Pub Date : 2024-06-07DOI: 10.1016/j.nonrwa.2024.104155
J.M. Mazón, J. Toledo
In this paper we study -Laplacian type diffusion equations in weighted graphs obtained as limit as to two types of -Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set and the set as models for sandpile growth in weighted graphs. Moreover, we also analyse the collapse of the initial condition when it does not belong to the stable sets or by means of an abstract result given in Bénilan (2003). We give an interpretation of the limit problems in terms of Monge–Kantorovich mass transport theory. Finally, we give some explicit solutions of simple examples that illustrate the dynamics of the sandpile growing or collapsing.
{"title":"Two models for sandpile growth in weighted graphs","authors":"J.M. Mazón, J. Toledo","doi":"10.1016/j.nonrwa.2024.104155","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104155","url":null,"abstract":"<div><p>In this paper we study <span><math><mi>∞</mi></math></span>-Laplacian type diffusion equations in weighted graphs obtained as limit as <span><math><mrow><mi>p</mi><mo>→</mo><mi>∞</mi></mrow></math></span> to two types of <span><math><mi>p</mi></math></span>-Laplacian evolution equations in such graphs. We propose these diffusion equations, that are governed by the subdifferential of a convex energy functionals associated to the indicator function of the set <span><math><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow><mrow><mi>G</mi></mrow></msubsup><mo>≔</mo><mfenced><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>V</mi><mo>,</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>)</mo></mrow><mspace></mspace><mo>:</mo><mspace></mspace><mo>|</mo><mi>u</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>−</mo><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo><mo>≤</mo><mn>1</mn><mspace></mspace><mspace></mspace><mi>i</mi><mi>f</mi><mspace></mspace><mspace></mspace><mi>x</mi><mo>∼</mo><mi>y</mi></mrow></mfenced></mrow></math></span> and the set <span><math><mrow><msubsup><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow><mrow><mi>w</mi></mrow></msubsup><mo>≔</mo><mfenced><mrow><mi>u</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>V</mi><mo>,</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>)</mo></mrow><mspace></mspace><mo>:</mo><mspace></mspace><mo>|</mo><mi>u</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>−</mo><mi>u</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>|</mo><mo>≤</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><msub><mrow><mi>w</mi></mrow><mrow><mi>x</mi><mi>y</mi></mrow></msub></mrow></msqrt></mrow></mfrac><mspace></mspace><mspace></mspace><mi>i</mi><mi>f</mi><mspace></mspace><mspace></mspace><mi>x</mi><mo>∼</mo><mi>y</mi></mrow></mfenced></mrow></math></span> as models for sandpile growth in weighted graphs. Moreover, we also analyse the collapse of the initial condition when it does not belong to the stable sets <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow><mrow><mi>G</mi></mrow></msubsup></math></span> or <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>∞</mi></mrow><mrow><mi>w</mi></mrow></msubsup></math></span> by means of an abstract result given in Bénilan (2003). We give an interpretation of the limit problems in terms of Monge–Kantorovich mass transport theory. Finally, we give some explicit solutions of simple examples that illustrate the dynamics of the sandpile growing or collapsing.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104155"},"PeriodicalIF":2.0,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1468121824000956/pdfft?md5=20687dd0e01f3c2727a6b21b5f35cead&pid=1-s2.0-S1468121824000956-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141292240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-05DOI: 10.1016/j.nonrwa.2024.104148
Zeyuan Liu , Lan Zhang
This paper is concerned with the global existence and large time behavior of classical solutions away from vacuum to the Cauchy problem of the 1D compressible quantum Navier–Stokes–Poisson equations with large initial perturbation. Moreover, we obtain the global strong/classical solution of Navier–Stokes–Poisson equations through the vanishing dispersion limit with certain convergence rates. We focus on the case that the viscosity depends on density linearly which extends the former results of constant viscosity in Zhang et al. (2022) by the second author. Some useful estimates are developed to deduce the uniform-in-time lower and upper bounds on the specific volume and the electric potential.
{"title":"Global well-posedness to the 1D compressible quantum Navier–Stokes–Poisson equations with large initial data","authors":"Zeyuan Liu , Lan Zhang","doi":"10.1016/j.nonrwa.2024.104148","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104148","url":null,"abstract":"<div><p>This paper is concerned with the global existence and large time behavior of classical solutions away from vacuum to the Cauchy problem of the 1D compressible quantum Navier–Stokes–Poisson equations with large initial perturbation. Moreover, we obtain the global strong/classical solution of Navier–Stokes–Poisson equations through the vanishing dispersion limit with certain convergence rates. We focus on the case that the viscosity depends on density linearly which extends the former results of constant viscosity in Zhang et al. (2022) by the second author. Some useful estimates are developed to deduce the uniform-in-time lower and upper bounds on the specific volume and the electric potential.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104148"},"PeriodicalIF":2.0,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141263911","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-04DOI: 10.1016/j.nonrwa.2024.104145
Jihong Zhao, Rong Jin, Hao Chen
In this paper, we study the Cauchy problem of the drift–diffusion system arising from semiconductor model. We prove that if a certain nonlinear function of the initial data is small enough, in a Besov type space, then there is a global solution to this drift–diffusion system. We also provide an example of initial data satisfying that nonlinear smallness condition, but whose norm be chosen arbitrarily large in .
