Pub Date : 2025-12-29DOI: 10.1016/j.jde.2025.114062
Fang Li, Bendong Lou
We consider a generalized degenerate diffusion equation with a reaction term , where A is a smooth function satisfying and for , and f is monostable in and bistable in . We first present a trichotomy result on the asymptotic behavior of solutions with compactly supported initial data. It states that, as , one of the following occurs: small-spreading (i.e., u tends to ), transition, or big-spreading (i.e., u tends to 1). Then we construct the classical and sharp traveling waves (a sharp wave is defined as a wave with a free boundary satisfying Darcy's law) for the generalized degenerate diffusion equation, and use them to characterize the spreading solution near its front.
考虑一个反应项为ut=[a (u)]xx+f(u)的广义退化扩散方程，其中a是满足a (0)= a '(0)=0和满足a (u), a ' (u), a″(u)>；0的光滑函数，f在[0，s1]内单稳定，在[s1,1]内双稳定。我们首先给出了具有紧支持初始数据的解的渐近行为的一个三分结果。它表明，当t→∞时，会出现以下情况之一：小扩散（即u趋向于s1）、过渡或大扩散（即u趋向于1）。然后构造广义简并扩散方程的经典锐行波（锐波定义为具有满足达西定律的自由边界的波），并利用它们来表征扩散方程前缘附近的扩散解。
{"title":"Asymptotic behavior of solutions of a degenerate diffusion equation with a multistable reaction","authors":"Fang Li, Bendong Lou","doi":"10.1016/j.jde.2025.114062","DOIUrl":"10.1016/j.jde.2025.114062","url":null,"abstract":"<div><div>We consider a generalized degenerate diffusion equation with a reaction term <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><msub><mrow><mo>[</mo><mi>A</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>]</mo></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>+</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span>, where <em>A</em> is a smooth function satisfying <span><math><mi>A</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mn>0</mn></math></span> and <span><math><mi>A</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mspace></mspace><msup><mrow><mi>A</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mspace></mspace><msup><mrow><mi>A</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>u</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> for <span><math><mi>u</mi><mo>></mo><mn>0</mn></math></span>, and <em>f</em> is monostable in <span><math><mo>[</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>]</mo></math></span> and bistable in <span><math><mo>[</mo><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. We first present a trichotomy result on the asymptotic behavior of solutions with compactly supported initial data. It states that, as <span><math><mi>t</mi><mo>→</mo><mo>∞</mo></math></span>, one of the following occurs: small-spreading (i.e., <em>u</em> tends to <span><math><msub><mrow><mi>s</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>), transition, or big-spreading (i.e., <em>u</em> tends to 1). Then we construct the classical and sharp traveling waves (a sharp wave is defined as a wave with a free boundary satisfying Darcy's law) for the generalized degenerate diffusion equation, and use them to characterize the spreading solution near its front.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114062"},"PeriodicalIF":2.3,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145881143","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 : 2025-12-29DOI: 10.1016/j.jde.2025.114069
Isaac A. García, Jaume Giné
We analyze the structure of the Poincaré map Π associated to a monodromic singularity of an analytic family of planar vector fields. We work under two assumptions. The first one is that the family possesses an inverse integrating factor that can be expanded in Laurent series centered at the singularity after a weighted polar blow-up fixed by the Newton diagram of the family. The second one is that we restrict our analysis to a subset of the monodromic parameter space that assures the non-existence of local curves with zero angular speed. The conclusions are that the asymptotic Dulac expansion of Π does not contain logarithmic terms, indeed it admits a formal power series expansion with a unique independent generalized Poincaré-Lyapunov quantity, which can be computed under some explicit conditions. Moreover we also give conditions that guarantee the analyticity of Π, in which case we show that the Bautin ideal is principal and therefore the cyclicity of the singularity with respect to perturbation within the family is zero.
