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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
{"title":"Evolutionary equations are G-compact","authors":"Krešimir Burazin, Marko Erceg, Marcus Waurick","doi":"10.1007/s00028-024-00971-w","DOIUrl":"https://doi.org/10.1007/s00028-024-00971-w","url":null,"abstract":"<p>We prove a compactness result related to <i>G</i>-convergence for autonomous evolutionary equations in the sense of Picard. Compared to previous work related to applications, we do not require any boundedness or regularity of the underlying spatial domain; nor do we assume any periodicity or ergodicity assumption on the potentially oscillatory part. In terms of abstract evolutionary equations, we remove any compactness assumptions of the resolvent modulo kernel of the spatial operator. To achieve the results, we introduced a slightly more general class of material laws. As a by-product, we also provide a criterion for <i>G</i>-convergence for time-dependent equations solely in terms of static equations.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"31 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935508","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
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":"<p>We consider a non-homogeneous parabolic equation with degenerate coefficients of the form <span>(u_t-L_{omega } u=u^p)</span>, where <span>(L_{omega }=omega ^{-1}mathrm div(omega nabla ))</span>. 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 <span>(u_t-textrm{div}(omega nabla u) =u^p)</span>, which is not self-adjoint, with two types of homogeneous weights: <span>(omega (x) = |x_1|^a)</span> and <span>(omega (x) = |x|^b)</span> where <span>(a,b&gt;0)</span>. 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 <span>(A_2)</span>, non-decreasing, and where the limits <span>(alpha :=lim _{|x'|rightarrow infty }(log omega (x))/(log |x'|))</span> and <span>(beta :=lim _{|x'|rightarrow 0}(log omega (x))/(log |x'|))</span> exist, where <span>(x' = (x_1, dots , x_n))</span> and <span>(1le nle N)</span>. The main result establishes that the Fujita exponent is given by <span>(p_F = 1+2/(N+alpha ))</span>. 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</p>","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":"<p>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 <i>X</i>. The leading operator <i>A</i> is assumed to generate a strongly continuous semigroup <i>S</i> on <i>X</i>, and the focus is on non-parabolic problems. The main result concerns convergence of the <i>uniform strong error</i></p><span>$$begin{aligned} textrm{E}_{k}^{infty } {:}{=}Big (mathbb {E}sup _{jin {0, ldots , N_k}} Vert U(t_j) - U^jVert _X^pBig )^{1/p} rightarrow 0quad (k rightarrow 0), end{aligned}$$</span><p>where <span>(p in [2,infty ))</span>, <i>U</i> is the mild solution, <span>(U^j)</span> is obtained from a time discretisation scheme, <i>k</i> is the step size, and <span>(N_k = T/k)</span> for final time <span>(T&gt;0)</span>. 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 <i>pointwise strong error</i></p><span>$$begin{aligned} textrm{E}_k {:}{=}bigg (sup _{jin {0,ldots ,N_k}}mathbb {E}Vert U(t_j) - U^{j}Vert _X^pbigg )^{1/p}, end{aligned}$$</span><p>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.</p>","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
{"title":"New lower bounds for the radius of analyticity for the mKdV equation and a system of mKdV-type equations","authors":"Renata O. Figueira, Mahendra Panthee","doi":"10.1007/s00028-024-00977-4","DOIUrl":"https://doi.org/10.1007/s00028-024-00977-4","url":null,"abstract":"<p>This paper is devoted to obtaining new lower bounds to the radius of spatial analyticity for the solutions of modified Korteweg–de Vries (mKdV) equation and a coupled system of mKdV-type equations, starting with real analytic initial data with a fixed radius of analyticity <span>(sigma _0)</span>. Specifically, we derive almost conserved quantities to prove that the local solution can be extended to a time interval [0, <i>T</i>] for any large <span>(T&gt;0)</span> in such a way that the radius of analyticity <span>(sigma (T))</span> decays no faster than <span>(cT^{-1})</span> for both the equations, where <i>c</i> is a positive constant. The results of this paper improve the ones obtained in Figueira and Panthee (Decay of the radius of spatial analyticity for the modified KdV equation and the nonlinear Schrödinger equation with third order dispersion, to appear in NoDEA, arXiv:2307.09096) and Figueira and Himonas (J Math Anal Appl 497(2):124917, 2021), respectively, for the mKdV equation and a mKdV-type system.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"109 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140935612","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
Three evolution problems modeling the interaction between acoustic waves and non-locally reacting surfaces 声波与非局部反应表面相互作用模型的三个演化问题
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-05-04 DOI: 10.1007/s00028-024-00974-7
Enzo Vitillaro
{"title":"Three evolution problems modeling the interaction between acoustic waves and non-locally reacting surfaces","authors":"Enzo Vitillaro","doi":"10.1007/s00028-024-00974-7","DOIUrl":"https://doi.org/10.1007/s00028-024-00974-7","url":null,"abstract":"<p>The paper deals with three evolution problems arising in the physical modeling of small amplitude acoustic phenomena occurring in a fluid, bounded by a surface of extended reaction. The first one is the widely studied wave equation with acoustic boundary conditions, but its derivation from the physical model is mathematically not fully satisfactory. The other two models studied in the paper, in the Lagrangian and Eulerian settings, are physically transparent. In the paper the first model is derived from the other two in a rigorous way, also for solutions merely belonging to the natural energy spaces.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"15 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140887489","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
Second-order sufficient conditions in optimal control of evolution systems 进化系统优化控制中的二阶充分条件
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-04-27 DOI: 10.1007/s00028-024-00968-5
H. Frankowska, E. M. Marchini, M. Mazzola
