Pub Date : 2024-12-03DOI: 10.1016/j.jde.2024.11.050
Yuyou Zhong , Qi Wang , Chungen Liu
In this paper, we study the existence and multiplicity of the symmetrical periodic solutions for some asymptotically linear distributed delay differential systems under a symmetric condition by using the symmetric -index theory.
{"title":"The existence and multiplicity of symmetrical periodic solutions for asymptotically linear distributed delay differential systems","authors":"Yuyou Zhong , Qi Wang , Chungen Liu","doi":"10.1016/j.jde.2024.11.050","DOIUrl":"10.1016/j.jde.2024.11.050","url":null,"abstract":"<div><div>In this paper, we study the existence and multiplicity of the symmetrical periodic solutions for some asymptotically linear distributed delay differential systems under a symmetric condition by using the symmetric <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-index theory.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"420 ","pages":"Pages 99-117"},"PeriodicalIF":2.4,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759015","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 : 2024-12-02DOI: 10.1016/j.jde.2024.11.038
Pascal Auscher , Pierre Portal
We establish well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this level of generality, and avoid any of the smoothness assumptions used in the literature, we introduce a notion of pathwise weak solution and develop a new harmonic analysis toolkit. The latter includes techniques to prove the boundedness of various maximal regularity operators on relevant spaces of square functions, the parabolic tent spaces . Applied to deterministic parabolic PDE in divergence form with real coefficients, our results also give the first extension of Lions maximal regularity theorem on to , for all in this generality.
{"title":"Stochastic and deterministic parabolic equations with bounded measurable coefficients in space and time: Well-posedness and maximal regularity","authors":"Pascal Auscher , Pierre Portal","doi":"10.1016/j.jde.2024.11.038","DOIUrl":"10.1016/j.jde.2024.11.038","url":null,"abstract":"<div><div>We establish well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this level of generality, and avoid any of the smoothness assumptions used in the literature, we introduce a notion of pathwise weak solution and develop a new harmonic analysis toolkit. The latter includes techniques to prove the boundedness of various maximal regularity operators on relevant spaces of square functions, the parabolic tent spaces <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>. Applied to deterministic parabolic PDE in divergence form with real coefficients, our results also give the first extension of Lions maximal regularity theorem on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>×</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>=</mo><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> to <span><math><msup><mrow><mi>T</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>, for all <span><math><mn>1</mn><mo>−</mo><mi>ε</mi><mo><</mo><mi>p</mi><mo>≤</mo><mo>∞</mo></math></span> in this generality.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"420 ","pages":"Pages 1-51"},"PeriodicalIF":2.4,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142759014","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 : 2024-11-29DOI: 10.1016/j.jde.2024.11.029
F. Feppon
We compute full asymptotic expansions of the permeability matrix of a laminar fluid flowing through a periodic array of small solid particles. The derivation considers obstacles with arbitrary shape in arbitrary space dimension. In the first step, we use hydrodynamics layer potential theory to obtain the asymptotic expansion of the velocity and pressure fields across the periodic array. The terms of these expansions can be computed through a procedure involving a cascade of exterior and interior problems. In the second step, we deduce the asymptotic expansion of the permeability matrix. The derivation requires evaluating Hadamard finite part integrals and tensors depending on the values of the fundamental solution or its derivatives on the faces of the unit cell. We verify that our expansions agree to the leading order with the expressions found by Hasimoto [24] in the case of spherical obstacles in two and three dimensions.
