{"title":"Characteristic scales of bounded L 2 sequences","authors":"Marko Erceg, M. Lazar","doi":"10.3233/ASY-181474","DOIUrl":"https://doi.org/10.3233/ASY-181474","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"1 1","pages":"171-192"},"PeriodicalIF":0.0,"publicationDate":"2018-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89244786","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}
We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or Green's functions on lattices, with respect to powers of time variable up to an arbitrary order and estimate the remainders uniformly on the whole lattice. Unlike in the 1D case, the asymptotics contains a time independent term. The derivation of its spatial asymptotics is the technical core of the paper. Besides numerical applications, the obtained results play a crucial role in the analysis of spatio-temporal patterns for reaction-diffusion equations on lattices, in particular rattling patterns for hysteretic diffusion systems.
{"title":"Asymptotics of the heat kernels on 2D lattices","authors":"P. Gurevich","doi":"10.3233/ASY-181498","DOIUrl":"https://doi.org/10.3233/ASY-181498","url":null,"abstract":"We obtain asymptotic expansions of the spatially discrete 2D heat kernels, or Green's functions on lattices, with respect to powers of time variable up to an arbitrary order and estimate the remainders uniformly on the whole lattice. Unlike in the 1D case, the asymptotics contains a time independent term. The derivation of its spatial asymptotics is the technical core of the paper. Besides numerical applications, the obtained results play a crucial role in the analysis of spatio-temporal patterns for reaction-diffusion equations on lattices, in particular rattling patterns for hysteretic diffusion systems.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"4 1","pages":"107-124"},"PeriodicalIF":0.0,"publicationDate":"2018-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86250631","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}
{"title":"Positive ground state solutions for a class of Schrödinger-Poisson systems in R 4 involving critical Sobolev exponent","authors":"Sofiane Khoutir, Haibo Chen","doi":"10.3233/ASY-181471","DOIUrl":"https://doi.org/10.3233/ASY-181471","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"29 1","pages":"91-109"},"PeriodicalIF":0.0,"publicationDate":"2018-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72639679","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}
{"title":"Recurrence of bounded solutions to a semilinear integro-differential equation perturbed by Lévy noise","authors":"Yong-Kui Chang, G. N’Guérékata, Zhi-Han Zhao","doi":"10.3233/ASY-181466","DOIUrl":"https://doi.org/10.3233/ASY-181466","url":null,"abstract":"","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"26 1 1","pages":"27-52"},"PeriodicalIF":0.0,"publicationDate":"2018-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90866628","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}
In this paper, we deal with the topological asymptotic analysis of an optimal control problem modeled by a coupled system. The control is a geometrical object and the cost is given by the misfit between a target function and the state, solution of the Helmholtz-Laplace coupled system. Higher-order topological derivatives are used to devise a non-iterative algorithm to compute the optimal control for the problem of interest. Numerical examples are presented in order to demonstrate the effectiveness of the proposed algorithm.
{"title":"Topological asymptotic analysis of an optimal control problem modeled by a coupled system","authors":"L. Fernandez, A. Novotny, R. Prakash","doi":"10.3233/ASY-181465","DOIUrl":"https://doi.org/10.3233/ASY-181465","url":null,"abstract":"In this paper, we deal with the topological asymptotic analysis of an optimal control problem modeled by a coupled system. The control is a geometrical object and the cost is given by the misfit between a target function and the state, solution of the Helmholtz-Laplace coupled system. Higher-order topological derivatives are used to devise a non-iterative algorithm to compute the optimal control for the problem of interest. Numerical examples are presented in order to demonstrate the effectiveness of the proposed algorithm.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"26 1","pages":"1-26"},"PeriodicalIF":0.0,"publicationDate":"2018-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73613610","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}
In this report we obtain higher order asymptotic expansions of solutions to wave equations with frictional and viscoelastic damping terms. Although the diffusion phenomena are dominant, differences between the solutions we deal with and those of heat equations can be seen by comparing the second order expansions of them. In order to analyze such effects we consider the weighted L1 initial data. We also give some lower bounds which show the optimality of obtained expansions.
{"title":"Moment conditions and lower bounds in expanding solutions of wave equations with double damping terms","authors":"R. Ikehata, Hironori Michihisa","doi":"10.3233/ASY-181516","DOIUrl":"https://doi.org/10.3233/ASY-181516","url":null,"abstract":"In this report we obtain higher order asymptotic expansions of solutions to wave equations with frictional and viscoelastic damping terms. Although the diffusion phenomena are dominant, differences between the solutions we deal with and those of heat equations can be seen by comparing the second order expansions of them. In order to analyze such effects we consider the weighted L1 initial data. We also give some lower bounds which show the optimality of obtained expansions.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"14 1","pages":"19-36"},"PeriodicalIF":0.0,"publicationDate":"2018-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80814658","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}
This article is about the $mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{rm div}left( A(ncdot) cdot nabla u right)$, where $A(x)$ is a $mathbb{Z}^d$-periodic, coercive, and H"older continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $nabla_x G_n(x,y)$, $nabla_y G_n(x,y)$ and $nabla_x nabla_y G_n(x,y)$ in dimensions $d geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.
{"title":"Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients","authors":"Marc Josien","doi":"10.3233/ASY-181504","DOIUrl":"https://doi.org/10.3233/ASY-181504","url":null,"abstract":"This article is about the $mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{rm div}left( A(ncdot) cdot nabla u right)$, where $A(x)$ is a $mathbb{Z}^d$-periodic, coercive, and H\"older continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $nabla_x G_n(x,y)$, $nabla_y G_n(x,y)$ and $nabla_x nabla_y G_n(x,y)$ in dimensions $d geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"112 1","pages":"227-246"},"PeriodicalIF":0.0,"publicationDate":"2018-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82135600","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}
Nikolaos S. Papageorgiou, Vicentiu D. Rădulescu, Dušan D. Repovš
We study a nonlinear Robin problem driven by the $p$-Laplacian and with a reaction term depending on the gradient (the convection term). Using the theory of nonlinear operators of monotone-type and the asymptotic analysis of a suitable perturbation of the original equation, we show the existence of a positive smooth solution.
{"title":"Positive solutions for nonvariational Robin problems","authors":"Nikolaos S. Papageorgiou, Vicentiu D. Rădulescu, Dušan D. Repovš","doi":"10.3233/ASY-181464","DOIUrl":"https://doi.org/10.3233/ASY-181464","url":null,"abstract":"We study a nonlinear Robin problem driven by the $p$-Laplacian and with a reaction term depending on the gradient (the convection term). Using the theory of nonlinear operators of monotone-type and the asymptotic analysis of a suitable perturbation of the original equation, we show the existence of a positive smooth solution.","PeriodicalId":8603,"journal":{"name":"Asymptot. Anal.","volume":"365 1","pages":"243-255"},"PeriodicalIF":0.0,"publicationDate":"2018-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78126865","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}
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