under homogeneous Neumann boundary conditions in a smoothly bounded domain (varOmega subset mathbb {R}^{n}(nge 1)), where ( chi , xi , alpha , beta , gamma , delta , rho , a_{1},a_{2},)(b_{1},b_{2}) are positive parameters, the functions (D_{i} in C^{2}([0,infty ))) and (S_{i}in C^{2}([0,infty ))) with (S_{i}(0)=0(i=1,2)). Firstly, under certain suitable conditions, we prove that the system admits a unique globally bounded classical solution when (nle 4). Moreover, we investigate the asymptotic stability and precise convergence rates of globally bounded solutions by constructing appropriate Lyapunov functionals. Finally, we present numerical simulations that not only support our theoretical results, but also involve new and interesting phenomena.
{"title":"On a quasilinear fully parabolic predator–prey model with indirect pursuit-evasion interaction","authors":"Chuanjia Wan, Pan Zheng, Wenhai Shan","doi":"10.1007/s00028-023-00931-w","DOIUrl":"https://doi.org/10.1007/s00028-023-00931-w","url":null,"abstract":"<p>In this paper, we study the quasilinear fully parabolic predator–prey model with indirect pursuit-evasion interaction </p><span>$$begin{aligned} begin{aligned} left{ begin{aligned}&u_t=nabla cdot left( D_{1}(u)nabla uright) -chi nabla cdot left( S_{1}(u)nabla zright) +uleft( alpha v-a_{1} -b_{1}uright) ,&x in varOmega , t>0, &v_t=nabla cdot left( D_{2}(v)nabla vright) +xi nabla cdot left( S_{2}(v)nabla {w}right) +vleft( a_{2} -b_{2} v-uright) ,&x in varOmega , t>0, &{w_t}=Delta w+beta {u}-gamma {w},&x in varOmega , t>0,&{z_t}=Delta z+delta {v}-rho z,&x in varOmega , t>0, end{aligned} right. end{aligned} end{aligned}$$</span><p>under homogeneous Neumann boundary conditions in a smoothly bounded domain <span>(varOmega subset mathbb {R}^{n}(nge 1))</span>, where <span>( chi , xi , alpha , beta , gamma , delta , rho , a_{1},a_{2},)</span> <span>(b_{1},b_{2})</span> are positive parameters, the functions <span>(D_{i} in C^{2}([0,infty )))</span> and <span>(S_{i}in C^{2}([0,infty )))</span> with <span>(S_{i}(0)=0(i=1,2))</span>. Firstly, under certain suitable conditions, we prove that the system admits a unique globally bounded classical solution when <span>(nle 4)</span>. Moreover, we investigate the asymptotic stability and precise convergence rates of globally bounded solutions by constructing appropriate Lyapunov functionals. Finally, we present numerical simulations that not only support our theoretical results, but also involve new and interesting phenomena.</p>","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"77 1","pages":""},"PeriodicalIF":1.4,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138542749","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}
Pub Date : 2023-11-09DOI: 10.1007/s00028-023-00925-8
Philippe G. LeFloch, Jesús Oliver, Yoshio Tsutsumi
{"title":"Boundedness of the conformal hyperboloidal energy for a wave-Klein–Gordon model","authors":"Philippe G. LeFloch, Jesús Oliver, Yoshio Tsutsumi","doi":"10.1007/s00028-023-00925-8","DOIUrl":"https://doi.org/10.1007/s00028-023-00925-8","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":" 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241265","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}
Pub Date : 2023-11-09DOI: 10.1007/s00028-023-00926-7
Florian Bechtold, Jörn Wichmann
Abstract We study an evolutionary p -Laplace problem whose potential is subject to a translation in time. Provided the trajectory along which the potential is translated admits a sufficiently regular local time, we establish existence of solutions to the problem for singular potentials for which a priori bounds in classical approaches break down, thereby establishing a pathwise regularization by noise phenomena for this nonlinear problem.
