Journal of Evolution Equations最新文献_第4页

Fujita exponent for non-local parabolic equation involving the Hardy–Leray potential 涉及哈代-勒雷势的非局部抛物方程的藤田指数

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-06-22 DOI: 10.1007/s00028-024-00984-5

Boumediene Abdellaoui, Giovanni Siclari, Ana Primo

{"title":"Fujita exponent for non-local parabolic equation involving the Hardy–Leray potential","authors":"Boumediene Abdellaoui, Giovanni Siclari, Ana Primo","doi":"10.1007/s00028-024-00984-5","DOIUrl":"https://doi.org/10.1007/s00028-024-00984-5","url":null,"abstract":"In this paper, we analyse the existence and non-existence of non-negative solutions to a non-local parabolic equation with a Hardy–Leray-type potential. More precisely, we consider the problem $$begin{aligned} {left{ begin{array}{ll} (w_t-Delta w)^s=frac{lambda }{|x|^{2s}} w+w^p +f, &{}quad text {in }mathbb {R}^Ntimes (0,+infty ), w(x,t)=0, &{}quad text {in }mathbb {R}^Ntimes (-infty ,0], end{array}right. } end{aligned}$$where (N> 2s), (0<s<1) and (0<lambda <Lambda _{N,s}), the optimal constant in the fractional Hardy–Leray inequality. In particular, we show the existence of a critical existence exponent (p_{+}(lambda , s)) and of a Fujita-type exponent (F(lambda ,s)) such that the following holds:<ul>\u0000<li>\u0000Let (p>p_+(lambda ,s)). Then there are not any non-negative supersolutions.\u0000</li>\u0000<li>\u0000Let (p<p_+(lambda ,s)). Then there exist local solutions, while concerning global solutions we need to distinguish two cases:<ul>\u0000<li>\u0000Let ( 1< ple F(lambda ,s)). Here we show that a weighted norm of any positive solution blows up in finite time.\u0000</li>\u0000<li>\u0000Let (F(lambda ,s)<p<p_+(lambda ,s)). Here we prove the existence of global solutions under suitable hypotheses.\u0000</li>\u0000</ul>\u0000</li>\u0000</ul>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"23 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503818","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On $$L^2$$ decay of weak solutions of several incompressible fluid models 论若干不可压缩流体模型弱解的 $$L^2$$ 衰减

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-06-22 DOI: 10.1007/s00028-024-00985-4

Huan Yu

引用次数: 0

Vertical maximal functions on manifolds with ends 有端流形上的垂直最大函数

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-06-09 DOI: 10.1007/s00028-024-00981-8

Himani Sharma, Adam Sikora

引用次数: 0

Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order 海森堡群上的半线性阻尼波方程与负阶索博列夫空间的初始数据

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-06-09 DOI: 10.1007/s00028-024-00976-5

Aparajita Dasgupta, Vishvesh Kumar, Shyam Swarup Mondal, Michael Ruzhansky

{"title":"Semilinear damped wave equations on the Heisenberg group with initial data from Sobolev spaces of negative order","authors":"Aparajita Dasgupta, Vishvesh Kumar, Shyam Swarup Mondal, Michael Ruzhansky","doi":"10.1007/s00028-024-00976-5","DOIUrl":"https://doi.org/10.1007/s00028-024-00976-5","url":null,"abstract":"In this paper, we focus on studying the Cauchy problem for semilinear damped wave equations involving the sub-Laplacian (mathcal {L}) on the Heisenberg group (mathbb {H}^n) with power type nonlinearity (|u|^p) and initial data taken from Sobolev spaces of negative order homogeneous Sobolev space (dot{H}^{-gamma }_{mathcal {L}}(mathbb {H}^n), gamma >0), on (mathbb {H}^n). In particular, in the framework of Sobolev spaces of negative order, we prove that the critical exponent is the exponent (p_{text {crit}}(Q, gamma )=1+frac{4}{Q+2gamma },) for (gamma in (0, frac{Q}{2})), where (Q:=2n+2) is the homogeneous dimension of (mathbb {H}^n). More precisely, we establish<ul>\u0000<li>\u0000A global-in-time existence of small data Sobolev solutions of lower regularity for (p>p_{text {crit}}(Q, gamma )) in the energy evolution space;\u0000</li>\u0000<li>\u0000A finite time blow-up of weak solutions for (1<p<p_{text {crit}}(Q, gamma )) under certain conditions on the initial data by using the test function method.\u0000</li>\u0000</ul> Furthermore, to precisely characterize the blow-up time, we derive sharp upper bound and lower bound estimates for the lifespan in the subcritical case.\u0000","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"23 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-06-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141503820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

On the second boundary value problem for a class of fully nonlinear flow III 关于一类全非线性流动的第二边界值问题 III

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-06-09 DOI: 10.1007/s00028-024-00983-6

