Pub Date : 2024-11-14DOI: 10.1016/j.jde.2024.11.011
Luiz Fernando de Oliveira Faria , Aldo Henrique de Souza Medeiros , Jeferson Camilo Silva
In this article, we investigate a class of nonlinear elliptic equations driven by the p-Laplacian operator in the entire space , known as the Emden-Fowler equation type. The complexity of the problem arises from the interplay of two distinct critical growth phenomena, characterized by both Sobolev and Hardy senses. We explore the existence of positive radial solutions, with the proof relying on variational methods. Due to multiple critical nonlinearities, the Mountain Pass Lemma does not yield critical points but only Palais-Smale sequences. The primary challenge lies in the asymptotic competition among the energies carried by these multiple critical nonlinearities.
{"title":"On the Emden-Fowler equation type involving double critical growth","authors":"Luiz Fernando de Oliveira Faria , Aldo Henrique de Souza Medeiros , Jeferson Camilo Silva","doi":"10.1016/j.jde.2024.11.011","DOIUrl":"10.1016/j.jde.2024.11.011","url":null,"abstract":"<div><div>In this article, we investigate a class of nonlinear elliptic equations driven by the <em>p</em>-Laplacian operator in the entire space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>, known as the Emden-Fowler equation type. The complexity of the problem arises from the interplay of two distinct critical growth phenomena, characterized by both Sobolev and Hardy senses. We explore the existence of positive radial solutions, with the proof relying on variational methods. Due to multiple critical nonlinearities, the Mountain Pass Lemma does not yield critical points but only Palais-Smale sequences. The primary challenge lies in the asymptotic competition among the energies carried by these multiple critical nonlinearities.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1861-1880"},"PeriodicalIF":2.4,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652792","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 : 2024-11-14DOI: 10.1016/j.jde.2024.11.004
Andrew Yang , Xu Zhao , Wenshu Zhou
This paper concerns with the one dimensional compressible isentropic Navier–Stokes equations with a free boundary separating fluid and vacuum when the viscosity coefficient depends on the density. Precisely, the pressure P and the viscosity coefficient μ are assumed to be proportional to and respectively, where ρ is the density, and γ and θ are constants. We establish the unique solvability in the framework of global classical solutions for this problem when . Since the previous results on this topic are limited to the case when , the result in this paper fills in the gap for . Note that the key estimate is to show that the density has a positive lower bound and the new ingredient of the proof relies on the study of the quasilinear parabolic equation for the viscosity coefficient by reducing the nonlocal terms in order to apply the comparison principle.
{"title":"Global classical solutions of free boundary problem of compressible Navier–Stokes equations with degenerate viscosity","authors":"Andrew Yang , Xu Zhao , Wenshu Zhou","doi":"10.1016/j.jde.2024.11.004","DOIUrl":"10.1016/j.jde.2024.11.004","url":null,"abstract":"<div><div>This paper concerns with the one dimensional compressible isentropic Navier–Stokes equations with a free boundary separating fluid and vacuum when the viscosity coefficient depends on the density. Precisely, the pressure <em>P</em> and the viscosity coefficient <em>μ</em> are assumed to be proportional to <span><math><msup><mrow><mi>ρ</mi></mrow><mrow><mi>γ</mi></mrow></msup></math></span> and <span><math><msup><mrow><mi>ρ</mi></mrow><mrow><mi>θ</mi></mrow></msup></math></span> respectively, where <em>ρ</em> is the density, and <em>γ</em> and <em>θ</em> are constants. We establish the unique solvability in the framework of global classical solutions for this problem when <span><math><mi>γ</mi><mo>≥</mo><mi>θ</mi><mo>></mo><mn>1</mn></math></span>. Since the previous results on this topic are limited to the case when <span><math><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, the result in this paper fills in the gap for <span><math><mi>θ</mi><mo>></mo><mn>1</mn></math></span>. Note that the key estimate is to show that the density has a positive lower bound and the new ingredient of the proof relies on the study of the quasilinear parabolic equation for the viscosity coefficient by reducing the nonlocal terms in order to apply the comparison principle.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1837-1860"},"PeriodicalIF":2.4,"publicationDate":"2024-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142652791","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 : 2024-11-12DOI: 10.1016/j.jde.2024.10.033
Bingyuan Huang , Yingshan Chen , Limei Zhu
We consider the initial boundary value problem for the compressible fluid-particle interaction system with cylindrical symmetry. We derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate when the shear viscosity goes to zero without any smallness assumption on the initial and boundary data.