{"title":"Principal Bautin ideal of monodromic singularities with inverse integrating factors","authors":"Isaac A. García, Jaume Giné","doi":"10.1016/j.jde.2025.114069","DOIUrl":"10.1016/j.jde.2025.114069","url":null,"abstract":"<div><div>We analyze the structure of the Poincaré map Π associated to a monodromic singularity of an analytic family of planar vector fields. We work under two assumptions. The first one is that the family possesses an inverse integrating factor that can be expanded in Laurent series centered at the singularity after a weighted polar blow-up fixed by the Newton diagram of the family. The second one is that we restrict our analysis to a subset of the monodromic parameter space that assures the non-existence of local curves with zero angular speed. The conclusions are that the asymptotic Dulac expansion of Π does not contain logarithmic terms, indeed it admits a formal power series expansion with a unique independent generalized Poincaré-Lyapunov quantity, which can be computed under some explicit conditions. Moreover we also give conditions that guarantee the analyticity of Π, in which case we show that the Bautin ideal is principal and therefore the cyclicity of the singularity with respect to perturbation within the family is zero.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114069"},"PeriodicalIF":2.3,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145882139","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 : 2025-12-23DOI: 10.1016/j.jde.2025.114050
Chris Guiver , Hartmut Logemann
For a large class of Lur'e systems with time-varying nonlinearities and feedthrough we consider several well-posedness issues, namely: existence, continuation, blow-up in finite-time, forward completeness and uniqueness of solutions. Lur'e systems with feedthrough are systems of forced, nonlinear ordinary differential equations coupled with a nonlinear algebraic equation determining the output of the system. The presence of feedthrough means that the algebraic equation is implicit in the output, and, in general, the output may not be expressible by an analytic formula in terms of the state and the input. Simple examples illustrate that the well-posedness properties of such systems are not necessarily guaranteed by assumptions sufficient for the corresponding well-posedness properties of Lur'e systems without feedthrough. We provide sufficient conditions for the well-posedness properties mentioned above, using global inversion theorems from real analysis and tools from non-smooth analysis and differential inclusions. The theory is illustrated with examples.
{"title":"Well-posedness of Lur'e systems with feedthrough","authors":"Chris Guiver , Hartmut Logemann","doi":"10.1016/j.jde.2025.114050","DOIUrl":"10.1016/j.jde.2025.114050","url":null,"abstract":"<div><div>For a large class of Lur'e systems with time-varying nonlinearities and feedthrough we consider several well-posedness issues, namely: existence, continuation, blow-up in finite-time, forward completeness and uniqueness of solutions. Lur'e systems with feedthrough are systems of forced, nonlinear ordinary differential equations coupled with a nonlinear algebraic equation determining the output of the system. The presence of feedthrough means that the algebraic equation is implicit in the output, and, in general, the output may not be expressible by an analytic formula in terms of the state and the input. Simple examples illustrate that the well-posedness properties of such systems are not necessarily guaranteed by assumptions sufficient for the corresponding well-posedness properties of Lur'e systems without feedthrough. We provide sufficient conditions for the well-posedness properties mentioned above, using global inversion theorems from real analysis and tools from non-smooth analysis and differential inclusions. The theory is illustrated with examples.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"456 ","pages":"Article 114050"},"PeriodicalIF":2.3,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145836913","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 : 2025-12-23DOI: 10.1016/j.jde.2025.114056
Jialu Tian , Yihong Du , Ping Liu , Wenjie Ni
This paper investigates a modified Leslie-Gower type predator-prey model that incorporates prey-taxis and a free boundary. In this model, the prey species (with density ) has population range and gradually expands its range over time through the right end (the free boundary), while the predator species (with density ) occupies the entire available space . We want to use this model to describe a typical ecological invasion scenario, namely a new prey species invades into a territory where a native predator with a broad diet spectrum already resides. The primary concern is whether the invasive prey population can persist and establish itself under such predation pressure. We first prove the existence, uniqueness, and uniform boundedness of the solution by overcoming several technical challenges posed by the coupling of prey-taxis, the unbounded spatial domain, and the free boundary. We then investigate the long-time dynamics, and identify two distinct scenarios: (i) Vanishing - the habitat of the prey expands but remains ultimately bounded, leading to the prey density going to 0 and the predator density converging to a positive constant equilibrium as time ; (ii) Spreading - the prey's range expands to the entire available space , ensuring the persistence of the prey species within any bounded subregion of , while the predator maintains a positive density. Finally, several sufficient conditions are obtained to guarantee the occurrence of spreading and vanishing, respectively.