Chong Wang, Rongli Huang, Jiguang Bao

引用次数: 0

Dispersive decay bound of small data solutions to Kawahara equation in a finite time scale 有限时间尺度内川原方程小数据解的分散衰减约束

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-05-24 DOI: 10.1007/s00028-024-00980-9

Jongwon Lee

引用次数: 0

Evolutionary equations are G-compact 进化方程是 G-紧凑的

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-05-11 DOI: 10.1007/s00028-024-00971-w

Krešimir Burazin, Marko Erceg, Marcus Waurick

引用次数: 0

Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights 具有非均质权重的半线性热方程全局实时解的藤田指数

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-05-11 DOI: 10.1007/s00028-024-00969-4

Tatsuki Kawakami, Yannick Sire, Jiayi Nikki Wang

{"title":"Fujita exponent for the global-in-time solutions to a semilinear heat equation with non-homogeneous weights","authors":"Tatsuki Kawakami, Yannick Sire, Jiayi Nikki Wang","doi":"10.1007/s00028-024-00969-4","DOIUrl":"https://doi.org/10.1007/s00028-024-00969-4","url":null,"abstract":"We consider a non-homogeneous parabolic equation with degenerate coefficients of the form (u_t-L_{omega } u=u^p), where (L_{omega }=omega ^{-1}mathrm div(omega nabla )). This paper establishes the existence/non-existence of global-in-time mild solutions based on a critical exponent, known as the Fujita exponent. Similar topics for a semilinear heat equation with degenerate coefficients are treated in Fujishima (Calc Var Partial Differ Equ 58:25, 2019). They considered an equation (u_t-textrm{div}(omega nabla u) =u^p), which is not self-adjoint, with two types of homogeneous weights: (omega (x) = |x_1|^a) and (omega (x) = |x|^b) where (a,b>0). In this paper we consider the case of a self-adjoint operator, and extend to more general weights that meet certain restrictions such as being in the Muckenhoupt class (A_2), non-decreasing, and where the limits (alpha :=lim _{|x'|rightarrow infty }(log omega (x))/(log |x'|)) and (beta :=lim _{|x'|rightarrow 0}(log omega (x))/(log |x'|)) exist, where (x' = (x_1, dots , x_n)) and (1le nle N). The main result establishes that the Fujita exponent is given by (p_F = 1+2/(N+alpha )). This means that the asymptotic behavior of the weight at infinity affects global existence of solutions and the one at the origin does not.\u0000","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"40 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935506","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Temporal approximation of stochastic evolution equations with irregular nonlinearities 具有不规则非线性的随机演化方程的时间逼近

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-05-09 DOI: 10.1007/s00028-024-00975-6

Katharina Klioba, Mark Veraar

{"title":"Temporal approximation of stochastic evolution equations with irregular nonlinearities","authors":"Katharina Klioba, Mark Veraar","doi":"10.1007/s00028-024-00975-6","DOIUrl":"https://doi.org/10.1007/s00028-024-00975-6","url":null,"abstract":"In this paper, we prove convergence for contractive time discretisation schemes for semi-linear stochastic evolution equations with irregular Lipschitz nonlinearities, initial values, and additive or multiplicative Gaussian noise on 2-smooth Banach spaces X. The leading operator A is assumed to generate a strongly continuous semigroup S on X, and the focus is on non-parabolic problems. The main result concerns convergence of the uniform strong error$$begin{aligned} textrm{E}_{k}^{infty } {:}{=}Big (mathbb {E}sup _{jin {0, ldots , N_k}} Vert U(t_j) - U^jVert _X^pBig )^{1/p} rightarrow 0quad (k rightarrow 0), end{aligned}$$where (p in [2,infty )), U is the mild solution, (U^j) is obtained from a time discretisation scheme, k is the step size, and (N_k = T/k) for final time (T>0). This generalises previous results to a larger class of admissible nonlinearities and noise, as well as rough initial data from the Hilbert space case to more general spaces. We present a proof based on a regularisation argument. Within this scope, we extend previous quantified convergence results for more regular nonlinearity and noise from Hilbert to 2-smooth Banach spaces. The uniform strong error cannot be estimated in terms of the simpler pointwise strong error$$begin{aligned} textrm{E}_k {:}{=}bigg (sup _{jin {0,ldots ,N_k}}mathbb {E}Vert U(t_j) - U^{j}Vert _X^pbigg )^{1/p}, end{aligned}$$which most of the existing literature is concerned with. Our results are illustrated for a variant of the Schrödinger equation, for which previous convergence results were not applicable.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"24 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equations mKdV 方程和 mKdV 型方程组解析半径的新下限

IF 1.4 3区数学

Journal of Evolution Equations Pub Date : 2024-05-07 DOI: 10.1007/s00028-024-00977-4

Renata O. Figueira, Mahendra Panthee

引用次数: 0