{"title":"Optimal convergence rate of the vanishing shear viscosity limit for a compressible fluid-particle interaction system","authors":"Bingyuan Huang , Yingshan Chen , Limei Zhu","doi":"10.1016/j.jde.2024.10.033","DOIUrl":"10.1016/j.jde.2024.10.033","url":null,"abstract":"<div><div>We consider the initial boundary value problem for the compressible fluid-particle interaction system with cylindrical symmetry. We derive explicit Prandtl type boundary layer equations and prove the global in time stability of the boundary layer profile together with the optimal convergence rate when the shear viscosity <span><math><mi>μ</mi><mo>=</mo><mi>κ</mi><msup><mrow><mi>ρ</mi></mrow><mrow><mi>β</mi></mrow></msup></math></span> goes to zero without any smallness assumption on the initial and boundary data.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1792-1823"},"PeriodicalIF":2.4,"publicationDate":"2024-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142655493","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 : 2024-11-06DOI: 10.1016/j.jde.2024.10.040
Qi Qiao , Xiang Zhang
For a chemotaxis-growth model with Allee effect, whose chemotactic sensitivity and diffusion coefficient of the chemical substance are both small, we prove existence of the positive traveling waves with slow wave speeds and their unstability and asymptotic stability with shift depending on the choice of the parameters of the system.
{"title":"Traveling waves to a chemotaxis-growth model with Allee effect","authors":"Qi Qiao , Xiang Zhang","doi":"10.1016/j.jde.2024.10.040","DOIUrl":"10.1016/j.jde.2024.10.040","url":null,"abstract":"<div><div>For a chemotaxis-growth model with Allee effect, whose chemotactic sensitivity and diffusion coefficient of the chemical substance are both small, we prove existence of the positive traveling waves with slow wave speeds and their unstability and asymptotic stability with shift depending on the choice of the parameters of the system.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1747-1770"},"PeriodicalIF":2.4,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593613","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 : 2024-11-06DOI: 10.1016/j.jde.2024.10.036
Daniel Alfonso Santiesteban , Ricardo Abreu Blaya , Juan Bory Reyes
Clifford analysis offers suited framework for a unified treatment of higher-dimensional phenomena. This paper is concerned with boundary value problems for higher order Dirac operators, which are directly related to the Lamé-Navier and iterated Laplace operators. The conditioning of the problems upon the boundaries of the considered domains ensures their well-posedness in the sense of Hadamard.
{"title":"On the well-posedness of boundary value problems for higher order Dirac operators in Rm","authors":"Daniel Alfonso Santiesteban , Ricardo Abreu Blaya , Juan Bory Reyes","doi":"10.1016/j.jde.2024.10.036","DOIUrl":"10.1016/j.jde.2024.10.036","url":null,"abstract":"<div><div>Clifford analysis offers suited framework for a unified treatment of higher-dimensional phenomena. This paper is concerned with boundary value problems for higher order Dirac operators, which are directly related to the Lamé-Navier and iterated Laplace operators. The conditioning of the problems upon the boundaries of the considered domains ensures their well-posedness in the sense of Hadamard.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1729-1746"},"PeriodicalIF":2.4,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593612","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 : 2024-11-06DOI: 10.1016/j.jde.2024.10.038
Yan-Hua Xing, Jian-Wen Sun
In this paper, we study the positive solutions of some nonlocal dispersal equations. We are interested in the new profiles of positive solutions with different reaction functions when spatial degeneracy occurs. It is shown that there can exist six kinds of asymptotic profiles for the nonlocal dispersal problem. Our study also provides the precise effect of reaction functions.
{"title":"Fine profiles of positive solutions for some nonlocal dispersal equations","authors":"Yan-Hua Xing, Jian-Wen Sun","doi":"10.1016/j.jde.2024.10.038","DOIUrl":"10.1016/j.jde.2024.10.038","url":null,"abstract":"<div><div>In this paper, we study the positive solutions of some nonlocal dispersal equations. We are interested in the new profiles of positive solutions with different reaction functions when spatial degeneracy occurs. It is shown that there can exist six kinds of asymptotic profiles for the nonlocal dispersal problem. Our study also provides the precise effect of reaction functions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1771-1791"},"PeriodicalIF":2.4,"publicationDate":"2024-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142593606","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 : 2024-11-05DOI: 10.1016/j.jde.2024.10.042
Rafael B. Gonzalez
We consider a class of first-order partial differential operators, acting on the space of ultradifferentiable periodic functions, and we describe their range by using the following conditions on the coefficients of the operators: the connectedness of certain sublevel sets, the dimension of the subspace generated by the imaginary part of the coefficients, and Diophantine conditions. In addition, we show that these properties are also linked to the regularity of the solutions. The results extend previous ones in Gevrey classes.
