Pub Date : 2024-09-25DOI: 10.1016/j.jde.2024.09.031
Xijun Deng , Stéphane Lafortune
In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter b. This family is called the “b-Novikov” since it reduces to the integrable Novikov equation in the case . By extending the corresponding linearized operator defined on functions in to one defined on weaker functions on , we prove spectral and linear instability on of peakons in the b-Novikov equations for any b. We also consider the stability on and show that the peakons are spectrally or linearly stable only in the case .
在本文中，我们关注的是具有立方非线性参数的峰值子方程的一参数族，其参数通常用字母 b 表示。这个族被称为 "b-Novikov"，因为它在 b=3 的情况下简化为可积分的 Novikov 方程。通过将定义在 H1(R) 中函数上的相应线性化算子扩展到定义在 L2(R) 上较弱函数上的算子，我们证明了 b-Novikov 方程中任何 b 的峰值子在 L2(R) 上的谱和线性不稳定性。
{"title":"Spectral instability of peakons for the b-family of Novikov equations","authors":"Xijun Deng , Stéphane Lafortune","doi":"10.1016/j.jde.2024.09.031","DOIUrl":"10.1016/j.jde.2024.09.031","url":null,"abstract":"<div><div>In this paper, we are concerned with a one-parameter family of peakon equations with cubic nonlinearity parametrized by a parameter usually denoted by the letter <em>b</em>. This family is called the “<em>b</em>-Novikov” since it reduces to the integrable Novikov equation in the case <span><math><mi>b</mi><mo>=</mo><mn>3</mn></math></span>. By extending the corresponding linearized operator defined on functions in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> to one defined on weaker functions on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>, we prove spectral and linear instability on <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> of peakons in the <em>b</em>-Novikov equations for any <em>b</em>. We also consider the stability on <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span> and show that the peakons are spectrally or linearly stable only in the case <span><math><mi>b</mi><mo>=</mo><mn>3</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142320377","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 : 2024-09-24DOI: 10.1016/j.jde.2024.09.033
Jaehoon Kang , Daehan Park
<div><div>In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>a</mi></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn></math></span></span></span> in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> spaces. Our spatial operator <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>a</mi></mrow></msup></math></span> is an integro-differential operator of the form<span><span><span><math><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></munder><mrow><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>⋅</mo><mi>y</mi><msub><mrow><mn>1</mn></mrow><mrow><mo>|</mo><mi>y</mi><mo>|</mo><mo>≤</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow><mi>a</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>y</mi><mo>)</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mo>|</mo><mi>y</mi><mo>|</mo><mo>)</mo><mi>d</mi><mi>y</mi><mo>.</mo></math></span></span></span> Here, <span><math><mi>a</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo></math></span> which yield <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>-regularity of solutions. Our assumptions on <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> are general so that <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo></math></span> may be comparable to <span><math><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup><mi>ℓ</mi><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> for a function <em>ℓ</em> which is slowly varying at infinity. For example, we can take <span><math><mi>ℓ</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mi>log</mi><mo></mo><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>)</mo></math></span> or <span><math><mi>ℓ</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>α<
{"title":"An Lq(Lp)-regularity theory for parabolic equations with integro-differential operators having low intensity kernels","authors":"Jaehoon Kang , Daehan Park","doi":"10.1016/j.jde.2024.09.033","DOIUrl":"10.1016/j.jde.2024.09.033","url":null,"abstract":"<div><div>In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>a</mi></mrow></msup><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn></math></span></span></span> in <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span> spaces. Our spatial operator <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>a</mi></mrow></msup></math></span> is an integro-differential operator of the form<span><span><span><math><munder><mo>∫</mo><mrow><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></mrow></munder><mrow><mo>(</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>)</mo><mo>−</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>−</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>⋅</mo><mi>y</mi><msub><mrow><mn>1</mn></mrow><mrow><mo>|</mo><mi>y</mi><mo>|</mo><mo>≤</mo><mn>1</mn></mrow></msub><mo>)</mo></mrow><mi>a</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>y</mi><mo>)</mo><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mo>|</mo><mi>y</mi><mo>|</mo><mo>)</mo><mi>d</mi><mi>y</mi><mo>.</mo></math></span></span></span> Here, <span><math><mi>a</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo></math></span> which yield <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>(</mo><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>)</mo></math></span>-regularity of solutions. Our assumptions on <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span> are general so that <span><math><msub><mrow><mi>j</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>(</mo><mi>r</mi><mo>)</mo></math></span> may be comparable to <span><math><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mi>d</mi></mrow></msup><mi>ℓ</mi><mo>(</mo><msup><mrow><mi>r</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math></span> for a function <em>ℓ</em> which is slowly varying at infinity. For example, we can take <span><math><mi>ℓ</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mi>log</mi><mo></mo><mo>(</mo><mn>1</mn><mo>+</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>α</mi></mrow></msup><mo>)</mo></math></span> or <span><math><mi>ℓ</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><msup><mrow><mi>r</mi></mrow><mrow><mi>α<","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315770","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 : 2024-09-24DOI: 10.1016/j.jde.2024.09.039
