Pub Date : 2024-10-03DOI: 10.1016/j.jde.2024.09.050
Yibin Feng , Yuanyuan Li , Lei Xu
Gaussian dual curvature measure is introduced and Gaussian dual Minkowski problem is studied. This problem amounts to solving a class of Monge-Ampère type equations on the unit sphere. Existence and uniqueness of solutions to the relevant Monge-Ampère type equations are obtained in the smooth category when , respectively. For , a complete solution to existence part of the Gaussian dual Minkowski problem is presented. For the case of , a weak solution to the Monge-Ampère type equation related to this problem is provided when given measure has the density f which is sandwiched between two positive constants belonging to the interval 0 to 1.
{"title":"Existence of solutions to the Gaussian dual Minkowski problem","authors":"Yibin Feng , Yuanyuan Li , Lei Xu","doi":"10.1016/j.jde.2024.09.050","DOIUrl":"10.1016/j.jde.2024.09.050","url":null,"abstract":"<div><div>Gaussian dual curvature measure is introduced and Gaussian dual Minkowski problem is studied. This problem amounts to solving a class of Monge-Ampère type equations on the unit sphere. Existence and uniqueness of solutions to the relevant Monge-Ampère type equations are obtained in the smooth category when <span><math><mi>q</mi><mo>≤</mo><mn>0</mn></math></span>, respectively. For <span><math><mi>q</mi><mo><</mo><mn>0</mn></math></span>, a complete solution to existence part of the Gaussian dual Minkowski problem is presented. For the case of <span><math><mi>q</mi><mo>=</mo><mn>0</mn></math></span>, a weak solution to the Monge-Ampère type equation related to this problem is provided when given measure has the density <em>f</em> which is sandwiched between two positive constants belonging to the interval 0 to 1.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 268-298"},"PeriodicalIF":2.4,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425398","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-03DOI: 10.1016/j.jde.2024.09.055
Tho Nguyen Duc
The purpose of this article is to study pseudospectral properties of the one-dimensional Schrödinger operator perturbed by a complex steplike potential. By constructing the resolvent kernel, we show that the pseudospectrum of this operator is trivial if and only if the imaginary part of the potential is constant. As a by-product, a new method to obtain a sharp resolvent estimate is developed, answering a concern of Henry and Krejčiřík, and a way to construct an optimal pseudomode is discovered, answering a concern of Krejčiřík and Siegl. This article also analyzes the impact of a complex point interaction on the spectrum and the resolvent norm.
{"title":"Schrödinger operator with a complex steplike potential","authors":"Tho Nguyen Duc","doi":"10.1016/j.jde.2024.09.055","DOIUrl":"10.1016/j.jde.2024.09.055","url":null,"abstract":"<div><div>The purpose of this article is to study pseudospectral properties of the one-dimensional Schrödinger operator perturbed by a complex steplike potential. By constructing the resolvent kernel, we show that the pseudospectrum of this operator is trivial if and only if the imaginary part of the potential is constant. As a by-product, a new method to obtain a sharp resolvent estimate is developed, answering a concern of Henry and Krejčiřík, and a way to construct an optimal pseudomode is discovered, answering a concern of Krejčiřík and Siegl. This article also analyzes the impact of a complex point interaction on the spectrum and the resolvent norm.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 299-356"},"PeriodicalIF":2.4,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425399","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-03DOI: 10.1016/j.jde.2024.09.053
C. Grotta-Ragazzo , F.J.S. Nascimento
This paper addresses a question posed by Carmen Chicone and proves that an analytic vector field with a non-degenerate global center can be transformed into a classical Newtonian equation .
Additionally, we establish a global Poincaré normal form for planar centers. We also demonstrate the global analytic integrability of the equation , where , under some additional conditions.
