Pub Date : 2024-10-23DOI: 10.1016/j.jde.2024.10.020
Yuehong Zhuang
This paper is concerned with a free boundary problem modeling tumor growth with time-dependent nutrient supply and inhibitor action. We highlight in this paper that the spatial domain occupied by the tumor is set to be n-dimensional for any , and it is taken into account that the nutrient supply and the inhibitor injection on the tumor surface are time-varying in this problem. The high-dimensional setting of the problem makes the proof of the existence of radial stationary solutions and the accurate determination of their numbers highly nontrivial, in which we have developed a new method that is different from the previous work by Cui and Friedman [11]. We can give a complete classification of the radial stationary solutions to this problem under different parameter conditions, and also explore the asymptotic behavior of the transient solution for small in the case that and have finite limits as .
{"title":"Analysis of a high-dimensional free boundary problem on tumor growth with time-dependent nutrient supply and inhibitor action","authors":"Yuehong Zhuang","doi":"10.1016/j.jde.2024.10.020","DOIUrl":"10.1016/j.jde.2024.10.020","url":null,"abstract":"<div><div>This paper is concerned with a free boundary problem modeling tumor growth with time-dependent nutrient supply and inhibitor action. We highlight in this paper that the spatial domain occupied by the tumor is set to be <em>n</em>-dimensional for any <span><math><mi>n</mi><mo>⩾</mo><mn>3</mn></math></span>, and it is taken into account that the nutrient supply <span><math><mi>ϕ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and the inhibitor injection <span><math><mi>ψ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> on the tumor surface are time-varying in this problem. The high-dimensional setting of the problem makes the proof of the existence of radial stationary solutions and the accurate determination of their numbers highly nontrivial, in which we have developed a new method that is different from the previous work by Cui and Friedman <span><span>[11]</span></span>. We can give a complete classification of the radial stationary solutions to this problem under different parameter conditions, and also explore the asymptotic behavior of the transient solution for small <span><math><mi>c</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>c</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> in the case that <span><math><mi>ϕ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> and <span><math><mi>ψ</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> have finite limits as <span><math><mi>t</mi><mo>→</mo><mo>∞</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1222-1259"},"PeriodicalIF":2.4,"publicationDate":"2024-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530866","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 : 2024-10-22DOI: 10.1016/j.jde.2024.10.019
Junying Chen, Ruixiang Xing
In this paper, we study a free boundary problem modeling the growth of 3-dimensional tumor cords. Since tumor cells grow freely in both the longitudinal and cross-sectional directions of blood vessels, the investigation of symmetry-breaking phenomena in both directions is biologically very reasonable. This forces the possible bifurcation value to be dependent on two variables m and n. Some monotonicity properties of the possible bifurcation value or obtained in Friedman and Hu (2008) [1] and He and Xing (2023) [2] no longer hold here, which brings a great challenge to the bifurcation analysis. The novelty of this paper lies in determining the order of for . Together with periodicity and symmetry, we propose an effective method to avoid the need for the monotonicity of . We give symmetry-breaking bifurcation results for every .
{"title":"Symmetry-breaking bifurcation analysis of a free boundary problem modeling 3-dimensional tumor cord growth","authors":"Junying Chen, Ruixiang Xing","doi":"10.1016/j.jde.2024.10.019","DOIUrl":"10.1016/j.jde.2024.10.019","url":null,"abstract":"<div><div>In this paper, we study a free boundary problem modeling the growth of 3-dimensional tumor cords. Since tumor cells grow freely in both the longitudinal and cross-sectional directions of blood vessels, the investigation of symmetry-breaking phenomena in both directions is biologically very reasonable. This forces the possible bifurcation value <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> to be dependent on two variables <em>m</em> and <em>n</em>. Some monotonicity properties of the possible bifurcation value <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> or <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span> obtained in Friedman and Hu (2008) <span><span>[1]</span></span> and He and Xing (2023) <span><span>[2]</span></span> no longer hold here, which brings a great challenge to the bifurcation analysis. The novelty of this paper lies in determining the order of <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> for <span><math><msqrt><mrow><msup><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msqrt></math></span>. Together with periodicity and symmetry, we propose an effective method to avoid the need for the monotonicity of <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. We give symmetry-breaking bifurcation results for every <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>></mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"415 ","pages":"Pages 829-854"},"PeriodicalIF":2.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527657","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 : 2024-10-22DOI: 10.1016/j.jde.2024.10.018
Hai Huang , Xianlong Fu
In this work, by using the theory of fundamental solution and resolvent operators, we investigate the existence of solutions for Bolza optimal control problems for a semi-linear neutral integro-differential equation with infinite delay. It is stressed that both the integral cost functional and the admissible set do not require convexity conditions other than the existing literature. Meanwhile, the existence of time optimal control to a target set is also considered and obtained by limit arguments. Finally, we provide a example to demonstrate the applications of our main results.
