph{change of sign of both self-similar exponents} at $m=m_s=(N-2)/(N+2)$, leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a \emph{perfect equilibrium} in the competition between the fast diffusion and the reaction effects.
We study the asymptotic behavior of solutions to the defocusing mass-subcritical Hartree NLS $iu_t + Δu = F(u) = (|x|^{-γ}*|u|^2)u$ on $\mathbb{R}^d$, $d\geq 2$, $\frac{4}{3} < γ< 2$. We show that the scattering problem associated to this equation is analytically well-posed in the weighted spaces $Σ= H^1\cap\mathcal{F}H^1$ and $\mathcal{F}H^1$. Furthermore, we show that the same problem fails to be analytically well-posed for data in $L^2$. This constitutes an infinite loss of regularity between the scattering problems in weighted spaces and in $L^2$. This further develops an earlier investiga…
On global well-posedness for nonlinear semirelativistic equations in some scaling subcritical and critical cases
研究了欧几里得空间上具有幂型非线性项的半相对论性方程的整体适定性。在二维$1 \leq s \leq 2$标度次临界情形下，局部适定性适定性来自于一个Schmidhartz估计。在高维$H^1$标度次临界情形下，径向解的局部适定性由加权的Schmidhartz估计得到。此外，在三维$H^1$标度临界情形下，径向解的局部适定性可由相应的一维问题的解的一致界得到.局部解可以通过先验估计来扩展。
On the Hardy-Hénon heat equation with an inverse square potential
Indian Institute of Science Education and Research Pune（印度科学教育研究所浦那分校）；Indian Institute of Science Education and Research Tirupati（印度科学教育研究所蒂鲁帕蒂分校）；Keio University（庆应义塾大学）；RIKEN（理化学研究所）
Large-data modified wave operators for the defocusing nonlinear Schrödinger equation in one space dimension with subcritical long-range nonlinearity
Research Institute for Interdisciplinary Science, Okayama University（冈山大学跨学科科学研究所）；Department of Mathematics, Graduate School of Science, The University of Osaka（大阪大学理学研究科数学系）
The Cauchy problem for the Hardy-Hénon parabolic equation is studied in the critical and subcritical regime in weighted Lebesgue spaces on the Euclidean space $\mathbb{R}^d$. Well-posedness for singular initial data and existence of non-radial forward self-similar solution of the problem are previously shown only for the Hardy and Fujita cases ($γ\le 0$) in earlier works. The weighted spaces enable us to treat the potential $|x|^γ$ as an increase or decrease of the weight, thereby we can prove well-posedness to the problem for all $γ$ with $-\min\{2,d\}<γ$ including the Hénon case ($γ>0$). As …
We consider a class of weighted Emden-Fowler equations \begin{equation} \tag{$\mathcal P_α$} \label{eqab} \left\{\begin{array}{ll} -Δu=V_α (x) \, u^p & \text{in} \,\,B,\\ u>0 & \text{in} \,\,B,\\ u=0 & \text{on}\,\,\partial B, \end{array}\right. \end{equation} posed on the unit ball $B=B(0,1)\subset \mathbb R^N$, $N \geq1$. We prove that symmetry breaking occurs for the groundstate solutions as the parameter $α\rightarrow \infty.$ The above problem reads as a possibly large perturbation of the classical Hénon equation. We consider a radial function $V_α$ having a spherical shell of zeroes at $…
We analyze the well-posedness of an anisotropic, nonlocal diffusion equation. Establishing an equivalence between weighted and unweighted anisotropic nonlocal diffusion operators in the vein of unified nonlocal vector calculus, we apply our analysis to a class of fractional-order operators and present rigorous estimates for the solution of the corresponding anisotropic anomalous diffusion equation. Furthermore, we extend our analysis to the anisotropic diffusion-advection equation and prove well-posedness for fractional orders s in [0.5,1). We also present an application of the advection-diffu…
This article establishes existence, non-existence and Liouville-type theorems for nonlinear equations of the form $$-div (|x|^{a} D u ) = f(x,u), ~ u > 0,\, \mbox{ in } Ω,$$ where $N \geq 3$, $Ω$ is an open domain in $\mathbb{R}^N$ containing the origin, $N-2+a > 0$ and $f$ satisfies structural conditions, including certain growth properties. The first main result is a non-existence theorem for boundary-value problems in bounded domains star-shaped with respect to the origin, provided $f$ exhibits supercritical growth. A consequence of this is the existence of positive entire solutions to the …
The aim of this paper is to study the critical elliptic equations with Stein-Weiss type convolution parts $$ \displaystyle-Δu =\frac{1}{|x|^α}\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{2_{α, μ}^{\ast}}}{|x-y|^μ|y|^α}dy\right) |u|^{2_{α, μ}^{\ast}-2}u,~~~x\in\mathbb{R}^{N}, $$ where the critical exponent is due to the weighted Hardy-Littlewood-Sobolev inequality and Sobolev embedding. We develop a nonlocal version of concentration-compactness principle to investigate the existence of solutions and study the regularity, symmetry of positive solutions by moving plane arguments. In the second part, …
By assuming certain local energy estimates on $(1+3)$-dimensional asymptotically flat space-time, we study the existence portion of the \emph{Strauss} type wave system. Firstly we give a kind of space-time estimates which are related to the local energy norm that appeared in \cite{MR2944027}. These estimates can be used to prove a series of weighted \emph{Strichartz} and \emph{KSS} type estimates, for wave equations on asymptotically flat space-time. Then we apply the space-time estimates to obtain the lower bound of the lifespan when the nonlinear exponents $p$ and $q\ge 2$. In particular, ou…
Global theory for NLS in the weighted spaces I: finite pseudo conformal energy
Center for Applied Mathematics Tianjin University（天津大学应用数学中心）；School of Mathematical Sciences and LPMC Nankai University（南开大学数学科学学院及低维物理和材料计算重点实验室）；Tianjin University（天津大学）
In this paper we prove that all small, uniformly in time $L^1\cap H^1$ bounded solutions to KdV and related perturbations must converge to zero, as time goes to infinity, locally in an increasing-in-time region of space of order $t^{1/2}$ around any compact set in space. This set is included in the linearly dominated dispersive region $x\ll t$. Moreover, we prove this result independently of the well-known supercritical character of KdV scattering. In particular, no standing breather-like nor solitary wave structures exists in this particular regime. For the proof, we make use of well-chosen w…