{"title":"Global normalizations for centers of planar vector fields","authors":"C. Grotta-Ragazzo , F.J.S. Nascimento","doi":"10.1016/j.jde.2024.09.053","DOIUrl":"10.1016/j.jde.2024.09.053","url":null,"abstract":"<div><div>This paper addresses a question posed by Carmen Chicone and proves that an analytic vector field with a non-degenerate global center can be transformed into a classical Newtonian equation <span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>=</mo><mo>−</mo><msup><mrow><mi>V</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo></math></span>.</div><div>Additionally, we establish a global Poincaré normal form for planar centers. We also demonstrate the global analytic integrability of the equation <span><math><mover><mrow><mi>x</mi></mrow><mrow><mo>¨</mo></mrow></mover><mo>=</mo><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mover><mrow><mi>u</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>)</mo></math></span>, where <span><math><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>)</mo><mo>=</mo><mi>F</mi><mo>(</mo><mi>u</mi><mo>,</mo><mo>−</mo><mi>v</mi><mo>)</mo></math></span>, under some additional conditions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"415 ","pages":"Pages 701-721"},"PeriodicalIF":2.4,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142417302","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.jde.2024.09.051
Henrique Borrin
In this paper, we study the Lagrangian structure of Vlasov-Maxwell system, that is, by using a suitable notion of flow, we prove that if the densities are integrable in spacetime, and the charge acceleration and (or ) are integrable functions in spacetime, then renormalized and distributional solutions of the system are the transport of the initial condition by its flow. We study more general vector fields, with wavelike structure in the sense that it has finite speed of propagation, generalizing the vector fields studied in [6]. The result is a extension of those obtained by Ambrosio, Colombo, and Figalli [2] for the Vlasov-Poisson system, and by the author and Marcon [5] for relativistic Vlasov-systems with quasistatic approximations of Maxwell's equations.
{"title":"Regular Lagrangian flow for wavelike vector fields and the Vlasov-Maxwell system","authors":"Henrique Borrin","doi":"10.1016/j.jde.2024.09.051","DOIUrl":"10.1016/j.jde.2024.09.051","url":null,"abstract":"<div><div>In this paper, we study the Lagrangian structure of Vlasov-Maxwell system, that is, by using a suitable notion of flow, we prove that if the densities <span><math><mi>ρ</mi><mo>,</mo><mspace></mspace><mi>j</mi></math></span> are integrable in spacetime, and the charge acceleration <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>j</mi></math></span> and <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mi>j</mi></math></span> (or <span><math><mi>∇</mi><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>j</mi></math></span>) are integrable functions in spacetime, then renormalized and distributional solutions of the system are the transport of the initial condition by its flow. We study more general vector fields, with wavelike structure in the sense that it has finite speed of propagation, generalizing the vector fields studied in <span><span>[6]</span></span>. The result is a extension of those obtained by Ambrosio, Colombo, and Figalli <span><span>[2]</span></span> for the Vlasov-Poisson system, and by the author and Marcon <span><span>[5]</span></span> for relativistic Vlasov-systems with quasistatic approximations of Maxwell's equations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 190-226"},"PeriodicalIF":2.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425396","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.jde.2024.09.049
Richard De la cruz , Wladimir Neves
In this paper, we study two systems with a time-variable coefficient and general time-gradually-degenerate damping. More explicitly, we construct the Riemann solutions to the time-variable coefficient Zeldovich approximation and time-variable coefficient pressureless gas systems both with general time-gradually-degenerate damping. Applying the method of similar variables and nonlinear viscosity, we obtain classical Riemann solutions and delta shock wave solutions.
{"title":"On a class of nonautonomous quasilinear systems with general time-gradually-degenerate damping","authors":"Richard De la cruz , Wladimir Neves","doi":"10.1016/j.jde.2024.09.049","DOIUrl":"10.1016/j.jde.2024.09.049","url":null,"abstract":"<div><div>In this paper, we study two systems with a time-variable coefficient and general time-gradually-degenerate damping. More explicitly, we construct the Riemann solutions to the time-variable coefficient Zeldovich approximation and time-variable coefficient pressureless gas systems both with general time-gradually-degenerate damping. Applying the method of similar variables and nonlinear viscosity, we obtain classical Riemann solutions and delta shock wave solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 52-81"},"PeriodicalIF":2.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.jde.2024.09.052
Minbo Yang , Jefferson Abrantes , Pedro Ubilla , Jiazheng Zhou
We establish global multiplicity of positive solutions (existence and nonexistence theory) for the following problem where , is a parameter, and f is a sublinear nonlinearity at ∞. In order to obtain our results we use a combination of the sub- super solution method and variational techniques. For instance, we need to implement a relevant result of type versus X local minimizer for some appropriate space X.