{"title":"A predator-prey model with prey-taxis and free boundary: Well-posedness and long-time dynamics","authors":"Jialu Tian , Yihong Du , Ping Liu , Wenjie Ni","doi":"10.1016/j.jde.2025.114056","DOIUrl":"10.1016/j.jde.2025.114056","url":null,"abstract":"<div><div>This paper investigates a modified Leslie-Gower type predator-prey model that incorporates prey-taxis and a free boundary. In this model, the prey species (with density <span><math><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>) has population range <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>]</mo><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span> and gradually expands its range over time through the right end <span><math><mi>x</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span> (the free boundary), while the predator species (with density <span><math><mi>v</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span>) occupies the entire available space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>. We want to use this model to describe a typical ecological invasion scenario, namely a new prey species invades into a territory where a native predator with a broad diet spectrum already resides. The primary concern is whether the invasive prey population can persist and establish itself under such predation pressure. We first prove the existence, uniqueness, and uniform boundedness of the solution by overcoming several technical challenges posed by the coupling of prey-taxis, the unbounded spatial domain, and the free boundary. We then investigate the long-time dynamics, and identify two distinct scenarios: (i) Vanishing - the habitat of the prey expands but remains ultimately bounded, leading to the prey density <span><math><mi>u</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> going to 0 and the predator density <span><math><mi>v</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo></math></span> converging to a positive constant equilibrium as time <span><math><mi>t</mi><mo>→</mo><mo>∞</mo></math></span>; (ii) Spreading - the prey's range <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mi>h</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>]</mo></math></span> expands to the entire available space <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, ensuring the persistence of the prey species within any bounded subregion of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, while the predator maintains a positive density. Finally, several sufficient conditions are obtained to guarantee the occurrence of spreading and vanishing, respectively.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114056"},"PeriodicalIF":2.3,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145837605","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 : 2025-12-23DOI: 10.1016/j.jde.2025.114068
Marina Domingues , Jaume Llibre , Luis Fernando Mello
Let be a polynomial map from the real plane to the real plane with a non-zero Jacobian determinant at any point of the real plane. We prove that if the higher homogeneous terms of the derivatives and do not have real linear factors in common then F is injective. The tool for proving this result is the qualitative theory of the differential systems.
{"title":"A new sufficient condition in order that the real Jacobian conjecture in R2 holds","authors":"Marina Domingues , Jaume Llibre , Luis Fernando Mello","doi":"10.1016/j.jde.2025.114068","DOIUrl":"10.1016/j.jde.2025.114068","url":null,"abstract":"<div><div>Let <span><math><mi>F</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>,</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>)</mo></math></span> be a polynomial map from the real plane to the real plane with a non-zero Jacobian determinant at any point of the real plane. We prove that if the higher homogeneous terms of the derivatives <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>x</mi></mrow></msub></math></span> and <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>y</mi></mrow></msub></math></span> do not have real linear factors in common then <em>F</em> is injective. The tool for proving this result is the qualitative theory of the differential systems.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 114068"},"PeriodicalIF":2.3,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145837395","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 : 2025-12-23DOI: 10.1016/j.jde.2025.114058
Genival da Silva
In this note, we present an alternative proof that weak solutions to belong to , where and . The first complete proof of this result was given in [1]; here, we give an alternative argument.