Lin Shi
In this paper, we investigate a class of stochastic semilinear parabolic equations subjected to multiplicative noise within singularly perturbed phase spaces. We first establish the existence and smoothness of stable foliations. Then we prove that the long-term behavior of each solution is determined by a solution residing on the pseudo-unstable manifold via a leaf of the stable foliation. Finally, we present the convergence of invariant foliations as the high dimensional region collapse to low dimensional region. In contrast to the convergence of pseudo-unstable manifolds, we introduce a novel technique to address challenges arising from the singularity of the stable term of hyperbolicity in the proof of convergence of stable manifolds and stable foliations as the space collapses.
{"title":"Limiting behavior of invariant foliations for SPDEs in singularly perturbed spaces","authors":"Lin Shi","doi":"10.1016/j.jde.2024.09.039","DOIUrl":"10.1016/j.jde.2024.09.039","url":null,"abstract":"<div><div>In this paper, we investigate a class of stochastic semilinear parabolic equations subjected to multiplicative noise within singularly perturbed phase spaces. We first establish the existence and smoothness of stable foliations. Then we prove that the long-term behavior of each solution is determined by a solution residing on the pseudo-unstable manifold via a leaf of the stable foliation. Finally, we present the convergence of <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup></math></span> invariant foliations as the high dimensional region collapse to low dimensional region. In contrast to the convergence of pseudo-unstable manifolds, we introduce a novel technique to address challenges arising from the singularity of the stable term of hyperbolicity in the proof of convergence of stable manifolds and stable foliations as the space collapses.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315771","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 : 2024-09-24DOI: 10.1016/j.jde.2024.09.036
Yinghua Li, Manrou Xie, Yuanxiang Yan
This paper is concerned with a simplified model for two-phase fluids with diffuse interface. The model couples the nonhomogeneous incompressible Navier-Stokes equations with the Allen-Cahn equation. The viscosity coefficient is allowed to depend both on the phase variable and on the density. Under some smallness assumptions on initial data, the global existence of unique strong solutions to the 3D Cauchy problem and the initial boundary value problem is established. Meanwhile, we obtain the exponential decay-in-time properties of the solutions. Here, the initial vacuum is allowed and no compatibility conditions are required for the initial data via time weighted techniques.
{"title":"Exponential stability of a diffuse interface model of incompressible two-phase flow with phase variable dependent viscosity and vacuum","authors":"Yinghua Li, Manrou Xie, Yuanxiang Yan","doi":"10.1016/j.jde.2024.09.036","DOIUrl":"10.1016/j.jde.2024.09.036","url":null,"abstract":"<div><div>This paper is concerned with a simplified model for two-phase fluids with diffuse interface. The model couples the nonhomogeneous incompressible Navier-Stokes equations with the Allen-Cahn equation. The viscosity coefficient is allowed to depend both on the phase variable and on the density. Under some smallness assumptions on initial data, the global existence of unique strong solutions to the 3D Cauchy problem and the initial boundary value problem is established. Meanwhile, we obtain the exponential decay-in-time properties of the solutions. Here, the initial vacuum is allowed and no compatibility conditions are required for the initial data via time weighted techniques.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142314634","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 : 2024-09-24DOI: 10.1016/j.jde.2024.09.032
Xuejun Pan , Hongying Shu , Lin Wang , Xiang-Sheng Wang , Jianshe Yu
In this paper, we investigate the existence and stability of periodic solutions of switching dynamical systems consisting of two sub-equations. We first establish a general criterion to determine the stability of periodic solutions; namely, we derive the conditions under which the periodic solution is locally asymptotically stable, globally asymptotically stable, or unstable. Next, we develop general theorems to count the number of periodic solutions and find the basins of attractions for the periodic solutions and the trivial solution, respectively. As applications, we analyze two biological models in recent literature. Our general theorems not only reproduce the existing results in a unified and simpler manner but also lead to new and complete dynamical results including bistability of the periodic solution and the trivial solution. Numerical examples are also given to illustrate our theoretical results.
