In this paper we consider a compact Riemannian manifold (M, g) of class C 1 $cap$ W 2,$infty$ and the damped wave or Schr"odinger equations on M , under the action of a damping function a = a(x). We establish the following fact: if the measure of the set {x $in$ M ; a(x) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.
{"title":"A remark on the logarithmic decay of the damped wave and Schrödinger equations on a compact Riemannian manifold","authors":"Iván Moyano, N. Burq","doi":"10.4171/pm/2107","DOIUrl":"https://doi.org/10.4171/pm/2107","url":null,"abstract":"In this paper we consider a compact Riemannian manifold (M, g) of class C 1 $cap$ W 2,$infty$ and the damped wave or Schr\"odinger equations on M , under the action of a damping function a = a(x). We establish the following fact: if the measure of the set {x $in$ M ; a(x) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44641899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Generalized fractional integral operators on variable exponent Morrey type spaces over metric measure spaces","authors":"T. Ohno, T. Shimomura","doi":"10.4171/pm/2092","DOIUrl":"https://doi.org/10.4171/pm/2092","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45347641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A note on the real support of Silva–Hasumi–Morimoto ultrahyperfunctions","authors":"M. Alves, D. Franco","doi":"10.4171/pm/2091","DOIUrl":"https://doi.org/10.4171/pm/2091","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41982921","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"The solution of a type of absolute value equations using two new matrix splitting iterative techniques","authors":"Rashid Ali, K. Pan","doi":"10.4171/pm/2089","DOIUrl":"https://doi.org/10.4171/pm/2089","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43219421","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Uniform regularity for the flow of a chemically reacting gaseous mixture","authors":"Jianzhu Sun, T. Tang","doi":"10.4171/pm/2088","DOIUrl":"https://doi.org/10.4171/pm/2088","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70896899","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the optional and orthogonal decompositions of a class of semimartingales","authors":"A. Berkaoui","doi":"10.4171/pm/2083","DOIUrl":"https://doi.org/10.4171/pm/2083","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45150550","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A positive cosmological constant $Lambda>0$ sets an upper limit for the area of marginally future-trapped surfaces enclosing a black hole (BH). Does this mean that the mass of the BH cannot increase beyond the corresponding limit? I analyze some simple spherically symmetric models where regions within a dynamical horizon keep gaining mass-energy so that eventually the $Lambda$ limit is surpassed. This shows that the black hole proper transmutes into a collapsing universe, and no observers will ever reach infinity, which dematerializes together with the event horizon and the `cosmological horizon'. The region containing the dynamical horizon cannot be causally influenced by the vast majority of the spacetime, its past being just a finite portion of the total, spatially infinite, spacetime. Thereby, a new type of horizon arises, but now relative to past null infinity: the boundary of the past of all marginally trapped spheres, which contains in particular one with the maximum area $4pi/Lambda$. The singularity is universal and extends mostly outside the collapsing matter. The resulting spacetimes models turn out to be inextendible and globally hyperbolic. It is remarkable that they cannot exist if $Lambda$ vanishes. Given the accepted value of $Lambda$ deduced from cosmological observations, such ultra-massive objects will need to contain a substantial portion of the total {it present} mass of the {it observable} Universe.