我们为以下问题建立了正解的全局多重性（存在与不存在理论）{Δu+λh(x)f(u)=0inRN,u>0inRN,u∈D1,2(RN)，其中 N≥3, λ>0 是参数，0≤h∈L∞(RN)，f 是∞处的亚线性非线性。为了得到我们的结果，我们结合使用了次超解方法和变分技术。例如，我们需要在某个合适的空间 X 上实现 D1,2(RN) 对 X 局部最小化的相关结果。
{"title":"Global multiplicity of positive solutions for a sublinear elliptic equation in RN","authors":"Minbo Yang , Jefferson Abrantes , Pedro Ubilla , Jiazheng Zhou","doi":"10.1016/j.jde.2024.09.052","DOIUrl":"10.1016/j.jde.2024.09.052","url":null,"abstract":"<div><div>We establish global multiplicity of positive solutions (existence and nonexistence theory) for the following problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>Δ</mi><mi>u</mi><mo>+</mo><mi>λ</mi><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>></mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>∈</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>N</mi><mo>≥</mo><mn>3</mn></math></span>, <span><math><mi>λ</mi><mo>></mo><mn>0</mn></math></span> is a parameter, <span><math><mn>0</mn><mo>≤</mo><mi>h</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></math></span> and <em>f</em> is a sublinear nonlinearity at ∞. In order to obtain our results we use a combination of the sub- super solution method and variational techniques. For instance, we need to implement a relevant result of type <span><math><msup><mrow><mi>D</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></math></span> versus <em>X</em> local minimizer for some appropriate space <em>X</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 159-189"},"PeriodicalIF":2.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425395","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.jde.2024.09.048
Calin I. Martin
We address here a question of fundamental importance in the analysis of nonlinear partial differential equations: when does a solution to a nonlinear partial differential equation develop singularities and what is the nature of those singularities? The particular type of singularity that we attend to here is wave breaking which is defined as the situation when the wave remains bounded up to the maximal existence time at which its slope becomes infinite. More specifically, our wave breaking result concerns the geophysical nonlinear water wave problem for an inviscid, incompressible, homogeneous fluid, written in cylindrical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions.
{"title":"A wave-breaking result for azimuthally varying water flows in cylindrical coordinates","authors":"Calin I. Martin","doi":"10.1016/j.jde.2024.09.048","DOIUrl":"10.1016/j.jde.2024.09.048","url":null,"abstract":"<div><div>We address here a question of fundamental importance in the analysis of nonlinear partial differential equations: when does a solution to a nonlinear partial differential equation develop singularities and what is the nature of those singularities? The particular type of singularity that we attend to here is wave breaking which is defined as the situation when the wave remains bounded up to the maximal existence time at which its slope becomes infinite. More specifically, our wave breaking result concerns the geophysical nonlinear water wave problem for an inviscid, incompressible, homogeneous fluid, written in cylindrical coordinates that are fixed at a point on the rotating Earth, together with the free surface and rigid bottom boundary conditions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 143-158"},"PeriodicalIF":2.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425443","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.jde.2024.09.045
Yunfeng Jia , Jingjing Wang , Yi Li
A predator-prey system with Holling-II functional response in the presence of additional food resource and degeneracy is proposed in this paper. The main objective is to show the effects of additional food resource and degeneracy on the dynamics for system. We mainly obtain that there exist two critical values induced by degeneracy and improved functional response respectively, such that the system permits positive solutions. Additionally, we also show that both providing additional resource to predator with high quantity or quality, and introducing degeneracy effect into system have positive impacts in improving the amount of predator, which is indeed an environmentally-friendly strategy in preserving biodiversity.
{"title":"Effects of additional resource and degeneracy on the dynamics for a diffusive predator-prey system","authors":"Yunfeng Jia , Jingjing Wang , Yi Li","doi":"10.1016/j.jde.2024.09.045","DOIUrl":"10.1016/j.jde.2024.09.045","url":null,"abstract":"<div><div>A predator-prey system with Holling-II functional response in the presence of additional food resource and degeneracy is proposed in this paper. The main objective is to show the effects of additional food resource and degeneracy on the dynamics for system. We mainly obtain that there exist two critical values induced by degeneracy and improved functional response respectively, such that the system permits positive solutions. Additionally, we also show that both providing additional resource to predator with high quantity or quality, and introducing degeneracy effect into system have positive impacts in improving the amount of predator, which is indeed an environmentally-friendly strategy in preserving biodiversity.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 227-267"},"PeriodicalIF":2.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425397","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.jde.2024.09.046
Ujjwal Koley , Kazuo Yamazaki
We consider a transport-diffusion equation forced by random noise of three types: additive, linear multiplicative in Itô's interpretation, and transport in Stratonovich's interpretation. Via convex integration modified to probabilistic setting, we prove existence of a divergence-free vector field with spatial regularity in Sobolev space and corresponding solution to a transport-diffusion equation with spatial regularity in Lebesgue space, and consequently non-uniqueness in law at the level of probabilistically strong solutions globally in time.