{"title":"On the periodic solutions of switching scalar dynamical systems","authors":"Xuejun Pan , Hongying Shu , Lin Wang , Xiang-Sheng Wang , Jianshe Yu","doi":"10.1016/j.jde.2024.09.032","DOIUrl":"10.1016/j.jde.2024.09.032","url":null,"abstract":"<div><div>In this paper, we investigate the existence and stability of periodic solutions of switching dynamical systems consisting of two sub-equations. We first establish a general criterion to determine the stability of periodic solutions; namely, we derive the conditions under which the periodic solution is locally asymptotically stable, globally asymptotically stable, or unstable. Next, we develop general theorems to count the number of periodic solutions and find the basins of attractions for the periodic solutions and the trivial solution, respectively. As applications, we analyze two biological models in recent literature. Our general theorems not only reproduce the existing results in a unified and simpler manner but also lead to new and complete dynamical results including bistability of the periodic solution and the trivial solution. Numerical examples are also given to illustrate our theoretical results.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315767","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 : 2024-09-24DOI: 10.1016/j.jde.2024.09.037
Duanzhi Zhang , Zhihao Zhao
This paper introduces a novel iteration inequality for the Maslov-type -index of iterated symplectic paths. Here, P is a fixed 2n-dimensional symplectic and orthogonal matrix satisfying . These advancements in index theory are then applied to investigate the multiplicity of subharmonic solutions in Hamiltonian systems exhibiting dihedral equivariance with period mτ. Notably, a criterion of geometric distinction is established for two subharmonic P-symmetric brake orbits with periods kmτ and lmτ within the set . This criterion is based on a lower bound estimate for the ratio . Specifically, for odd k, the lower bound must be not less than , while for even k, it must be not less than .
{"title":"Maslov-type (L,P)-index and subharmonic P-symmetric brake solutions for Hamiltonian systems","authors":"Duanzhi Zhang , Zhihao Zhao","doi":"10.1016/j.jde.2024.09.037","DOIUrl":"10.1016/j.jde.2024.09.037","url":null,"abstract":"<div><div>This paper introduces a novel iteration inequality for the Maslov-type <span><math><mo>(</mo><mi>L</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span>-index of iterated symplectic paths. Here, <em>P</em> is a fixed 2<em>n</em>-dimensional symplectic and orthogonal matrix satisfying <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>=</mo><mi>I</mi></math></span>. These advancements in index theory are then applied to investigate the multiplicity of subharmonic solutions in Hamiltonian systems exhibiting dihedral equivariance with period <em>mτ</em>. Notably, a criterion of geometric distinction is established for two subharmonic <em>P</em>-symmetric brake orbits with periods <em>kmτ</em> and <em>lmτ</em> within the set <span><math><mo>{</mo><mi>k</mi><mi>m</mi><mi>τ</mi><mspace></mspace><mo>|</mo><mspace></mspace><mi>k</mi><mo>≡</mo><mn>1</mn><mtext> (mod </mtext><mi>m</mi><mo>)</mo><mo>}</mo></math></span>. This criterion is based on a lower bound estimate for the ratio <span><math><mi>l</mi><mo>/</mo><mi>k</mi></math></span>. Specifically, for odd <em>k</em>, the lower bound must be not less than <span><math><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>dim</mi><mo></mo><mi>ker</mi><mo></mo><mo>(</mo><mi>P</mi><mo>−</mo><mi>I</mi><mo>)</mo><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mi>m</mi><mo>+</mo><mn>1</mn></math></span>, while for even <em>k</em>, it must be not less than <span><math><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mi>dim</mi><mo></mo><mi>ker</mi><mo></mo><mo>(</mo><mi>P</mi><mo>−</mo><mi>I</mi><mo>)</mo><mo>+</mo><mi>n</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mi>m</mi><mo>+</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315768","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 : 2024-09-24DOI: 10.1016/j.jde.2024.09.034
Antonín Češík , Sebastian Schwarzacher
We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much larger) for acceleration. The main applications are from elastodynamics, namely so-called generalized solids, undergoing large deformations. The evolution follows an underlying variational structure exploited by step-wise minimization. We show for a large family of (elastic) energies that the introduced scheme is stable; allowing for non-linearities of highest order. If the highest order can be assumed to be linear, we show that the limit solutions are regular and that the minimizing movements scheme converges with optimal linear rate. Thus this work extends numerical time-step minimization methods to the realm of hyperbolic problems.