{"title":"Ultra-massive spacetimes","authors":"J. Senovilla","doi":"10.4171/pm/2095","DOIUrl":"https://doi.org/10.4171/pm/2095","url":null,"abstract":"A positive cosmological constant $Lambda>0$ sets an upper limit for the area of marginally future-trapped surfaces enclosing a black hole (BH). Does this mean that the mass of the BH cannot increase beyond the corresponding limit? I analyze some simple spherically symmetric models where regions within a dynamical horizon keep gaining mass-energy so that eventually the $Lambda$ limit is surpassed. This shows that the black hole proper transmutes into a collapsing universe, and no observers will ever reach infinity, which dematerializes together with the event horizon and the `cosmological horizon'. The region containing the dynamical horizon cannot be causally influenced by the vast majority of the spacetime, its past being just a finite portion of the total, spatially infinite, spacetime. Thereby, a new type of horizon arises, but now relative to past null infinity: the boundary of the past of all marginally trapped spheres, which contains in particular one with the maximum area $4pi/Lambda$. The singularity is universal and extends mostly outside the collapsing matter. The resulting spacetimes models turn out to be inextendible and globally hyperbolic. It is remarkable that they cannot exist if $Lambda$ vanishes. Given the accepted value of $Lambda$ deduced from cosmological observations, such ultra-massive objects will need to contain a substantial portion of the total {it present} mass of the {it observable} Universe.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46665728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct globally hyperbolic spacetimes such that each slice ${t=t_0}$ of the universal time $t$ is a model space of constant curvature $k(t_0)$ which may not only vary with $t_0inmathbb{R}$ but also change its sign. The metric is smooth and slightly different to FLRW spacetimes, namely, $g=-dt^2+dr^2+ S_{k(t)}^2(r) g_{mathbb{S}^{n-1}}$, where $g_{mathbb{S}^{n-1}}$ is the metric of the standard sphere, $S_{k(t)}(r)=sin(sqrt{k(t)}, r)/sqrt{k(t)}$ when $k(t)geq 0$ and $S_{k(t)}(r)=sinh(sqrt{-k(t)}, r)/sqrt{-k(t)}$ when $k(t)leq 0$. In the open case, the $t$-slices are (non-compact) Cauchy hypersurfaces of curvature $k(t)leq 0$, thus homeomorphic to $mathbb{R}^n$; a typical example is $k(t)=-t^2$ (i.e., $S_{k(t)}(r)=sinh(tr)/t$). In the closed case, $k(t)>0$ somewhere, a slight extension of the class shows how the topology of the $t$-slices changes. This makes at least one comoving observer to disappear in finite time $t$ showing some similarities with an inflationary expansion. Anyway, the spacetime is foliated by Cauchy hypersurfaces homeomorphic to spheres, not all of them $t$-slices.
{"title":"A class of cosmological models with spatially constant sign-changing curvature","authors":"M. S'anchez","doi":"10.4171/pm/2099","DOIUrl":"https://doi.org/10.4171/pm/2099","url":null,"abstract":"We construct globally hyperbolic spacetimes such that each slice ${t=t_0}$ of the universal time $t$ is a model space of constant curvature $k(t_0)$ which may not only vary with $t_0inmathbb{R}$ but also change its sign. The metric is smooth and slightly different to FLRW spacetimes, namely, $g=-dt^2+dr^2+ S_{k(t)}^2(r) g_{mathbb{S}^{n-1}}$, where $g_{mathbb{S}^{n-1}}$ is the metric of the standard sphere, $S_{k(t)}(r)=sin(sqrt{k(t)}, r)/sqrt{k(t)}$ when $k(t)geq 0$ and $S_{k(t)}(r)=sinh(sqrt{-k(t)}, r)/sqrt{-k(t)}$ when $k(t)leq 0$. In the open case, the $t$-slices are (non-compact) Cauchy hypersurfaces of curvature $k(t)leq 0$, thus homeomorphic to $mathbb{R}^n$; a typical example is $k(t)=-t^2$ (i.e., $S_{k(t)}(r)=sinh(tr)/t$). In the closed case, $k(t)>0$ somewhere, a slight extension of the class shows how the topology of the $t$-slices changes. This makes at least one comoving observer to disappear in finite time $t$ showing some similarities with an inflationary expansion. Anyway, the spacetime is foliated by Cauchy hypersurfaces homeomorphic to spheres, not all of them $t$-slices.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43222470","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"$L$-parabolic linear Weingarten spacelike submanifolds immersed in an Einstein manifold","authors":"Railane Antonia, H. D. de Lima","doi":"10.4171/pm/2082","DOIUrl":"https://doi.org/10.4171/pm/2082","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47157150","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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