{"title":"A new proof of the Cp′-conjecture in the plane via a priori estimates","authors":"Genival da Silva","doi":"10.1016/j.jde.2025.114058","DOIUrl":"10.1016/j.jde.2025.114058","url":null,"abstract":"<div><div>In this note, we present an alternative proof that weak solutions to<span><span><span><math><mo>−</mo><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mi>u</mi><mo>=</mo><mi>f</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><msub><mrow><mi>B</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></math></span></span></span> belong to <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><msup><mrow><mi>p</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>, where <span><math><mi>p</mi><mo>></mo><mn>2</mn></math></span> and <span><math><mi>Ω</mi><mo>⊆</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>. The first complete proof of this result was given in <span><span>[1]</span></span>; here, we give an alternative argument.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"460 ","pages":"Article 114058"},"PeriodicalIF":2.3,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145802213","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 : 2025-12-23DOI: 10.1016/j.jde.2025.114041
Shuang Li, Yaping Wu
This paper is concerned with the asymptotic stability of traveling waves with the critical speed for a double degenerate Fisher type equation. By applying point-wise Green's function approach with detailed and special integral and semigroup estimates, we prove the linear and nonlinear asymptotic stability of the wave with the critical speed in some polynomially weighted spaces; and show that the decay of the semigroup can be in order of or faster than , and the decay of the perturbation of the wave in time can be in order of for some . In this paper, we also prove the spectral stability of the wave with the critical speed in some exponentially and polynomially weighted spaces.
{"title":"Asymptotic stability of traveling waves with critical speed for a double degenerate Fisher type equation","authors":"Shuang Li, Yaping Wu","doi":"10.1016/j.jde.2025.114041","DOIUrl":"10.1016/j.jde.2025.114041","url":null,"abstract":"<div><div>This paper is concerned with the asymptotic stability of traveling waves with the critical speed for a double degenerate Fisher type equation. By applying point-wise Green's function approach with detailed and special integral and semigroup estimates, we prove the linear and nonlinear asymptotic stability of the wave with the critical speed in some polynomially weighted spaces; and show that the decay of the semigroup can be in order of <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msup></math></span> or faster than <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span>, and the decay of the perturbation of the wave in time can be in order of <span><math><msup><mrow><mi>t</mi></mrow><mrow><mo>−</mo><mi>θ</mi></mrow></msup></math></span> for some <span><math><mi>θ</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>. In this paper, we also prove the spectral stability of the wave with the critical speed in some exponentially and polynomially weighted spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114041"},"PeriodicalIF":2.3,"publicationDate":"2025-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145838806","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 : 2025-12-22DOI: 10.1016/j.jde.2025.114063
Pengyan Ding , Baoxia Jin , Zhijian Yang
<div><div>In this paper, we are concerned with the well-posedness, regularity and longtime dynamics of the quasi-linear hyperbolic equation with structural damping on a 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>:<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>Δ</mi><mi>ϕ</mi><mo>(</mo><mi>Δ</mi><mi>u</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></math></span></span></span> together with the perturbed dissipative index <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span> and the hinged boundary condition. We show that (i) When the growth order <em>p</em> of the nonlinearity <em>ϕ</em> is up to the optimal subcritical range: <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mn>2</mn><mo>(</mo><mi>α</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><msup><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo>(</mo><mi>α</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfrac></math></span>, the model is well-posed in phase space <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, and its weak solution has additionally partial regularity as <span><math><mi>t</mi><mo>></mo><mn>0</mn></math></span>, especially when <span><math><mi>g</mi><mo>=</mo><mn>0</mn></math></span>, it has in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> a trivial global and exponential attractor, respectively. And when <span><math><mi>g</mi><mo>≠</mo><mn>0</mn></math></span>, the model in <em>N</em>-dimension case still possesses <span><math><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>δ</mi></mrow></msub><mo>×</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>α</mi><mo>−</mo><mi>δ</mi></mrow></msub><mo>)</mo></math></span>-global and exponential attractors, respectively. (ii) In particular, when <span><math><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>α</mi><mo>∈</mo><mo>(</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, without any polynomial growth restriction for <em>ϕ</em>, the weak solution has stronger complete regularity as <span><math><mi>t</mi><mo>></mo><mn>0</mn></math></span>, which guarantees that it is just the strong one. Furthermore, the related solution se