{"title":"Stability and convergence of in time approximations of hyperbolic elastodynamics via stepwise minimization","authors":"Antonín Češík , Sebastian Schwarzacher","doi":"10.1016/j.jde.2024.09.034","DOIUrl":"10.1016/j.jde.2024.09.034","url":null,"abstract":"<div><div>We study step-wise time approximations of non-linear hyperbolic initial value problems. The technique used here is a generalization of the minimizing movements method, using two time-scales: one for velocity, the other (potentially much larger) for acceleration. The main applications are from elastodynamics, namely so-called generalized solids, undergoing large deformations. The evolution follows an underlying variational structure exploited by step-wise minimization. We show for a large family of (elastic) energies that the introduced scheme is stable; allowing for non-linearities of highest order. If the highest order can be assumed to be linear, we show that the limit solutions are regular and that the minimizing movements scheme converges with optimal linear rate. Thus this work extends numerical time-step minimization methods to the realm of hyperbolic problems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142315769","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 : 2024-09-23DOI: 10.1016/j.jde.2024.09.035
Qing Li , Yanyan Zhang
This study examines nonnegative solutions to the problem where , , and are constants. The possible asymptotic behaviors of at and are classified according to . In particular, the results show that for some , exhibits only “isotropic” behavior at and . However, in other cases, may exhibit the “anisotropic” behavior at or . Furthermore, the relation between the limit at and the limit at for a global solution is investigated.
{"title":"Study on the behaviors of rupture solutions for a class of elliptic MEMS equations in R2","authors":"Qing Li , Yanyan Zhang","doi":"10.1016/j.jde.2024.09.035","DOIUrl":"10.1016/j.jde.2024.09.035","url":null,"abstract":"<div><div>This study examines nonnegative solutions to the problem<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><mi>Δ</mi><mi>u</mi><mo>=</mo><mfrac><mrow><mi>λ</mi><mo>|</mo><mi>x</mi><msup><mrow><mo>|</mo></mrow><mrow><mi>α</mi></mrow></msup></mrow><mrow><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup></mrow></mfrac><mspace></mspace></mtd><mtd><mtext> in</mtext><mspace></mspace><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn><mspace></mspace><mtext>and</mtext><mspace></mspace><mi>u</mi><mo>></mo><mn>0</mn><mspace></mspace></mtd><mtd><mtext> in</mtext><mspace></mspace><mspace></mspace><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>∖</mo><mo>{</mo><mn>0</mn><mo>}</mo><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <span><math><mi>λ</mi><mo>></mo><mn>0</mn></math></span>, <span><math><mi>α</mi><mo>></mo><mo>−</mo><mn>2</mn></math></span>, and <span><math><mi>p</mi><mo>></mo><mn>0</mn></math></span> are constants. The possible asymptotic behaviors of <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> at <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mn>0</mn></math></span> and <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mo>∞</mo></math></span> are classified according to <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>. In particular, the results show that for some <span><math><mo>(</mo><mi>α</mi><mo>,</mo><mi>p</mi><mo>)</mo></math></span>, <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> exhibits only “isotropic” behavior at <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mn>0</mn></math></span> and <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mo>∞</mo></math></span>. However, in other cases, <span><math><mi>u</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span> may exhibit the “anisotropic” behavior at <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mn>0</mn></math></span> or <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mo>∞</mo></math></span>. Furthermore, the relation between the limit at <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mn>0</mn></math></span> and the limit at <span><math><mo>|</mo><mi>x</mi><mo>|</mo><mo>=</mo><mo>∞</mo></math></span> for a global solution is investigated.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312203","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 : 2024-09-23DOI: 10.1016/j.jde.2024.09.027
Shuyan Qiu , Li Luo , Xinyu Tu