{"title":"Existence and regularity of ultradifferentiable periodic solutions to certain vector fields","authors":"Rafael B. Gonzalez","doi":"10.1016/j.jde.2024.10.042","DOIUrl":"10.1016/j.jde.2024.10.042","url":null,"abstract":"<div><div>We consider a class of first-order partial differential operators, acting on the space of ultradifferentiable periodic functions, and we describe their range by using the following conditions on the coefficients of the operators: the connectedness of certain sublevel sets, the dimension of the subspace generated by the imaginary part of the coefficients, and Diophantine conditions. In addition, we show that these properties are also linked to the regularity of the solutions. The results extend previous ones in Gevrey classes.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1696-1728"},"PeriodicalIF":2.4,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142586638","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 : 2024-11-05DOI: 10.1016/j.jde.2024.10.030
Yuanzhen Shao , Gieri Simonett , Mathias Wilke
We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold with boundary. The motion on is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to pure or partial slip boundary conditions of Navier type on . We establish existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. Moreover, we show that the set of equilibria consists of Killing vector fields on that satisfy corresponding boundary conditions, and we prove that all equilibria are (locally) stable. In case is two-dimensional we show that solutions with divergence free initial condition in exist globally and converge to an equilibrium exponentially fast.
我们考虑的是不可压缩粘性流体在有边界的紧凑黎曼流形 M 上的运动。M 上的运动由不可压缩纳维-斯托克斯方程建模，流体受 ∂M 上纳维类型的纯滑移或部分滑移边界条件的影响。我们建立了临界空间中初始数据的强解和弱解（变分法）的存在性和唯一性。此外，我们证明了均衡集由 M 上满足相应边界条件的基林向量场组成，并证明了所有均衡都是（局部）稳定的。在 M 为二维的情况下，我们证明了在 L2(M;TM) 中具有无发散初始条件的解是全局存在的，并且以指数速度收敛到均衡点。
{"title":"The Navier-Stokes equations on manifolds with boundary","authors":"Yuanzhen Shao , Gieri Simonett , Mathias Wilke","doi":"10.1016/j.jde.2024.10.030","DOIUrl":"10.1016/j.jde.2024.10.030","url":null,"abstract":"<div><div>We consider the motion of an incompressible viscous fluid on a compact Riemannian manifold <span><math><mi>M</mi></math></span> with boundary. The motion on <span><math><mi>M</mi></math></span> is modeled by the incompressible Navier-Stokes equations, and the fluid is subject to pure or partial slip boundary conditions of Navier type on <span><math><mo>∂</mo><mi>M</mi></math></span>. We establish existence and uniqueness of strong as well as weak (variational) solutions for initial data in critical spaces. Moreover, we show that the set of equilibria consists of Killing vector fields on <span><math><mi>M</mi></math></span> that satisfy corresponding boundary conditions, and we prove that all equilibria are (locally) stable. In case <span><math><mi>M</mi></math></span> is two-dimensional we show that solutions with divergence free initial condition in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>M</mi><mo>;</mo><mi>T</mi><mi>M</mi><mo>)</mo></math></span> exist globally and converge to an equilibrium exponentially fast.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1602-1659"},"PeriodicalIF":2.4,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587331","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 : 2024-11-05DOI: 10.1016/j.jde.2024.10.043
Bang-Sheng Han, Meng-Xue Chang, Hong-Lei Wei, Yinghui Yang
This paper is concerned with curved fronts for Belousov-Zhabotinskii reaction-diffusion system in external domains with a compact obstacle K and aims to investigate the large time dynamics of an entire solution emanating from a pyramidal traveling wave. By constructing several super- and sub-solutions with desirable characteristics, some favorable properties of the pyramidal traveling wave are obtained. We show that by providing propagation completely of the entire solution, the pyramidal traveling wave will converge to the same shape of the pyramidal traveling wave after far behind the obstacle.
