Pub Date : 2025-02-07DOI: 10.1016/j.jde.2025.02.001
Yoshinori Nishii
We consider the Cauchy problem for cubic nonlinear Klein-Gordon equations in one space dimension. We give the -decay estimate for the small data solution and show that it decays faster than the free solution if the cubic nonlinearity has the suitable dissipative structure.
{"title":"On the decay estimate for small solutions to nonlinear Klein-Gordon equations with dissipative structure","authors":"Yoshinori Nishii","doi":"10.1016/j.jde.2025.02.001","DOIUrl":"10.1016/j.jde.2025.02.001","url":null,"abstract":"<div><div>We consider the Cauchy problem for cubic nonlinear Klein-Gordon equations in one space dimension. We give the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-decay estimate for the small data solution and show that it decays faster than the free solution if the cubic nonlinearity has the suitable dissipative structure.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"424 ","pages":"Pages 815-832"},"PeriodicalIF":2.4,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349144","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 : 2025-02-07DOI: 10.1016/j.jde.2025.02.008
Xiao Yan , Hua Nie , Yanling Li
This study explores a nonlocal reaction-diffusion-advection system that models interactions between two competing phytoplankton species in a water column, incorporating crowding effects. We introduce a special cone based on the cumulative distributions of population densities and establish a comparison principle under the order induced by . This results in strong monotonicity within the semiflow generated by the system. We then analyze the dynamics of the system in terms of the advection rates of the two phytoplankton species using monotone dynamical system theory. We identify critical curves that categorize competition outcomes into competitive exclusion, coexistence, and/or bistability. The position and shape of these critical curves can vary significantly depending on key parameters, such as death rates. Furthermore, we derive global results for specific scenarios using a perturbation approach. These findings highlight the crucial role of advection rates and death rates in shaping dynamics within two-species phytoplankton communities.
{"title":"Dynamics of a nonlocal phytoplankton competition model with crowding effects","authors":"Xiao Yan , Hua Nie , Yanling Li","doi":"10.1016/j.jde.2025.02.008","DOIUrl":"10.1016/j.jde.2025.02.008","url":null,"abstract":"<div><div>This study explores a nonlocal reaction-diffusion-advection system that models interactions between two competing phytoplankton species in a water column, incorporating crowding effects. We introduce a special cone <span><math><mi>K</mi></math></span> based on the cumulative distributions of population densities and establish a comparison principle under the order induced by <span><math><mi>K</mi></math></span>. This results in strong monotonicity within the semiflow generated by the system. We then analyze the dynamics of the system in terms of the advection rates of the two phytoplankton species using monotone dynamical system theory. We identify critical curves that categorize competition outcomes into competitive exclusion, coexistence, and/or bistability. The position and shape of these critical curves can vary significantly depending on key parameters, such as death rates. Furthermore, we derive global results for specific scenarios using a perturbation approach. These findings highlight the crucial role of advection rates and death rates in shaping dynamics within two-species phytoplankton communities.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 719-760"},"PeriodicalIF":2.4,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348343","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 : 2025-02-07DOI: 10.1016/j.jde.2025.02.006
Nausica Aldeghi, Jonathan Rohleder
We consider the eigenvalue problem for the Laplacian with mixed Dirichlet and Neumann boundary conditions. For a certain class of bounded, simply connected planar domains we prove monotonicity properties of the first eigenfunction. As a consequence, we establish a variant of the hot spots conjecture for mixed boundary conditions. Moreover, we obtain an inequality between the lowest eigenvalue of this mixed problem and the lowest eigenvalue of the corresponding dual problem where the Dirichlet and Neumann boundary conditions are interchanged. The proofs are based on a novel variational principle, which we establish.