{"title":"Nonconvex optimal control problems for semi-linear neutral integro-differential systems with infinite delay","authors":"Hai Huang , Xianlong Fu","doi":"10.1016/j.jde.2024.10.018","DOIUrl":"10.1016/j.jde.2024.10.018","url":null,"abstract":"<div><div>In this work, by using the theory of fundamental solution and resolvent operators, we investigate the existence of solutions for Bolza optimal control problems for a semi-linear neutral integro-differential equation with infinite delay. It is stressed that both the integral cost functional and the admissible set do not require convexity conditions other than the existing literature. Meanwhile, the existence of time optimal control to a target set is also considered and obtained by limit arguments. Finally, we provide a example to demonstrate the applications of our main results.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1137-1177"},"PeriodicalIF":2.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530864","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 : 2024-10-22DOI: 10.1016/j.jde.2024.10.017
Diego Maldonado
Harnack inequalities for nonnegative strong solutions to nondivergence-form elliptic PDEs with degeneracies or singularities of logarithmic type are proved. The results are developed within the Monge-Ampère real-analytic and geometric tools associated to certain convex functions.
{"title":"On certain degenerate and singular elliptic PDEs IV: Nondivergence-form operators with logarithmic degeneracies or singularities","authors":"Diego Maldonado","doi":"10.1016/j.jde.2024.10.017","DOIUrl":"10.1016/j.jde.2024.10.017","url":null,"abstract":"<div><div>Harnack inequalities for nonnegative strong solutions to nondivergence-form elliptic PDEs with degeneracies or singularities of logarithmic type are proved. The results are developed within the Monge-Ampère real-analytic and geometric tools associated to certain convex functions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1099-1136"},"PeriodicalIF":2.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530875","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 : 2024-10-22DOI: 10.1016/j.jde.2024.10.024
Yan-long Fang , Alden Waters
The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in operator norm in terms of time t and powers of h. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem.
本文的目的是为麦克斯韦方程在完全导电球外部的高频色散估计建立一般原则。我们为相应的麦克斯韦传播子构建了全新的广义特征函数。我们证明了与电场相对应的传播子在 L1-L∞ 算子规范中具有以时间 t 和 h 的幂为单位的全局衰减率。由于亥姆霍兹分解定理的存在，对于球外部的麦克斯韦方程，一般来说，迪里夏特-拉普拉斯波算子 L1-L∞ 规范估计值不应成立。
{"title":"Dispersive estimates for Maxwell's equations in the exterior of a sphere","authors":"Yan-long Fang , Alden Waters","doi":"10.1016/j.jde.2024.10.024","DOIUrl":"10.1016/j.jde.2024.10.024","url":null,"abstract":"<div><div>The goal of this article is to establish general principles for high frequency dispersive estimates for Maxwell's equation in the exterior of a perfectly conducting ball. We construct entirely new generalized eigenfunctions for the corresponding Maxwell propagator. We show that the propagator corresponding to the electric field has a global rate of decay in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> operator norm in terms of time <em>t</em> and powers of <em>h</em>. In particular we show that some, but not all, polarizations of electromagnetic waves scatter at the same rate as the usual wave operator. The Dirichlet Laplacian wave operator <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>−</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span> norm estimate should not be expected to hold in general for Maxwell's equations in the exterior of a ball because of the Helmholtz decomposition theorem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"415 ","pages":"Pages 855-885"},"PeriodicalIF":2.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142527658","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 : 2024-10-22DOI: 10.1016/j.jde.2024.10.022
Mohammad Akil , Genni Fragnelli , Ibtissam Issa
In this paper we study the stability of two different problems. The first one is a one-dimensional degenerate wave equation with degenerate damping, incorporating a drift term and a leading operator in non-divergence form. In the second problem we consider a system that couples degenerate and non-degenerate wave equations, connected through transmission, and subject to a single dissipation law at the boundary of the non-degenerate equation. In both scenarios, we derive exponential stability results.