{"title":"A pathwise regularization by noise phenomenon for the evolutionary p-Laplace equation","authors":"Florian Bechtold, Jörn Wichmann","doi":"10.1007/s00028-023-00926-7","DOIUrl":"https://doi.org/10.1007/s00028-023-00926-7","url":null,"abstract":"Abstract We study an evolutionary p -Laplace problem whose potential is subject to a translation in time. Provided the trajectory along which the potential is translated admits a sufficiently regular local time, we establish existence of solutions to the problem for singular potentials for which a priori bounds in classical approaches break down, thereby establishing a pathwise regularization by noise phenomena for this nonlinear problem.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":" 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135192783","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}
Pub Date : 2023-11-06DOI: 10.1007/s00028-023-00924-9
Abderrahim Charkaoui, Nour Eddine Alaa
{"title":"An $$L^1$$-theory for a nonlinear temporal periodic problem involving p(x)-growth structure with a strong dependence on gradients","authors":"Abderrahim Charkaoui, Nour Eddine Alaa","doi":"10.1007/s00028-023-00924-9","DOIUrl":"https://doi.org/10.1007/s00028-023-00924-9","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135635034","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}
Pub Date : 2023-11-06DOI: 10.1007/s00028-023-00920-z
Giovanni Franzina, Bruno Volzone
Abstract Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.
摘要采用Brasco (Adv Math 394:108029, 2022)的方法，研究了光滑边界开有界集上有符号分数阶多孔介质方程的长时间行为。考虑齐次外部狄利克雷边界条件。我们证明了如果初始基准具有足够小的能量，那么一旦适当地重新标度，解收敛于次线性分数阶Lane-Emden方程的非平凡常符号解。在此基础上给出了一个非局部的充分能判据，这对于确定极限轮廓的正解或负解具有重要意义。
{"title":"Large time behavior of signed fractional porous media equations on bounded domains","authors":"Giovanni Franzina, Bruno Volzone","doi":"10.1007/s00028-023-00920-z","DOIUrl":"https://doi.org/10.1007/s00028-023-00920-z","url":null,"abstract":"Abstract Following the methodology of Brasco (Adv Math 394:108029, 2022), we study the long-time behavior for the signed fractional porous medium equation in open bounded sets with smooth boundary. Homogeneous exterior Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution, once suitably rescaled, converges to a nontrivial constant sign solution of a sublinear fractional Lane–Emden equation. Furthermore, we give a nonlocal sufficient energetic criterion on the initial datum, which is important to identify the exact limit profile, namely the positive solution or the negative one.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135635714","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}
Pub Date : 2023-10-28DOI: 10.1007/s00028-023-00923-w
Ruben Jakob
{"title":"The Willmore flow of Hopf-tori in the 3-sphere","authors":"Ruben Jakob","doi":"10.1007/s00028-023-00923-w","DOIUrl":"https://doi.org/10.1007/s00028-023-00923-w","url":null,"abstract":"","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"140 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136157601","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}
Pub Date : 2023-10-26DOI: 10.1007/s00028-023-00916-9
Stefano Pagliarani, Giacomo Lucertini, Andrea Pascucci
Abstract We consider a class of degenerate equations in non-divergence form satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin diffusions.
{"title":"Optimal regularity for degenerate Kolmogorov equations in non-divergence form with rough-in-time coefficients","authors":"Stefano Pagliarani, Giacomo Lucertini, Andrea Pascucci","doi":"10.1007/s00028-023-00916-9","DOIUrl":"https://doi.org/10.1007/s00028-023-00916-9","url":null,"abstract":"Abstract We consider a class of degenerate equations in non-divergence form satisfying a parabolic Hörmander condition, with coefficients that are measurable in time and Hölder continuous in the space variables. By utilizing a generalized notion of strong solution, we establish the existence of a fundamental solution and its optimal Hölder regularity, as well as Gaussian estimates. These results are key to study the backward Kolmogorov equations associated to a class of Langevin diffusions.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"75 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136381954","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}
Pub Date : 2023-10-26DOI: 10.1007/s00028-023-00922-x
Torebek, Berikbol T.