{"title":"On the first eigenvalue and eigenfunction of the Laplacian with mixed boundary conditions","authors":"Nausica Aldeghi, Jonathan Rohleder","doi":"10.1016/j.jde.2025.02.006","DOIUrl":"10.1016/j.jde.2025.02.006","url":null,"abstract":"<div><div>We consider the eigenvalue problem for the Laplacian with mixed Dirichlet and Neumann boundary conditions. For a certain class of bounded, simply connected planar domains we prove monotonicity properties of the first eigenfunction. As a consequence, we establish a variant of the hot spots conjecture for mixed boundary conditions. Moreover, we obtain an inequality between the lowest eigenvalue of this mixed problem and the lowest eigenvalue of the corresponding dual problem where the Dirichlet and Neumann boundary conditions are interchanged. The proofs are based on a novel variational principle, which we establish.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 689-718"},"PeriodicalIF":2.4,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348342","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 : 2025-02-07DOI: 10.1016/j.jde.2025.02.005
Jiawei Wang , Junyan Zhang
We prove the incompressible limit of compressible ideal magnetohydrodynamic (MHD) flows in a reference domain where the magnetic field is tangential to the boundary. Unlike the case of transversal magnetic fields, the linearized problem of our case is not well-posed in standard Sobolev space , while the incompressible problem is still well-posed in . The key observation to overcome the difficulty is a hidden structure contributed by Lorentz force in the vorticity analysis, which reveals that one should trade one normal derivative for two tangential derivatives together with a gain of Mach number weight . Thus, the energy functional should be defined by using suitable anisotropic Sobolev spaces. The weights of Mach number should be carefully chosen according to the number of tangential derivatives, such that the energy estimates are uniform in Mach number. Besides, part of the proof is similar to the study of compressible water waves, so our result opens the possibility to study the incompressible limit of free-boundary problems in ideal MHD.
{"title":"Incompressible limit of compressible ideal MHD flows inside a perfectly conducting wall","authors":"Jiawei Wang , Junyan Zhang","doi":"10.1016/j.jde.2025.02.005","DOIUrl":"10.1016/j.jde.2025.02.005","url":null,"abstract":"<div><div>We prove the incompressible limit of compressible ideal magnetohydrodynamic (MHD) flows in a reference domain where the magnetic field is tangential to the boundary. Unlike the case of transversal magnetic fields, the linearized problem of our case is not well-posed in standard Sobolev space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msup><mspace></mspace><mo>(</mo><mi>m</mi><mo>≥</mo><mn>2</mn><mo>)</mo></math></span>, while the incompressible problem is still well-posed in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span>. The key observation to overcome the difficulty is a hidden structure contributed by Lorentz force in the vorticity analysis, which reveals that one should trade one normal derivative for two tangential derivatives together with a gain of Mach number weight <span><math><msup><mrow><mi>ε</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Thus, the energy functional should be defined by using suitable anisotropic Sobolev spaces. The weights of Mach number should be carefully chosen according to the number of tangential derivatives, such that the energy estimates are uniform in Mach number. Besides, part of the proof is similar to the study of compressible water waves, so our result opens the possibility to study the incompressible limit of free-boundary problems in ideal MHD.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"425 ","pages":"Pages 846-894"},"PeriodicalIF":2.4,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143319074","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 : 2025-02-06DOI: 10.1016/j.jde.2025.02.007
Jinbiao Wu , Biteng Xu , Liangquan Zhang
This paper investigates the risk-sensitive recursive utility control problem, where the system is governed by both regular controls and singular controls. The main feature of this problem is that the cost functional is given by a backward stochastic differential equation (BSDE) with quadratic growth, driven by a discontinuous semimartingale. We examine the existence and uniqueness of solutions to the BSDE, as well as the comparison theorem and stability. From this, we derive the continuity of the value function with respect to the initial state. Additionally, using the dynamic programming principle (DPP), we demonstrate that the value function is the unique viscosity solution to the Hamilton-Jacobi-Bellman (HJB) inequality. Finally, we establish the connection between the DPP and the maximum principle for the risk-sensitive recursive utility singular control problem.