{"title":"Second-order sufficient conditions in optimal control of evolution systems","authors":"H. Frankowska, E. M. Marchini, M. Mazzola","doi":"10.1007/s00028-024-00968-5","DOIUrl":"https://doi.org/10.1007/s00028-024-00968-5","url":null,"abstract":"<p>This paper concerns second-order sufficient conditions of optimality for a class of infinite dimensional optimal control problems under control constraints and end-point constraints. The distinctive feature of our formulation is the use of variational analysis techniques. Our approach is based on a suitable decomposition of controls, using second-order tangents, and it is developed in the case of functional constraints. An adaptation of the tools in the case of pointwise constraints on the controls is proposed too. We provide applications to concrete optimal control problems involving PDEs.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"175 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140808988","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
Solvability of the Cauchy problem for fractional semilinear parabolic equations in critical and doubly critical cases 临界和双临界情况下分数半线性抛物方程考希问题的可解性
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-04-26 DOI: 10.1007/s00028-024-00967-6
Yasuhito Miyamoto, Masamitsu Suzuki
{"title":"Solvability of the Cauchy problem for fractional semilinear parabolic equations in critical and doubly critical cases","authors":"Yasuhito Miyamoto, Masamitsu Suzuki","doi":"10.1007/s00028-024-00967-6","DOIUrl":"https://doi.org/10.1007/s00028-024-00967-6","url":null,"abstract":"<p>Let <span>(0&lt;theta le 2)</span>, <span>(Nge 1)</span> and <span>(T&gt;0)</span>. We are concerned with the Cauchy problem for the fractional semilinear parabolic equation </p><span>$$begin{aligned} {left{ begin{array}{ll} partial _t u+(-Delta )^{theta /2}u=f(u) &amp;{} text {in} {{mathbb {R}}^N}times (0,T), u(x,0)=u_0 (x)ge 0 &amp;{} text {in} {{mathbb {R}}^N}. end{array}right. } end{aligned}$$</span><p>Here, <span>(fin C[0,infty ))</span> denotes a rather general growing nonlinearity and <span>(u_0)</span> may be unbounded. We study local in time solvability in the so-called critical and doubly critical cases. In particular, when <span>(f(u)=u^{1+{theta }/{N}}left[ log (u+e)right] ^{a})</span>, we obtain a sharp integrability condition on <span>(u_0)</span> which explicitly determines local in time existence/nonexistence of a nonnegative solution.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"69 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806112","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
Strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii type: the time-dependent case 具有 Nemytskii 型系数的 McKean-Vlasov SDEs 的强解：随时间变化的情况
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-04-22 DOI: 10.1007/s00028-024-00970-x
Sebastian Grube
{"title":"Strong solutions to McKean–Vlasov SDEs with coefficients of Nemytskii type: the time-dependent case","authors":"Sebastian Grube","doi":"10.1007/s00028-024-00970-x","DOIUrl":"https://doi.org/10.1007/s00028-024-00970-x","url":null,"abstract":"<p>We consider a large class of nonlinear FPKEs with coefficients of Nemytskii type depending <i>explicitly</i> on time and space, for which it is known that there exists a sufficiently Sobolev-regular Schwartz-distributional solution <span>(uin L^1cap L^infty )</span>. We show that there exists a unique strong solution to the associated McKean–Vlasov SDE with time marginal law densities <i>u</i>. In particular, every weak solution of this equation with time marginal law densities <i>u</i> can be written as a functional of the driving Brownian motion. Moreover, plugging any Brownian motion into this very functional produces a weak solution with time marginal law densities <i>u</i>.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"50 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805879","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
Propagation of anisotropic Gabor singularities for Schrödinger type equations 薛定谔方程的各向异性 Gabor 奇点传播
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-04-02 DOI: 10.1007/s00028-024-00963-w
{"title":"Propagation of anisotropic Gabor singularities for Schrödinger type equations","authors":"","doi":"10.1007/s00028-024-00963-w","DOIUrl":"https://doi.org/10.1007/s00028-024-00963-w","url":null,"abstract":"<h3>Abstract</h3> <p>We show results on propagation of anisotropic Gabor wave front sets for solutions to a class of evolution equations of Schrödinger type. The Hamiltonian is assumed to have a real-valued principal symbol with the anisotropic homogeneity <span> <span>(a(lambda x, lambda ^sigma xi ) = lambda ^{1+sigma } a(x,xi ))</span> </span> for <span> <span>(lambda &gt; 0)</span> </span> where <span> <span>(sigma &gt; 0)</span> </span> is a rational anisotropy parameter. We prove that the propagator is continuous on anisotropic Shubin–Sobolev spaces. The main result says that the propagation of the anisotropic Gabor wave front set follows the Hamilton flow of the principal symbol.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"298 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592728","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
Large deviation principle for multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motions 分数布朗运动驱动的多尺度分布依赖性随机微分方程的大偏差原理
IF 1.4 3区 数学
Journal of Evolution Equations Pub Date : 2024-04-02 DOI: 10.1007/s00028-024-00960-z
Guangjun Shen, Huan Zhou, Jiang-Lun Wu
{"title":"Large deviation principle for multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motions","authors":"Guangjun Shen, Huan Zhou, Jiang-Lun Wu","doi":"10.1007/s00028-024-00960-z","DOIUrl":"https://doi.org/10.1007/s00028-024-00960-z","url":null,"abstract":"<p>In this paper, we are concerned with multi-scale distribution-dependent stochastic differential equations driven by fractional Brownian motion (with Hurst index <span>(H&gt;frac{1}{2})</span>) and standard Brownian motion, simultaneously. Our aim is to establish a large deviation principle for the multi-scale distribution-dependent stochastic differential equations. This is done via the weak convergence approach and our proof is based heavily on the fractional calculus.\u0000</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"16 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2024-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140592195","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
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