<div><div>In this paper, we consider the two-species chemotaxis-competition system with loop and nonlocal kinetics<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>11</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>12</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>z</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><mi>u</mi><mo>+</mo><mi>w</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>21</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>w</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>22</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>w</mi><mi>∇</mi><mi>z</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>z</mi><mo>−</mo><mi>z</mi><mo>+</mo><mi>u</mi><mo>+</mo><mi>w</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> subject to homogeneous Neumann boundary conditions in a smooth bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>></mo><mn>0</mn><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>u</mi><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>u</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>w</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi><mi>d</mi><mi>
{"title":"Global dynamics for a two-species chemotaxis-competition system with loop and nonlocal kinetics","authors":"Shuyan Qiu , Li Luo , Xinyu Tu","doi":"10.1016/j.jde.2024.09.027","DOIUrl":"10.1016/j.jde.2024.09.027","url":null,"abstract":"<div><div>In this paper, we consider the two-species chemotaxis-competition system with loop and nonlocal kinetics<span><span><span><math><mrow><mo>{</mo><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>11</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>12</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>u</mi><mi>∇</mi><mi>z</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>v</mi><mo>+</mo><mi>u</mi><mo>+</mo><mi>w</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>21</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>w</mi><mi>∇</mi><mi>v</mi><mo>)</mo><mo>−</mo><msub><mrow><mi>χ</mi></mrow><mrow><mn>22</mn></mrow></msub><mi>∇</mi><mo>⋅</mo><mo>(</mo><mi>w</mi><mi>∇</mi><mi>z</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>f</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd><mn>0</mn><mo>=</mo><mi>Δ</mi><mi>z</mi><mo>−</mo><mi>z</mi><mo>+</mo><mi>u</mi><mo>+</mo><mi>w</mi><mo>,</mo><mspace></mspace></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> subject to homogeneous Neumann boundary conditions in a smooth bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>(</mo><mi>n</mi><mo>≥</mo><mn>1</mn><mo>)</mo></math></span>, where <span><math><msub><mrow><mi>χ</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>></mo><mn>0</mn><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, <span><math><msub><mrow><mi>f</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>u</mi><mrow><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>u</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>w</mi><mo>−</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>3</mn></mrow></msub><msub><mrow><mo>∫</mo></mrow><mrow><mi>Ω</mi></mrow></msub><mi>u</mi><mi>d</mi><mi>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142310593","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 : 2024-09-23DOI: 10.1016/j.jde.2024.09.025
Anne Bronzi , Ricardo Guimarães , Cecilia Mondaini
We analyze finite-time blowup scenarios of locally self-similar type for the inviscid generalized surface quasi-geostrophic equation (gSQG) in . Under an growth assumption on the self-similar profile and its gradient, we identify appropriate ranges of the self-similar parameter where the profile is either identically zero, and hence blowup cannot occur, or its asymptotic behavior can be characterized, for suitable . Our results extend the work by Xue [38] regarding the SQG equation, and also partially recover the results proved by Cannone and Xue [3] concerning globally self-similar solutions of the gSQG equation.
{"title":"On the locally self-similar blowup for the generalized SQG equation","authors":"Anne Bronzi , Ricardo Guimarães , Cecilia Mondaini","doi":"10.1016/j.jde.2024.09.025","DOIUrl":"10.1016/j.jde.2024.09.025","url":null,"abstract":"<div><div>We analyze finite-time blowup scenarios of locally self-similar type for the inviscid generalized surface quasi-geostrophic equation (gSQG) in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. Under an <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> growth assumption on the self-similar profile and its gradient, we identify appropriate ranges of the self-similar parameter where the profile is either identically zero, and hence blowup cannot occur, or its <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> asymptotic behavior can be characterized, for suitable <span><math><mi>r</mi><mo>,</mo><mi>p</mi></math></span>. Our results extend the work by Xue <span><span>[38]</span></span> regarding the SQG equation, and also partially recover the results proved by Cannone and Xue <span><span>[3]</span></span> concerning globally self-similar solutions of the gSQG equation.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":null,"pages":null},"PeriodicalIF":2.4,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142310594","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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