{"title":"Risk-sensitive singular control for stochastic recursive systems and Hamilton-Jacobi-Bellman inequality","authors":"Jinbiao Wu , Biteng Xu , Liangquan Zhang","doi":"10.1016/j.jde.2025.02.007","DOIUrl":"10.1016/j.jde.2025.02.007","url":null,"abstract":"<div><div>This paper investigates the risk-sensitive recursive utility control problem, where the system is governed by both regular controls and singular controls. The main feature of this problem is that the cost functional is given by a backward stochastic differential equation (BSDE) with quadratic growth, driven by a discontinuous semimartingale. We examine the existence and uniqueness of solutions to the BSDE, as well as the comparison theorem and stability. From this, we derive the continuity of the value function with respect to the initial state. Additionally, using the dynamic programming principle (DPP), we demonstrate that the value function is the unique viscosity solution to the Hamilton-Jacobi-Bellman (HJB) inequality. Finally, we establish the connection between the DPP and the maximum principle for the risk-sensitive recursive utility singular control problem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 641-675"},"PeriodicalIF":2.4,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143278735","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 : 2025-02-06DOI: 10.1016/j.jde.2025.01.083
David Hoff
We derive a cancellation property satisfied by certain combinations of derivatives of the Green's functions for the Laplace operator corresponding to Dirichlet and Neumann boundary conditions. This cancellation property is expressed in terms of pointwise bounds independent of distances to the boundary and generalizes Newton's third law concerning equal and opposite forces. We give an application to fluid mechanics and we include a self-contained exposition of the construction of the Green's functions and the derivations of pointwise bounds for their general derivatives up to an order determined by the regularity of the domain.
{"title":"Cancellation properties and pointwise bounds for the Green's functions for the Laplace operator","authors":"David Hoff","doi":"10.1016/j.jde.2025.01.083","DOIUrl":"10.1016/j.jde.2025.01.083","url":null,"abstract":"<div><div>We derive a cancellation property satisfied by certain combinations of derivatives of the Green's functions for the Laplace operator corresponding to Dirichlet and Neumann boundary conditions. This cancellation property is expressed in terms of pointwise bounds independent of distances to the boundary and generalizes Newton's third law concerning equal and opposite forces. We give an application to fluid mechanics and we include a self-contained exposition of the construction of the Green's functions and the derivations of pointwise bounds for their general derivatives up to an order determined by the regularity of the domain.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 601-640"},"PeriodicalIF":2.4,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143278734","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 : 2025-02-06DOI: 10.1016/j.jde.2025.02.002
Ziping Lei , Puchun Zhou
Combinatorial Calabi flows are introduced by Ge in his Ph.D. thesis (Combinatorial methods and geometric equations, Peking University, Beijing, 2012), and have been studied extensively in Euclidean and hyperbolic background geometry. In this paper, we introduce the combinatorial Calabi flow in spherical background geometry for finding ideal circle patterns with prescribed total geodesic curvatures. We prove that the solution of combinatorial Calabi flow exists for all time and converges if and only if there exists an ideal circle pattern with prescribed total geodesic curvatures. We also show that if it converges, it will converge exponentially fast to the desired metric, which provides an effective algorithm to find certain ideal circle patterns. To our knowledge, it is the first combinatorial Calabi flow in spherical background geometry.
{"title":"Combinatorial Calabi flows for ideal circle patterns in spherical background geometry","authors":"Ziping Lei , Puchun Zhou","doi":"10.1016/j.jde.2025.02.002","DOIUrl":"10.1016/j.jde.2025.02.002","url":null,"abstract":"<div><div>Combinatorial Calabi flows are introduced by Ge in his Ph.D. thesis (Combinatorial methods and geometric equations, Peking University, Beijing, 2012), and have been studied extensively in Euclidean and hyperbolic background geometry. In this paper, we introduce the combinatorial Calabi flow in spherical background geometry for finding ideal circle patterns with prescribed total geodesic curvatures. We prove that the solution of combinatorial Calabi flow exists for all time and converges if and only if there exists an ideal circle pattern with prescribed total geodesic curvatures. We also show that if it converges, it will converge exponentially fast to the desired metric, which provides an effective algorithm to find certain ideal circle patterns. To our knowledge, it is the first combinatorial Calabi flow in spherical background geometry.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 676-688"},"PeriodicalIF":2.4,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143278736","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 : 2025-02-06DOI: 10.1016/j.jde.2025.02.009
Zaizheng Li , Susanna Terracini
