arXiv: Analysis of PDEs最新文献_第5页

Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system 奇异Gierer-Meinhardt系统阴影极限模型的扩散诱导爆破解

arXiv: Analysis of PDEs Pub Date : 2020-10-19 DOI: 10.1142/s0218202521500305

G. K. Duong, N. Kavallaris, H. Zaag

{"title":"Diffusion-induced blowup solutions for the shadow limit model of a singular Gierer–Meinhardt system","authors":"G. K. Duong, N. Kavallaris, H. Zaag","doi":"10.1142/s0218202521500305","DOIUrl":"https://doi.org/10.1142/s0218202521500305","url":null,"abstract":"In the current paper, we provide a thorough investigation of the blowing up behaviour induced via diffusion of the solution of the following non local problem begin{equation*} left{begin{array}{rcl} partial_t u &=& Delta u - u + displaystyle{frac{u^p}{ left(mathop{,rlap{-}!!int}nolimits_Omega u^r dr right)^gamma }}quadtext{in}quad Omega times (0,T), [0.2cm] frac{ partial u}{ partial nu} & = & 0 text{ on } Gamma = partial Omega times (0,T), u(0) & = & u_0, end{array} right. end{equation*} where $Omega$ is a bounded domain in $mathbb{R}^N$ with smooth boundary $partial Omega;$ such problem is derived as the shadow limit of a singular Gierer-Meinhardt system, cf. cite{KSN17, NKMI2018}. Under the Turing type condition $$ frac{r}{p-1} < frac{N}{2}, gamma r ne p-1, $$ we construct a solution which blows up in finite time and only at an interior point $x_0$ of $Omega,$ i.e. $$ u(x_0, t) sim (theta^*)^{-frac{1}{p-1}} left[kappa (T-t)^{-frac{1}{p-1}} right], $$ where $$ theta^* := lim_{t to T} left(mathop{,rlap{-}!!int}nolimits_Omega u^r dr right)^{- gamma} text{ and } kappa = (p-1)^{-frac{1}{p-1}}. $$ More precisely, we also give a description on the final asymptotic profile at the blowup point $$ u(x,T) sim ( theta^* )^{-frac{1}{p-1}} left[ frac{(p-1)^2}{8p} frac{|x-x_0|^2}{ |ln|x-x_0||} right]^{ -frac{1}{p-1}} text{ as } x to 0, $$ and thus we unveil the form of the Turing patterns occurring in that case due to driven-diffusion instability. The applied technique for the construction of the preceding blowing up solution mainly relies on the approach developed in cite{MZnon97} and cite{DZM3AS19}.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91130707","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 6

Smooth traveling-wave solutions to the inviscid surface quasi-geostrophic equations 无粘表面准地转方程的光滑行波解

arXiv: Analysis of PDEs Pub Date : 2020-10-18 DOI: 10.5802/CRMATH.159

Ludovic Godard-Cadillac

引用次数: 15

$L^p$ estimates for wave equations with specific $C^{0,1}$ coefficients 具有特定系数$C^{0,1}$的波动方程的$L^p$估计

arXiv: Analysis of PDEs Pub Date : 2020-10-16 DOI: 10.5445/IR/1000124653

D. Frey, Pierre Portal

引用次数: 9

Hölder regularity of Hamilton-Jacobi equations with stochastic forcing Hölder随机强迫下Hamilton-Jacobi方程的正则性

arXiv: Analysis of PDEs Pub Date : 2020-10-13 DOI: 10.1090/TRAN/8435

P. Cardaliaguet, B. Seeger

引用次数: 1

Large $|k|$ behavior of d-bar problems for domains with a smooth boundary 光滑边界域上d-bar问题的大$|k|$行为

arXiv: Analysis of PDEs Pub Date : 2020-10-09 DOI: 10.4171/ecr/18-1/15

C. Klein, J. Sjostrand, N. Stoilov

引用次数: 1

A blow-up result for the wave equation with localized initial data: the scale-invariant damping and mass term with combined nonlinearities 具有局部初始数据的波动方程的一个爆破结果:结合非线性的标度不变阻尼和质量项

arXiv: Analysis of PDEs Pub Date : 2020-10-08 DOI: 10.22541/au.160395665.59674549/v1

M. Hamouda, M. Hamza

{"title":"A blow-up result for the wave equation with localized initial data: the scale-invariant damping and mass term with combined nonlinearities","authors":"M. Hamouda, M. Hamza","doi":"10.22541/au.160395665.59674549/v1","DOIUrl":"https://doi.org/10.22541/au.160395665.59674549/v1","url":null,"abstract":"We are interested in this article in studying the damped wave equation with localized initial data, in the textit{scale-invariant case} with mass term and two combined nonlinearities. More precisely, we consider the following equation: $$ (E) {1cm} u_{tt}-Delta u+frac{mu}{1+t}u_t+frac{nu^2}{(1+t)^2}u=|u_t|^p+|u|^q, quad mbox{in} mathbb{R}^Ntimes[0,infty), $$ with small initial data. Under some assumptions on the mass and damping coefficients, $nu$ and $mu>0$, respectively, we show that blow-up region and the lifespan bound of the solution of $(E)$ remain the same as the ones obtained in cite{Our2} in the case of a mass-free wave equation, it i.e. $(E)$ with $nu=0$. \u0000Furthermore, using in part the computations done for $(E)$, we enhance the result in cite{Palmieri} on the Glassey conjecture for the solution of $(E)$ with omitting the nonlinear term $|u|^q$. Indeed, the blow-up region is extended from $p in (1, p_G(N+sigma)]$, where $sigma$ is given by (1.12) below, to $p in (1, p_G(N+mu)]$ yielding, hence, a better estimate of the lifespan when $(mu-1)^2-4nu^2<1$. Otherwise, the two results coincide. Finally, we may conclude that the mass term {it has no influence} on the dynamics of $(E)$ (resp. $(E)$ without the nonlinear term $|u|^q$), and the conjecture we made in cite{Our2} on the threshold between the blow-up and the global existence regions obtained holds true here.","PeriodicalId":8445,"journal":{"name":"arXiv: Analysis of PDEs","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79092784","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 8

On Korn-Maxwell-Sobolev Inequalities 关于科恩-麦克斯韦-索博列夫不等式

arXiv: Analysis of PDEs Pub Date : 2020-10-07 DOI: 10.1016/J.JMAA.2021.125226

F. Gmeineder, Daniel Spector

引用次数: 10

Some non-homogeneous Gagliardo–Nirenberg inequalities and application to a biharmonic non-linear Schrödinger equation 一些非齐次Gagliardo-Nirenberg不等式及其在双调和非线性Schrödinger方程中的应用

arXiv: Analysis of PDEs Pub Date : 2020-10-04 DOI: 10.5445/IR/1000124276

Antonio J. Fern'andez, L. Jeanjean, Rainer Mandel, M. Mariş

引用次数: 5

(Dis)connectedness of nonlocal minimal surfaces in a cylinder and a stickiness property (1)圆柱非局部最小曲面的非连通性及粘滞性

arXiv: Analysis of PDEs Pub Date : 2020-10-02 DOI: 10.1090/proc/15796

S. Dipierro, F. Onoue, E. Valdinoci