{"title":"Full asymptotic expansion of the permeability matrix of a dilute periodic porous medium","authors":"F. Feppon","doi":"10.1016/j.jde.2024.11.029","DOIUrl":"10.1016/j.jde.2024.11.029","url":null,"abstract":"<div><div>We compute full asymptotic expansions of the permeability matrix of a laminar fluid flowing through a periodic array of small solid particles. The derivation considers obstacles with arbitrary shape in arbitrary space dimension. In the first step, we use hydrodynamics layer potential theory to obtain the asymptotic expansion of the velocity and pressure fields across the periodic array. The terms of these expansions can be computed through a procedure involving a cascade of exterior and interior problems. In the second step, we deduce the asymptotic expansion of the permeability matrix. The derivation requires evaluating Hadamard finite part integrals and tensors depending on the values of the fundamental solution or its derivatives on the faces of the unit cell. We verify that our expansions agree to the leading order with the expressions found by Hasimoto <span><span>[24]</span></span> in the case of spherical obstacles in two and three dimensions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 178-237"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745975","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 : 2024-11-29DOI: 10.1016/j.jde.2024.11.033
J.F. Carreño-Diaz, E.I. Kaikina
We consider the initial-boundary value problem for the Ginzburg-Landau equation with fractional Laplacian on a upper-right quarter plane.
We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the Sobolev spaces to the case of a Neumann type of boundary data. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.
{"title":"Neumann problem for fractional Ginzburg-Landau equation on a upper- right quarter plane","authors":"J.F. Carreño-Diaz, E.I. Kaikina","doi":"10.1016/j.jde.2024.11.033","DOIUrl":"10.1016/j.jde.2024.11.033","url":null,"abstract":"<div><div>We consider the initial-boundary value problem for the Ginzburg-Landau equation with fractional Laplacian on a upper-right quarter plane.</div><div>We study the main questions of the theory of IBV- problems for nonlocal equations: the existence and uniqueness of a solution, the asymptotic behavior of the solution for large time and the influence of initial and boundary data on the basic properties of the solution. We generalize the concept of the well-posedness of IBV- problem in the Sobolev spaces to the case of a Neumann type of boundary data. We also give optimal relations between the orders of the Sobolev spaces to which the initial and boundary data belong. The lower order compatibility conditions between initial and boundary data are also discussed.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 258-304"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745977","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 : 2024-11-29DOI: 10.1016/j.jde.2024.11.031
Tiago Carvalho , André do Amaral Antunes
It is well known that many results obtained for piecewise smooth vector fields do not have an analogous for smooth vector fields and vice-versa. These differences are generated by the non-uniqueness of trajectory passing through a point. Inspired by the classical fact that one-dimensional discrete dynamic systems can produce chaotic behavior, we construct a conjugation between shift maps and piecewise smooth vector fields presenting homoclinic loops which are associated to symbols in such a way that the flow restricted to a homoclinic loop is codified with a symbol. The construction of the topological conjugation between the quoted piecewise smooth vector fields and the respective shift spaces needs several technicality which were solved considering a specific family of piecewise smooth vector fields (Theorem A) and then generalizing the result for an entire class of piecewise smooth vector fields (Theorem B). By means of the results obtained and the techniques employed, a new perspective on the study of piecewise smooth vector fields is brought to light and, through already established results for discrete dynamic systems, we will be able to obtain results regarding piecewise smooth vector fields.
{"title":"Symbolic dynamics of planar piecewise smooth vector fields","authors":"Tiago Carvalho , André do Amaral Antunes","doi":"10.1016/j.jde.2024.11.031","DOIUrl":"10.1016/j.jde.2024.11.031","url":null,"abstract":"<div><div>It is well known that many results obtained for piecewise smooth vector fields do not have an analogous for smooth vector fields and vice-versa. These differences are generated by the non-uniqueness of trajectory passing through a point. Inspired by the classical fact that one-dimensional discrete dynamic systems can produce chaotic behavior, we construct a conjugation between shift maps and piecewise smooth vector fields presenting homoclinic loops which are associated to symbols in such a way that the flow restricted to a homoclinic loop is codified with a symbol. The construction of the topological conjugation between the quoted piecewise smooth vector fields and the respective shift spaces needs several technicality which were solved considering a specific family of piecewise smooth vector fields (<span><span>Theorem A</span></span>) and then generalizing the result for an entire class of piecewise smooth vector fields (<span><span>Theorem B</span></span>). By means of the results obtained and the techniques employed, a new perspective on the study of piecewise smooth vector fields is brought to light and, through already established results for discrete dynamic systems, we will be able to obtain results regarding piecewise smooth vector fields.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 150-174"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746149","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 : 2024-11-29DOI: 10.1016/j.jde.2024.11.021
Istvan Kadar
Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not critical according to our classification. Furthermore, we gave two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in [13].