{"title":"Stability for degenerate wave equations with drift under simultaneous degenerate damping","authors":"Mohammad Akil , Genni Fragnelli , Ibtissam Issa","doi":"10.1016/j.jde.2024.10.022","DOIUrl":"10.1016/j.jde.2024.10.022","url":null,"abstract":"<div><div>In this paper we study the stability of two different problems. The first one is a one-dimensional degenerate wave equation with degenerate damping, incorporating a drift term and a leading operator in non-divergence form. In the second problem we consider a system that couples degenerate and non-degenerate wave equations, connected through transmission, and subject to a single dissipation law at the boundary of the non-degenerate equation. In both scenarios, we derive exponential stability results.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1178-1221"},"PeriodicalIF":2.4,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530865","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 : 2024-10-18DOI: 10.1016/j.jde.2024.10.012
Fei Wang , Yichun Zhu
We give a detailed description of formation of the boundary layers in the inviscid limit problem. To be more specific, we prove that the magnitude of the vorticity near the boundary is growing to the size of and the width of the layer is spreading out to be proportional the in a finite time period. In fact, the growth time scaling is almost ν.
{"title":"The growth mechanism of boundary layers for the 2D Navier-Stokes equations","authors":"Fei Wang , Yichun Zhu","doi":"10.1016/j.jde.2024.10.012","DOIUrl":"10.1016/j.jde.2024.10.012","url":null,"abstract":"<div><div>We give a detailed description of formation of the boundary layers in the inviscid limit problem. To be more specific, we prove that the magnitude of the vorticity near the boundary is growing to the size of <span><math><mn>1</mn><mo>/</mo><msqrt><mrow><mi>ν</mi></mrow></msqrt></math></span> and the width of the layer is spreading out to be proportional the <span><math><msqrt><mrow><mi>ν</mi></mrow></msqrt></math></span> in a finite time period. In fact, the growth time scaling is almost <em>ν</em>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 973-1014"},"PeriodicalIF":2.4,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530869","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 : 2024-10-18DOI: 10.1016/j.jde.2024.10.011
Yupei Li, Wei Luo
In this paper, we investigate the asymptotic behavior of solutions to the axisymmetric stationary Navier-Stokes equations. We assume that the flow is periodic in -direction and has no swirl. Under the general integrability condition, we prove the pointwise decay estimate of the vorticity ω and obtain the Liouville-type theorem.
{"title":"Asymptotic behavior for stationary Navier-Stokes equations","authors":"Yupei Li, Wei Luo","doi":"10.1016/j.jde.2024.10.011","DOIUrl":"10.1016/j.jde.2024.10.011","url":null,"abstract":"<div><div>In this paper, we investigate the asymptotic behavior of solutions to the axisymmetric stationary Navier-Stokes equations. We assume that the flow is periodic in <span><math><msub><mrow><mi>x</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span>-direction and has no swirl. Under the general integrability condition, we prove the pointwise decay estimate of the vorticity <em>ω</em> and obtain the Liouville-type theorem.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1052-1070"},"PeriodicalIF":2.4,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530871","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 : 2024-10-18DOI: 10.1016/j.jde.2024.10.013
M.J. Álvarez , B. Coll , A. Gasull , R. Prohens
In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, , being non-negative integers and . More concretely, if and denotes the maximum number of limit cycles of the above equations, we show that for , and that for some values of N this new lower bound is . We also present examples with many limit cycles and different configurations. Finally, we show that and study in detail the quadratic case with three monomials proving in some of them non-existence, uniqueness or existence of exactly two limit cycles.