本文讨论了有界域Ω∧RN(N≥1):utt+Δ2u+（−Δ）αut+Δϕ（Δu）=g(x)上具有结构阻尼的拟线性双曲型方程的适定性、正则性和长时间动力学，并讨论了微扰耗散指数α∈(1,2)和边界条件。我们证明了(i)当非线性φ的生长阶数p达到最优次临界范围：1≤p<；pα:=N+2(α−1)(N−2(α−1))+时，模型在相空间Hα+1中是适定的，其弱解在t>；0时具有额外的部分正则性，特别是当g=0时，它在Hα+1中分别具有平凡的全局吸引子和指数吸引子。当g≠0时，n维模型仍然分别具有（Hα+1,Vα+1 - δ×Vα−δ）-全局吸引子和指数吸引子。（ii）特别地，当N=1,α∈(3/2,2)时，对于φ没有任何多项式生长限制，弱解在t>；0时具有更强的完全正则性，这保证了它正是强解。此外，对于每个α∈(3/2,2)，相关解半群Sα(t)分别具有强(Hα+1,Y2α)-全局吸引子和强(Hα+1,Y2α)-指数吸引子，其中Y2α为强解空间。（iii）建立了一个抽象判据，证明了强(Hα+1,Y2α)-全局吸引子族{Aα}α∈(3/2,2)在Y2α0拓扑上任意点α0∈(3/2,2)处是上半连续的。本文开发的方法允许将长期等效约束φ ' (s) ~ |s|p−1改进为亚临界范围内的右侧多项式生长约束：0≤φ ' (s)≤C(1+|s|p−1)，其中1≤p<；pα，并在一维情况下获得更好的结果。
{"title":"Well-posedness, regularity and longtime dynamics for the quasi-linear hyperbolic equation with structural damping","authors":"Pengyan Ding , Baoxia Jin , Zhijian Yang","doi":"10.1016/j.jde.2025.114063","DOIUrl":"10.1016/j.jde.2025.114063","url":null,"abstract":"<div><div>In this paper, we are concerned with the well-posedness, regularity and longtime dynamics of the quasi-linear hyperbolic equation with structural damping on a 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>:<span><span><span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi><mi>t</mi></mrow></msub><mo>+</mo><msup><mrow><mi>Δ</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>u</mi><mo>+</mo><msup><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow><mrow><mi>α</mi></mrow></msup><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>+</mo><mi>Δ</mi><mi>ϕ</mi><mo>(</mo><mi>Δ</mi><mi>u</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>,</mo></math></span></span></span> together with the perturbed dissipative index <span><math><mi>α</mi><mo>∈</mo><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>]</mo></math></span> and the hinged boundary condition. We show that (i) When the growth order <em>p</em> of the nonlinearity <em>ϕ</em> is up to the optimal subcritical range: <span><math><mn>1</mn><mo>≤</mo><mi>p</mi><mo><</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>α</mi></mrow></msub><mo>:</mo><mo>=</mo><mfrac><mrow><mi>N</mi><mo>+</mo><mn>2</mn><mo>(</mo><mi>α</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mrow><msup><mrow><mo>(</mo><mi>N</mi><mo>−</mo><mn>2</mn><mo>(</mo><mi>α</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><mrow><mo>+</mo></mrow></msup></mrow></mfrac></math></span>, the model is well-posed in phase space <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>, and its weak solution has additionally partial regularity as <span><math><mi>t</mi><mo>></mo><mn>0</mn></math></span>, especially when <span><math><mi>g</mi><mo>=</mo><mn>0</mn></math></span>, it has in <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> a trivial global and exponential attractor, respectively. And when <span><math><mi>g</mi><mo>≠</mo><mn>0</mn></math></span>, the model in <em>N</em>-dimension case still possesses <span><math><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>α</mi><mo>+</mo><mn>1</mn><mo>−</mo><mi>δ</mi></mrow></msub><mo>×</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>α</mi><mo>−</mo><mi>δ</mi></mrow></msub><mo>)</mo></math></span>-global and exponential attractors, respectively. (ii) In particular, when <span><math><mi>N</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>α</mi><mo>∈</mo><mo>(</mo><mn>3</mn><mo>/</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></math></span>, without any polynomial growth restriction for <em>ϕ</em>, the weak solution has stronger complete regularity as <span><math><mi>t</mi><mo>></mo><mn>0</mn></math></span>, which guarantees that it is just the strong one. Furthermore, the related solution se","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"458 ","pages":"Article 114063"},"PeriodicalIF":2.3,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145837606","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 : 2025-12-22DOI: 10.1016/j.jde.2025.114038
Changchang Guan , Shi-Liang Wu , Shigui Ruan
This paper is concerned with a time periodic Lotka-Volterra diffusion system with strong competition. We study the long time behavior of bounded solutions for the system that lie between two stable semi-trivial periodic solutions of the corresponding kinetic system. By transforming the competitive system into an equivalent cooperative system on , we first demonstrate local stability of a pair of diverging periodic traveling fronts. Then, by establishing a new Liouville-type theorem for solutions of the wave profile system and applying the truncation method, we prove asymptotic stability of these diverging periodic traveling fronts in the -norm. Based on this result, by investigating the behavior of solutions with a one-parameter family of initial data, we present the trichotomy of parameter-dependent solutions: propagation for large parameter values, extinction for small parameter values, and transition from propagation to extinction for intermediate parameter values. Finally, we explore some properties of the threshold solution.