本文关注具有紧凑障碍物 K 的外部域 Ω=RN﹨K 中的贝洛索夫-扎博金斯基反应扩散系统的曲线前沿，旨在研究由金字塔行波发出的整个解的大时间动力学。通过构建几个具有理想特性的超解和子解，我们获得了金字塔行波的一些有利特性。我们证明，通过提供整个解的完全传播，金字塔行波在远离障碍物后将收敛到金字塔行波的相同形状。
{"title":"Curved fronts for a Belousov-Zhabotinskii system in exterior domains","authors":"Bang-Sheng Han, Meng-Xue Chang, Hong-Lei Wei, Yinghui Yang","doi":"10.1016/j.jde.2024.10.043","DOIUrl":"10.1016/j.jde.2024.10.043","url":null,"abstract":"<div><div>This paper is concerned with curved fronts for Belousov-Zhabotinskii reaction-diffusion system in external domains <span><math><mi>Ω</mi><mo>=</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>﹨</mo><mi>K</mi></math></span> with a compact obstacle <em>K</em> and aims to investigate the large time dynamics of an entire solution emanating from a pyramidal traveling wave. By constructing several super- and sub-solutions with desirable characteristics, some favorable properties of the pyramidal traveling wave are obtained. We show that by providing propagation completely of the entire solution, the pyramidal traveling wave will converge to the same shape of the pyramidal traveling wave after far behind the obstacle.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1660-1695"},"PeriodicalIF":2.4,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142587332","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 : 2024-11-04DOI: 10.1016/j.jde.2024.10.023
N. Chernyavskaya , L. Shuster
We consider the boundary problem(1)(2) under the following conditions:
1)
;
2)
equation (1) is correctly solvable in , .
We obtain necessary and sufficient requirements for the functions r and q under which, regardless of the choice of a function , , the solution of equation (1) satisfies (2).
我们考虑以下条件下的边界问题(1)-(r(x)y′(x))′+q(x)y(x)=f(x),x∈R,(2)lim|x|→∞y(x)=0：1)r>0,1r∈L1loc(R),q∈L1loc(R);2)equation (1) is correctly solvable in Lp(R), p∈(1,∞).我们得到了函数 r 和 q 的必要条件和充分条件，在这些条件下，无论选择哪个函数 f∈Lp(R),p∈(1,∞)，方程 (1) 的解 y∈Lp(R) 都满足 (2)。
{"title":"Necessary and sufficient conditions for the solvability of a singular Dirichlet boundary problem for the Sturm-Liouville equation of general form","authors":"N. Chernyavskaya , L. Shuster","doi":"10.1016/j.jde.2024.10.023","DOIUrl":"10.1016/j.jde.2024.10.023","url":null,"abstract":"<div><div>We consider the boundary problem<span><span><span>(1)</span><span><math><mrow><mo>−</mo><msup><mrow><mo>(</mo><mi>r</mi><mo>(</mo><mi>x</mi><mo>)</mo><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>x</mi><mo>)</mo><mo>)</mo></mrow><mrow><mo>′</mo></mrow></msup><mo>+</mo><mi>q</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>x</mi><mo>∈</mo><mi>R</mi><mo>,</mo></mrow></math></span></span></span><span><span><span>(2)</span><span><math><mrow><munder><mi>lim</mi><mrow><mo>|</mo><mi>x</mi><mo>|</mo><mo>→</mo><mo>∞</mo></mrow></munder><mo></mo><mi>y</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><mn>0</mn></mrow></math></span></span></span> under the following conditions:<ul><li><span>1)</span><span><div><span><math><mi>r</mi><mo>></mo><mn>0</mn><mo>,</mo><mspace></mspace><mfrac><mrow><mn>1</mn></mrow><mrow><mi>r</mi></mrow></mfrac><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>loc</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>q</mi><mo>∈</mo><msubsup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>loc</mi></mrow></msubsup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>;</div></span></li><li><span>2)</span><span><div>equation <span><span>(1)</span></span> is correctly solvable in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>.</div></span></li></ul> We obtain necessary and sufficient requirements for the functions <em>r</em> and <em>q</em> under which, regardless of the choice of a function <span><math><mi>f</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, <span><math><mi>p</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, the solution <span><math><mi>y</mi><mo>∈</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>R</mi><mo>)</mo></math></span> of equation <span><span>(1)</span></span> satisfies <span><span>(2)</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1564-1601"},"PeriodicalIF":2.4,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578070","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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