This paper studies the large-time behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. This equation is a natural extension of the heat equations with combined nonlinearities considered by Jleli-Samet-Souplet (Proc AMS, 2020). Firstly, we focus on an interesting phenomenon of discontinuity of the critical exponents. In particular, we will fill the gap in the results of Jleli-Samet-Souplet for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define another critical exponent depending on the forcing term.
{"title":"Critical exponents for the p-Laplace heat equations with combined nonlinearities","authors":"Torebek, Berikbol T.","doi":"10.1007/s00028-023-00922-x","DOIUrl":"https://doi.org/10.1007/s00028-023-00922-x","url":null,"abstract":"This paper studies the large-time behavior of solutions to the quasilinear inhomogeneous parabolic equation with combined nonlinearities. This equation is a natural extension of the heat equations with combined nonlinearities considered by Jleli-Samet-Souplet (Proc AMS, 2020). Firstly, we focus on an interesting phenomenon of discontinuity of the critical exponents. In particular, we will fill the gap in the results of Jleli-Samet-Souplet for the critical case. We are also interested in the influence of the forcing term on the critical behavior of the considered problem, so we will define another critical exponent depending on the forcing term.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136377036","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}
Pub Date : 2023-10-24DOI: 10.1007/s00028-023-00914-x
Ophélie Cuvillier, Francesco Fanelli, Elena Salguero
In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain $$mathbb {T}^d$$ , for space dimensions $$d=2,3$$ . We admit the average turbulent kinetic energy k to vanish in part of the domain, i.e. we consider the case $$k ge 0$$ ; in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces $$H^s$$ , for any $$s>1+d/2$$ . We expect this regularity to be optimal, due to the degeneracy of the system when $$k approx 0$$ . We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the nonlinear terms involved in the computations.
本文研究了周期域$$mathbb {T}^d$$中空间维为$$d=2,3$$的Kolmogorov湍流双方程模型的适定性。我们承认平均湍流动能k在部分区域内消失，即我们考虑$$k ge 0$$;在这种情况下，方程的抛物线结构变得简并。对于该系统，我们证明了Sobolev空间$$H^s$$中对于任意$$s>1+d/2$$的局部适定性结果。我们期望这个规则是最优的，因为当$$k approx 0$$。我们还证明了一个延拓准则，并给出了解的寿命的下界。结果的证明是基于Littlewood-Paley分析和环面上的准微分演算，以及计算中涉及的非线性项的精确换易子分解。
{"title":"Well-posedness of the Kolmogorov two-equation model of turbulence in optimal Sobolev spaces","authors":"Ophélie Cuvillier, Francesco Fanelli, Elena Salguero","doi":"10.1007/s00028-023-00914-x","DOIUrl":"https://doi.org/10.1007/s00028-023-00914-x","url":null,"abstract":"In this paper, we study the well-posedness of the Kolmogorov two-equation model of turbulence in a periodic domain $$mathbb {T}^d$$ , for space dimensions $$d=2,3$$ . We admit the average turbulent kinetic energy k to vanish in part of the domain, i.e. we consider the case $$k ge 0$$ ; in this situation, the parabolic structure of the equations becomes degenerate. For this system, we prove a local well-posedness result in Sobolev spaces $$H^s$$ , for any $$s>1+d/2$$ . We expect this regularity to be optimal, due to the degeneracy of the system when $$k approx 0$$ . We also prove a continuation criterion and provide a lower bound for the lifespan of the solutions. The proof of the results is based on Littlewood-Paley analysis and paradifferential calculus on the torus, together with a precise commutator decomposition of the nonlinear terms involved in the computations.","PeriodicalId":51083,"journal":{"name":"Journal of Evolution Equations","volume":"106 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135220046","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}
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