<div><div>We investigate the existence of rotating spirals for three-component competition-diffusion systems in <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>β</mi><mi>α</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>β</mi><mi>γ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo></mtd><mtd><mtext>in</mtext><mspace></mspace><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>β</mi><mi>γ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>β</mi><mi>α</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo></mtd><mtd><mtext>in</mtext><mspace></mspace><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>β</mi><mi>α</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mi>β</mi><mi>γ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo></mtd><mtd><mtext>in</mtext><mspace></mspace><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mtext>x</mtext><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mtext>x</mtext><mo>)</mo><mo>,</mo><mi>
{"title":"Rotating spirals for three-component competition systems","authors":"Zaizheng Li , Susanna Terracini","doi":"10.1016/j.jde.2025.02.009","DOIUrl":"10.1016/j.jde.2025.02.009","url":null,"abstract":"<div><div>We investigate the existence of rotating spirals for three-component competition-diffusion systems in <span><math><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>:<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>β</mi><mi>α</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>β</mi><mi>γ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo></mtd><mtd><mtext>in</mtext><mspace></mspace><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>β</mi><mi>γ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mi>β</mi><mi>α</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo></mtd><mtd><mtext>in</mtext><mspace></mspace><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mi>Δ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>f</mi><mo>(</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>)</mo><mo>−</mo><mi>β</mi><mi>α</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mi>β</mi><mi>γ</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>,</mo></mtd><mtd><mtext>in</mtext><mspace></mspace><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mtext>x</mtext><mo>,</mo><mn>0</mn><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mtext>x</mtext><mo>)</mo><mo>,</mo><mi>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"426 ","pages":"Pages 853-875"},"PeriodicalIF":2.4,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143338010","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 : 2025-02-05DOI: 10.1016/j.jde.2025.01.091
Qing Liu, Erbol Zhanpeisov
We study a class of fully nonlinear boundary-degenerate elliptic equations, for which we prove that is the only solution. Although no boundary conditions are posed together with the equations, we show that the operator degeneracy actually generates an implicit boundary condition. Under appropriate assumptions on the degeneracy rate and regularity of the operator, we then prove that there exist no bounded solutions other than the trivial one. Our method is based on the arguments for uniqueness of viscosity solutions to state constraint problems for Hamilton-Jacobi equations. We obtain similar results for fully nonlinear degenerate parabolic equations. Several concrete examples of equations that satisfy the assumptions are also given.
{"title":"Liouville-type theorems for fully nonlinear elliptic and parabolic equations with boundary degeneracy","authors":"Qing Liu, Erbol Zhanpeisov","doi":"10.1016/j.jde.2025.01.091","DOIUrl":"10.1016/j.jde.2025.01.091","url":null,"abstract":"<div><div>We study a class of fully nonlinear boundary-degenerate elliptic equations, for which we prove that <span><math><mi>u</mi><mo>≡</mo><mn>0</mn></math></span> is the only solution. Although no boundary conditions are posed together with the equations, we show that the operator degeneracy actually generates an implicit boundary condition. Under appropriate assumptions on the degeneracy rate and regularity of the operator, we then prove that there exist no bounded solutions other than the trivial one. Our method is based on the arguments for uniqueness of viscosity solutions to state constraint problems for Hamilton-Jacobi equations. We obtain similar results for fully nonlinear degenerate parabolic equations. Several concrete examples of equations that satisfy the assumptions are also given.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 510-537"},"PeriodicalIF":2.4,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143170957","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 : 2025-02-05DOI: 10.1016/j.jde.2025.01.092
Yoshiyuki Kagei , Hiroshi Takeda
This paper studies a nonlinear viscoelastic equation with a time periodic external force on the three dimensional whole space. The existence of a time periodic solution is proved by using a spectral decomposition and the Poincaré map when the external force is small enough. Based on the regularity estimates of the time periodic solution derived from the smoothing effect of the semigroup, a stability result is obtained with time decay estimates of perturbations.
{"title":"Existence and stability of time periodic solutions to nonlinear elastic wave equations with viscoelastic terms","authors":"Yoshiyuki Kagei , Hiroshi Takeda","doi":"10.1016/j.jde.2025.01.092","DOIUrl":"10.1016/j.jde.2025.01.092","url":null,"abstract":"<div><div>This paper studies a nonlinear viscoelastic equation with a time periodic external force on the three dimensional whole space. The existence of a time periodic solution is proved by using a spectral decomposition and the Poincaré map when the external force is small enough. Based on the regularity estimates of the time periodic solution derived from the smoothing effect of the semigroup, a stability result is obtained with time decay estimates of perturbations.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"427 ","pages":"Pages 478-509"},"PeriodicalIF":2.4,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143171787","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}
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