{"title":"Non-uniqueness in law of transport-diffusion equation forced by random noise","authors":"Ujjwal Koley , Kazuo Yamazaki","doi":"10.1016/j.jde.2024.09.046","DOIUrl":"10.1016/j.jde.2024.09.046","url":null,"abstract":"<div><div>We consider a transport-diffusion equation forced by random noise of three types: additive, linear multiplicative in Itô's interpretation, and transport in Stratonovich's interpretation. Via convex integration modified to probabilistic setting, we prove existence of a divergence-free vector field with spatial regularity in Sobolev space and corresponding solution to a transport-diffusion equation with spatial regularity in Lebesgue space, and consequently non-uniqueness in law at the level of probabilistically strong solutions globally in time.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 82-142"},"PeriodicalIF":2.4,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142415958","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-27DOI: 10.1016/j.jde.2024.09.044
Filippo Giuliani
In this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to the case of square (and rational) tori. We weaken the regularity assumptions on the convolution potentials, required in a previous work by Guardia (2014) [11] for the square case, to obtain the -instability () of the elliptic equilibrium . We also provide the existence of solutions with arbitrarily small norm which achieve a prescribed growth, say , , within a time T satisfying polynomial estimates, namely for some .
在本文中，我们证明了二维无理环上具有卷积势的立方 NLS 方程的解的存在性，随着时间的推移，这些解的索波列夫规范会发生任意大的增长。我们的结果也适用于平方（和有理）环的情况。我们弱化了 Guardia（2014）[11] 之前针对正方形情形的工作中所要求的卷积势的正则性假设，从而得到了椭圆均衡 u=0 的 Hs-不稳定性 (s>1)。我们还提供了具有任意小 L2 准则的解 u(t)的存在性，这些解在满足多项式估计（即 0<T<Kc for some c>0）的时间 T 内实现了规定增长，即‖u(T)‖Hs≥K‖u(0)‖Hs, K≫1。
{"title":"Sobolev instability in the cubic NLS equation with convolution potentials on irrational tori","authors":"Filippo Giuliani","doi":"10.1016/j.jde.2024.09.044","DOIUrl":"10.1016/j.jde.2024.09.044","url":null,"abstract":"<div><div>In this paper we prove the existence of solutions to the cubic NLS equation with convolution potentials on two dimensional irrational tori undergoing an arbitrarily large growth of Sobolev norms as time evolves. Our results apply also to the case of square (and rational) tori. We weaken the regularity assumptions on the convolution potentials, required in a previous work by Guardia (2014) <span><span>[11]</span></span> for the square case, to obtain the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></math></span>-instability (<span><math><mi>s</mi><mo>></mo><mn>1</mn></math></span>) of the elliptic equilibrium <span><math><mi>u</mi><mo>=</mo><mn>0</mn></math></span>. We also provide the existence of solutions <span><math><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> with arbitrarily small <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> norm which achieve a prescribed growth, say <span><math><msub><mrow><mo>‖</mo><mi>u</mi><mo>(</mo><mi>T</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub><mo>≥</mo><mi>K</mi><msub><mrow><mo>‖</mo><mi>u</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup></mrow></msub></math></span>, <span><math><mi>K</mi><mo>≫</mo><mn>1</mn></math></span>, within a time <em>T</em> satisfying polynomial estimates, namely <span><math><mn>0</mn><mo><</mo><mi>T</mi><mo><</mo><msup><mrow><mi>K</mi></mrow><mrow><mi>c</mi></mrow></msup></math></span> for some <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1-27"},"PeriodicalIF":2.4,"publicationDate":"2024-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142326436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号