{"title":"More limit cycles for complex differential equations with three monomials","authors":"M.J. Álvarez , B. Coll , A. Gasull , R. Prohens","doi":"10.1016/j.jde.2024.10.013","DOIUrl":"10.1016/j.jde.2024.10.013","url":null,"abstract":"<div><div>In this paper we improve, by almost doubling, the existing lower bound for the number of limit cycles of the family of complex differential equations with three monomials, <span><math><mover><mrow><mi>z</mi></mrow><mrow><mo>˙</mo></mrow></mover><mo>=</mo><mi>A</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>k</mi></mrow></msup><msup><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>l</mi></mrow></msup><mo>+</mo><mi>B</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>n</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mrow><mi>z</mi></mrow><mrow><mi>p</mi></mrow></msup><msup><mrow><mover><mrow><mi>z</mi></mrow><mrow><mo>¯</mo></mrow></mover></mrow><mrow><mi>q</mi></mrow></msup></math></span>, being <span><math><mi>k</mi><mo>,</mo><mi>l</mi><mo>,</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi></math></span> non-negative integers and <span><math><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><mi>C</mi></math></span>. More concretely, if <span><math><mi>N</mi><mo>=</mo><mi>max</mi><mo></mo><mo>(</mo><mi>k</mi><mo>+</mo><mi>l</mi><mo>,</mo><mi>m</mi><mo>+</mo><mi>n</mi><mo>,</mo><mi>p</mi><mo>+</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>∈</mo><mi>N</mi><mo>∪</mo><mo>{</mo><mo>∞</mo><mo>}</mo></math></span> denotes the maximum number of limit cycles of the above equations, we show that for <span><math><mi>N</mi><mo>≥</mo><mn>4</mn></math></span>, <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mi>N</mi><mo>)</mo><mo>≥</mo><mi>N</mi><mo>−</mo><mn>3</mn></math></span> and that for some values of <em>N</em> this new lower bound is <span><math><mi>N</mi><mo>+</mo><mn>1</mn></math></span>. We also present examples with many limit cycles and different configurations. Finally, we show that <span><math><msub><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>(</mo><mn>2</mn><mo>)</mo><mo>≥</mo><mn>2</mn></math></span> and study in detail the quadratic case with three monomials proving in some of them non-existence, uniqueness or existence of exactly two limit cycles.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1071-1098"},"PeriodicalIF":2.4,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530874","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 : 2024-10-18DOI: 10.1016/j.jde.2024.09.042
Noureddine Igbida
Our aim is to study existence, uniqueness and the limit, as , of the solution of the porous medium equation with linear drift in bounded domain with Dirichlet boundary condition. We treat the problem without any sign restriction on the solution with an outpointing vector field V on the boundary and a general source term g (including the continuous Lipschitz case). Under reasonably sharp Sobolev assumptions on V, we show uniform -convergence towards the solution of reaction-diffusion Hele-Shaw flow with linear drift.
我们的目的是研究具有线性漂移的多孔介质方程 ∂tu-Δum+∇⋅(uV)=g(t,x,u) 的解的存在性、唯一性以及 m→∞ 时的极限。我们在处理这个问题时，不对解作任何符号限制，在边界上有一个外指向向量场 V 和一个一般源项 g（包括连续 Lipschitz 情况）。在 V 的合理尖锐 Sobolev 假设下，我们展示了对具有线性漂移的反应扩散 Hele-Shaw 流解的均匀 L1 收敛性。
{"title":"L1-theory for incompressible limit of reaction-diffusion porous medium flow with linear drift","authors":"Noureddine Igbida","doi":"10.1016/j.jde.2024.09.042","DOIUrl":"10.1016/j.jde.2024.09.042","url":null,"abstract":"<div><div>Our aim is to study existence, uniqueness and the limit, as <span><math><mi>m</mi><mo>→</mo><mo>∞</mo></math></span>, of the solution of the porous medium equation with linear drift <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mi>Δ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>+</mo><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mspace></mspace><mi>V</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>u</mi><mo>)</mo></math></span> in bounded domain with Dirichlet boundary condition. We treat the problem without any sign restriction on the solution with an outpointing vector field <em>V</em> on the boundary and a general source term <em>g</em> (including the continuous Lipschitz case). Under reasonably sharp Sobolev assumptions on <em>V</em>, we show uniform <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span>-convergence towards the solution of reaction-diffusion Hele-Shaw flow with linear drift.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"416 ","pages":"Pages 1015-1051"},"PeriodicalIF":2.4,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530870","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}
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