{"title":"Small data non-linear wave equation numerology: The role of asymptotics","authors":"Istvan Kadar","doi":"10.1016/j.jde.2024.11.021","DOIUrl":"10.1016/j.jde.2024.11.021","url":null,"abstract":"<div><div>Systems of wave equations may fail to be globally well posed, even for small initial data. Attempts to classify systems into well and ill-posed categories work by identifying structural properties of the equations that can work as indicators of well-posedness. The most famous of these are the null and weak null conditions. As noted by Keir, related formulations may fail to properly capture the effect of undifferentiated terms in systems of wave equations. We show that this is because null conditions are good for categorising behaviour close to null infinity, but not at timelike infinity. In this paper, we propose an alternative condition for semilinear equations that work for undifferentiated non-linearities as well. We illustrate the strength of this new condition by proving global well and ill-posedness statements for some systems of equation that are not <em>critical</em> according to our classification. Furthermore, we gave two examples of systems satisfying the weak null condition with global ill-posedness due to undifferentiated terms, thereby disproving the weak null conjecture as stated in <span><span>[13]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 305-373"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745978","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 : 2024-11-29DOI: 10.1016/j.jde.2024.11.028
Yingzhi Du, Tao Luo
For the initial boundary problem of the incompressible MHD equations in a bounded domain with general curved boundary in 3D with the general Navier-slip boundary conditions for the velocity field and the perfect conducting condition for the magnetic field, we establish the uniform regularity of conormal Sobolev norms and Lipschitz norms to addressing the anisotropic regularity of tangential and normal directions, which enable us to prove the vanishing dissipation limit as the viscosity and the magnetic diffusion coefficients tend to zero. We overcome the difficulties caused by the intricate interaction of boundary curvature, velocity field and magnetic fields and resolve the issue caused by the problem that the viscosity and the magnetic diffusion coefficients are not required to equal.
{"title":"Uniform regularity for incompressible MHD equations in a bounded domain with curved boundary in 3D","authors":"Yingzhi Du, Tao Luo","doi":"10.1016/j.jde.2024.11.028","DOIUrl":"10.1016/j.jde.2024.11.028","url":null,"abstract":"<div><div>For the initial boundary problem of the incompressible MHD equations in a bounded domain with general curved boundary in 3D with the general Navier-slip boundary conditions for the velocity field and the perfect conducting condition for the magnetic field, we establish the uniform regularity of conormal Sobolev norms and Lipschitz norms to addressing the anisotropic regularity of tangential and normal directions, which enable us to prove the vanishing dissipation limit as the viscosity and the magnetic diffusion coefficients tend to zero. We overcome the difficulties caused by the intricate interaction of boundary curvature, velocity field and magnetic fields and resolve the issue caused by the problem that the viscosity and the magnetic diffusion coefficients are not required to equal.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 175-252"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746150","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 : 2024-11-29DOI: 10.1016/j.jde.2024.11.024
Prashanta Garain , Erik Lindgren , Alireza Tavakoli
In this paper, we are concerned with the Hölder regularity for solutions of the nonlocal evolutionary equation Here, is the fractional p-Laplacian, and . We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. Our results complement the previous results for the superquadratic case when .
