Journal of Differential Equations最新文献

{"title":"Blowup dynamics for the mass critical half-wave equation in 2D","authors":"Vladimir Georgiev ,&nbsp;Yuan Li","doi":"10.1016/j.jde.2024.10.031","DOIUrl":"10.1016/j.jde.2024.10.031","url":null,"abstract":"<div><div>We consider the two-dimensional half-wave equation <span><math><mi>i</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>D</mi><mi>u</mi><mo>−</mo><mo>|</mo><mi>u</mi><mo>|</mo><mi>u</mi></math></span>. For the initial data <span><math><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo><mo>∈</mo><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>, <span><math><mi>s</mi><mo>∈</mo><mrow><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>, we obtain the existence of non-radial ground state mass blow-up solutions with the blow-up rate <span><math><msub><mrow><mo>‖</mo><msup><mrow><mi>D</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>∼</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo>|</mo><mi>t</mi><mo>|</mo></mrow></mfrac></math></span> as <span><math><mi>t</mi><mo>→</mo><msup><mrow><mn>0</mn></mrow><mrow><mo>−</mo></mrow></msup></math></span>. This work extends the recent study by Georgiev and Li (2022) <span><span>[9]</span></span>, which focused on constructing radial ground state mass blow-up solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560870","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}

{"title":"Boundedness in a chemotaxis system with weak singular sensitivity and logistic kinetics in any dimensional setting","authors":"Halil Ibrahim Kurt","doi":"10.1016/j.jde.2024.10.027","DOIUrl":"10.1016/j.jde.2024.10.027","url":null,"abstract":"&lt;div&gt;&lt;div&gt;This paper deals with the following parabolic-elliptic chemotaxis competition system with weak singular sensitivity and logistic source&lt;span&gt;&lt;span&gt;&lt;span&gt;(0.1)&lt;/span&gt;&lt;span&gt;&lt;math&gt;&lt;mrow&gt;&lt;mo&gt;{&lt;/mo&gt;&lt;mtable&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;t&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mo&gt;⋅&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mi&gt;∇&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;Δ&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;−&lt;/mo&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;mtr&gt;&lt;mtd&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;ν&lt;/mi&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;/mtd&gt;&lt;mtd&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;∂&lt;/mo&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;/mtd&gt;&lt;/mtr&gt;&lt;/mtable&gt;&lt;/mrow&gt;&lt;/math&gt;&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; where &lt;span&gt;&lt;math&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;mo&gt;⊂&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;R&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;≥&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt; is a smooth bounded domain, the parameters &lt;span&gt;&lt;math&gt;&lt;mi&gt;χ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;r&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;α&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mspace&gt;&lt;/mspace&gt;&lt;mi&gt;β&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt; are positive constants and &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;.&lt;/div&gt;&lt;div&gt;It is well known that for parabolic-elliptic chemotaxis systems including singularity, a uniform-in-time positive pointwise lower bound for &lt;em&gt;v&lt;/em&gt; is vitally important for establishing the global boundedness of classical solutions since the cross-diffusive term becomes unbounded near &lt;span&gt;&lt;math&gt;&lt;mi&gt;v&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;. To this end, a key step in the literature is to establish a proper positive lower bound for the mass functional &lt;span&gt;&lt;math&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mo&gt;∫&lt;/mo&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;Ω&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mi&gt;u&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;, which, due to the presence of logistic kinetics, is not preserved and hence it turns in for &lt;em&gt;v&lt;/em&gt;. In contrast to this approach, in this article, the boundedness of classical solutions of (0.1) is obtained without using the uniformly positive lower bound of &lt;em&gt;v&lt;/em&gt;.&lt;/div&gt;&lt;div&gt;Among others, it has been proven that without establishing a uniform-in-time positive pointwise lower bound for &lt;em&gt;v&lt;/em&gt;, if &lt;span&gt;&lt;math&gt;&lt;mi&gt;λ&lt;/mi&gt;&lt;mo&gt;∈&lt;/mo&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;, then there exists &lt;span&gt;&lt;math&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;μ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mo&gt;⁎&lt;/mo&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt; such that for all suitably smooth in","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552500","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}

{"title":"On the propagation of flatness for second order hypoelliptic operators","authors":"Paolo Albano","doi":"10.1016/j.jde.2024.10.034","DOIUrl":"10.1016/j.jde.2024.10.034","url":null,"abstract":"<div><div>For a class of hypoelliptic operators with real-analytic coefficients, we provide a criterion ensuring a partial analyticity result. As a consequence, even when the “elliptic” strong unique continuation (i.e. a solution of the homogeneous equation which vanishes of infinite order at a point is zero near such a point) fails, a weaker form of “propagation” of zeroes still holds.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552498","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}