{"title":"Long time behavior for a periodic Lotka-Volterra reaction-diffusion system with strong competition II: The threshold phenomenon","authors":"Changchang Guan , Shi-Liang Wu , Shigui Ruan","doi":"10.1016/j.jde.2025.114038","DOIUrl":"10.1016/j.jde.2025.114038","url":null,"abstract":"<div><div>This paper is concerned with a time periodic Lotka-Volterra diffusion system with strong competition. We study the long time behavior of bounded solutions for the system that lie between two stable semi-trivial periodic solutions of the corresponding kinetic system. By transforming the competitive system into an equivalent cooperative system on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, we first demonstrate local stability of a pair of diverging periodic traveling fronts. Then, by establishing a new Liouville-type theorem for solutions of the wave profile system and applying the truncation method, we prove asymptotic stability of these diverging periodic traveling fronts in the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></math></span>-norm. Based on this result, by investigating the behavior of solutions with a one-parameter family of initial data, we present the trichotomy of parameter-dependent solutions: propagation for large parameter values, extinction for small parameter values, and transition from propagation to extinction for intermediate parameter values. Finally, we explore some properties of the threshold solution.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"459 ","pages":"Article 114038"},"PeriodicalIF":2.3,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145838859","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}
The branching structure of metric graphs can influence the propagation of front waves in reaction-diffusion models describing species invasion within river or road networks. In this article, we deal with the bistable reaction-diffusion equation on a tree-shaped metric graph with two junctions, modeling the invasion of a species as . We consider a front-like entire solution whose front comes from infinity along a branch and investigate its asymmetric behavior after it passes through the first junction, which branches in two directions. Under suitable conditions, we prove that the front propagation of the entire solution is blocked on one branch at the second junction, while on the other branch it asymptotically converges to the traveling front profile far from the junction. To achieve this, we construct a stationary solution allowing the desired asymptotic behavior and analyze how this behavior depends on the length of the branch connecting the two junctions.
{"title":"Asymmetric front propagation for the bistable reaction-diffusion equation on a metric graph","authors":"Toru Kan , Yoshihisa Morita , Ken-Ichi Nakamura , Chang-Hong Wu","doi":"10.1016/j.jde.2025.114039","DOIUrl":"10.1016/j.jde.2025.114039","url":null,"abstract":"<div><div>The branching structure of metric graphs can influence the propagation of front waves in reaction-diffusion models describing species invasion within river or road networks. In this article, we deal with the bistable reaction-diffusion equation on a tree-shaped metric graph with two junctions, modeling the invasion of a species as <span><math><mi>t</mi><mo>→</mo><mo>−</mo><mo>∞</mo></math></span>. We consider a front-like entire solution whose front comes from infinity along a branch and investigate its asymmetric behavior after it passes through the first junction, which branches in two directions. Under suitable conditions, we prove that the front propagation of the entire solution is blocked on one branch at the second junction, while on the other branch it asymptotically converges to the traveling front profile far from the junction. To achieve this, we construct a stationary solution allowing the desired asymptotic behavior and analyze how this behavior depends on the length of the branch connecting the two junctions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"457 ","pages":"Article 114039"},"PeriodicalIF":2.3,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145837394","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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