{"title":"Higher Hölder regularity for a subquadratic nonlocal parabolic equation","authors":"Prashanta Garain , Erik Lindgren , Alireza Tavakoli","doi":"10.1016/j.jde.2024.11.024","DOIUrl":"10.1016/j.jde.2024.11.024","url":null,"abstract":"<div><div>In this paper, we are concerned with the Hölder regularity for solutions of the nonlocal evolutionary equation<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup><mi>u</mi><mo>=</mo><mn>0</mn><mo>.</mo></math></span></span></span> Here, <span><math><msup><mrow><mo>(</mo><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>s</mi></mrow></msup></math></span> is the fractional <em>p</em>-Laplacian, <span><math><mn>0</mn><mo><</mo><mi>s</mi><mo><</mo><mn>1</mn></math></span> and <span><math><mn>1</mn><mo><</mo><mi>p</mi><mo><</mo><mn>2</mn></math></span>. We establish Hölder regularity with explicit Hölder exponents. We also include the inhomogeneous equation with a bounded inhomogeneity. In some cases, the obtained Hölder exponents are almost sharp. Our results complement the previous results for the superquadratic case when <span><math><mi>p</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 253-290"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746151","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 : 2024-11-29DOI: 10.1016/j.jde.2024.11.032
Feimin Huang , Jiajin Shi , Yi Wang
We study Riemann problem for the two-dimensional (2D) pressureless Euler system with planar Riemann initial data. It is proved that there exist infinitely many bounded admissible weak solutions to the 2D Riemann problem by the method of convex integration. Meanwhile, the corresponding one-dimensional (1D) Riemann problem admits a unique measure-valued solution (so-called δ-shock) under the Oleĭnik's entropy condition and an additional energy condition, which implies the non-existence of 1D bounded admissible weak solutions with energy condition (cf. [19]).
{"title":"Non-uniqueness of admissible weak solutions to the two-dimensional pressureless Euler system","authors":"Feimin Huang , Jiajin Shi , Yi Wang","doi":"10.1016/j.jde.2024.11.032","DOIUrl":"10.1016/j.jde.2024.11.032","url":null,"abstract":"<div><div>We study Riemann problem for the two-dimensional (2D) pressureless Euler system with planar Riemann initial data. It is proved that there exist infinitely many bounded admissible weak solutions to the 2D Riemann problem by the method of convex integration. Meanwhile, the corresponding one-dimensional (1D) Riemann problem admits a unique measure-valued solution (so-called <em>δ</em>-shock) under the Oleĭnik's entropy condition and an additional energy condition, which implies the non-existence of 1D bounded admissible weak solutions with energy condition (cf. <span><span>[19]</span></span>).</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"418 ","pages":"Pages 238-257"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745976","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 : 2024-11-29DOI: 10.1016/j.jde.2024.11.030
Priscila Leal da Silva , Igor Leite Freire , Nazime Sales Filho
We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From the symmetries, we obtain invariant solutions that provide explicit metrics for the surfaces. These solutions are unbounded and often appear in mirrored pairs. We introduce the “collage” method, which uses conserved quantities to remove unbounded parts and smoothly join the solutions, leading to weak solutions consistent with the conserved quantities. As a result we get pseudo-peakons, which are smoother than Camassa-Holm peakons. Additionally, we apply a Miura-type transformation to relate our equation to the Degasperis-Procesi equation, allowing us to recover peakon and shock-peakon solutions for it from the solutions of the other equation.
{"title":"An integrable pseudospherical equation with pseudo-peakon solutions","authors":"Priscila Leal da Silva , Igor Leite Freire , Nazime Sales Filho","doi":"10.1016/j.jde.2024.11.030","DOIUrl":"10.1016/j.jde.2024.11.030","url":null,"abstract":"<div><div>We study an integrable equation whose solutions define a triad of one-forms describing a surface with Gaussian curvature -1. We identify a local group of diffeomorphisms that preserve these solutions and establish conserved quantities. From the symmetries, we obtain invariant solutions that provide explicit metrics for the surfaces. These solutions are unbounded and often appear in mirrored pairs. We introduce the “collage” method, which uses conserved quantities to remove unbounded parts and smoothly join the solutions, leading to weak solutions consistent with the conserved quantities. As a result we get pseudo-peakons, which are smoother than Camassa-Holm peakons. Additionally, we apply a Miura-type transformation to relate our equation to the Degasperis-Procesi equation, allowing us to recover peakon and shock-peakon solutions for it from the solutions of the other equation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"419 ","pages":"Pages 291-323"},"PeriodicalIF":2.4,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142746152","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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