{"title":"Minimal P-cyclic periodic brake orbits in semi-positive Hamiltonian system","authors":"Ruiliang Gao ,&nbsp;Xiaorui Li ,&nbsp;Duanzhi Zhang","doi":"10.1016/j.jde.2024.10.021","DOIUrl":"10.1016/j.jde.2024.10.021","url":null,"abstract":"<div><div>In this paper, the authors study the minimal period problems for brake orbits in two types of first order Hamiltonian systems in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>, namely the superquadratic type and asymptotically linear type. In both cases the Hamiltonian systems are assumed to be reversible, semi-positive, and symmetric with respect to certain orthogonal symplectic linear transformation <em>P</em> generating a <em>p</em>-order cyclic subgroup acting freely on <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo></math></span>. The authors prove that if <span><math><mi>P</mi><mo>=</mo><mi>R</mi><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo><mo>⋄</mo><mo>…</mo><mo>⋄</mo><mi>R</mi><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>π</mi><mo>]</mo></math></span> for each <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math></span>, then for each <span><math><mi>T</mi><mo>&gt;</mo><mn>0</mn></math></span> there exists a <em>pT</em>-periodic <em>P</em>-cyclic brake orbit with minimal period belonging to an finite set with the form<span><span><span><math><mrow><mo>{</mo><mfrac><mrow><mi>p</mi><mi>T</mi></mrow><mrow><mi>l</mi><mi>p</mi><mo>+</mo><mi>q</mi></mrow></mfrac><mo>:</mo><mi>l</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi>Z</mi><mo>,</mo><mspace></mspace><mn>0</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>,</mo><mspace></mspace><mrow><mi>gcd</mi></mrow><mo>(</mo><mi>q</mi><mo>,</mo><mi>p</mi><mo>)</mo><mo>=</mo><mn>1</mn><mo>,</mo><mspace></mspace><mn>1</mn><mo>≤</mo><mi>q</mi><mo>≤</mo><mi>p</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></math></span></span></span> for both cases, which is an generalization of the results in <span><span>[10]</span></span>. The main tools involved are the iteration inequalities for Maslov-type indices, the saddle point reduction method and the Galerkin approximation method under the corresponding Lagrangian boundary condition.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552497","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}

{"title":"Orbital stability of smooth solitons in H1 ∩ W1,4 for the modified Camassa-Holm equation","authors":"Qian Zhang,&nbsp;Guangming Zhu","doi":"10.1016/j.jde.2024.10.032","DOIUrl":"10.1016/j.jde.2024.10.032","url":null,"abstract":"<div><div>We analyze the stability of smooth solitary waves in the modified Camassa-Holm equation, a quasilinear, integrable model for shallow water wave propagation. Through phase portrait analysis, we identify a unique smooth solitary wave within a certain range of the dispersive parameter. Using variational methods, we prove the orbital stability of this wave under small disturbances, solving a minimization problem with constraints. We strengthen the <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>∩</mo><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>4</mn></mrow></msup></math></span> stability result in Li and Liu (2021) <span><span>[8]</span></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552499","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}

{"title":"The optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations","authors":"Yanlin Liu","doi":"10.1016/j.jde.2024.10.026","DOIUrl":"10.1016/j.jde.2024.10.026","url":null,"abstract":"<div><div>In this paper, we derive the optimal time-decay estimates for 2-D inhomogeneous Navier-Stokes equations. In particular, we prove that <span><math><msub><mrow><mo>‖</mo><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msubsup><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mi>p</mi><mo>,</mo><mn>1</mn></mrow><mrow><mi>θ</mi></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow></msub><mo>=</mo><mi>O</mi><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>p</mi></mrow></mfrac><mo>−</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mfrac><mrow><mi>θ</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></math></span> as <span><math><mi>t</mi><mo>→</mo><mo>∞</mo></math></span> for any <span><math><mi>p</mi><mo>∈</mo><mo>[</mo><mn>2</mn><mo>,</mo><mo>∞</mo><mo>[</mo><mo>,</mo><mspace></mspace><mi>θ</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span> if initially <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>0</mn></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><msubsup><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>˙</mo></mrow></mover></mrow><mrow><mn>2</mn><mo>,</mo><mo>∞</mo></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msubsup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. This is optimal even for the classical homogeneous Navier-Stokes equations. Different with Schonbek and Wiegner's Fourier splitting device, our method here seems more direct, and can adapt to many other equations as well. Moreover, our method allows us to work in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-based spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530867","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}

{"title":"Sequential stability of weak martingale solutions to stochastic compressible Navier-Stokes equations with viscosity vanishing on vacuum","authors":"Zdzisław Brzeźniak ,&nbsp;Gaurav Dhariwal ,&nbsp;Ewelina Zatorska","doi":"10.1016/j.jde.2024.10.016","DOIUrl":"10.1016/j.jde.2024.10.016","url":null,"abstract":"<div><div>In this paper, we investigate the compressible Navier-Stokes equations with degenerate, density-dependent, viscosity coefficient driven by multiplicative stochastic noise. We consider three-dimensional periodic domain and prove that the family of weak martingale solutions is sequentially compact.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530868","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}

{"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":null,"pages":null},"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}

{"title":"Nonconvex optimal control problems for semi-linear neutral integro-differential systems with infinite delay","authors":"Hai Huang ,&nbsp;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":null,"pages":null},"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}

{"title":"Symmetry-breaking bifurcation analysis of a free boundary problem modeling 3-dimensional tumor cord growth","authors":"Junying Chen,&nbsp;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>&gt;</mo><mn>0</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"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}