{"title":"Global existence of solutions for the drift–diffusion system with large initial data in Ḃ−2∞,∞ (Rd)","authors":"Jihong Zhao, Rong Jin, Hao Chen","doi":"10.1016/j.nonrwa.2024.104145","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104145","url":null,"abstract":"<div><p>In this paper, we study the Cauchy problem of the drift–diffusion system arising from semiconductor model. We prove that if a certain nonlinear function of the initial data is small enough, in a Besov type space, then there is a global solution to this drift–diffusion system. We also provide an example of initial data satisfying that nonlinear smallness condition, but whose norm be chosen arbitrarily large in <span><math><mrow><msubsup><mrow><mover><mrow><mi>B</mi></mrow><mrow><mo>̇</mo></mrow></mover></mrow><mrow><mi>∞</mi><mo>,</mo><mi>∞</mi></mrow><mrow><mo>−</mo><mn>2</mn></mrow></msubsup><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104145"},"PeriodicalIF":2.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141250578","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-04DOI: 10.1016/j.nonrwa.2024.104147
Kyunghan Choi, Yong-Jung Kim
The paper focuses on the pattern formation of a chemotactic cell aggregation model with a mechanism that density suppresses motility. The model exhibits four types of cell aggregation patterns: single-point peaks, hot spots, cold spots, and stripes, depending on the parameters and mean density. The analysis is performed in two ways. First, traditional instability analysis reveals the existence of two critical densities. This local analysis shows patterns emerge if the initial mean density lies between the two values. Second, a phase separation method using van der Waals’ double well potential reveals that pattern formation is possible in a bigger parameter regime that includes the one identified by the local analysis. This non-local analysis shows that pattern formation occurs beyond the parameter regimes of the classical local instability analysis.
{"title":"Chemotactic cell aggregation viewed as instability and phase separation","authors":"Kyunghan Choi, Yong-Jung Kim","doi":"10.1016/j.nonrwa.2024.104147","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104147","url":null,"abstract":"<div><p>The paper focuses on the pattern formation of a chemotactic cell aggregation model with a mechanism that density suppresses motility. The model exhibits four types of cell aggregation patterns: single-point peaks, hot spots, cold spots, and stripes, depending on the parameters and mean density. The analysis is performed in two ways. First, traditional instability analysis reveals the existence of two critical densities. This local analysis shows patterns emerge if the initial mean density lies between the two values. Second, a phase separation method using van der Waals’ double well potential reveals that pattern formation is possible in a bigger parameter regime that includes the one identified by the local analysis. This non-local analysis shows that pattern formation occurs beyond the parameter regimes of the classical local instability analysis.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104147"},"PeriodicalIF":2.0,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141250579","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-31DOI: 10.1016/j.nonrwa.2024.104143
Yi Cheng , Xin Wang , Baowei Feng , Donal O’ Regan
This paper considers the stabilization problem of the von Kármán beam equation with a combined boundary control of nonlinear delays and nonlinear non-delays. The combined boundary controls are applied at the transverse and longitudinal boundaries of the von Kármán beam, respectively. In this paper the nonlinear semigroup method is adopted in the investigation for the establishment of the well-posedness of the resulting closed-loop system. Constructing an appropriate energy-like function, the exponential decay rate of energy of the closed-loop system is demonstrated by a generalized Gronwall-type integral inequality and the integral multiplier technique.
本文研究了具有非线性延迟和非线性非延迟组合边界控制的 von Kármán 梁方程的稳定问题。组合边界控制分别应用于 von Kármán 梁的横向和纵向边界。本文在研究中采用了非线性半群法，以建立闭环系统的良好拟合。通过构造一个适当的类能量函数，利用广义格伦沃尔积分不等式和积分乘法器技术证明了闭环系统能量的指数衰减率。
{"title":"Semigroup well-posedness and exponential stability for the von Kármán beam equation under the combined boundary control of nonlinear delays and non-delays","authors":"Yi Cheng , Xin Wang , Baowei Feng , Donal O’ Regan","doi":"10.1016/j.nonrwa.2024.104143","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104143","url":null,"abstract":"<div><p>This paper considers the stabilization problem of the von Kármán beam equation with a combined boundary control of nonlinear delays and nonlinear non-delays. The combined boundary controls are applied at the transverse and longitudinal boundaries of the von Kármán beam, respectively. In this paper the nonlinear semigroup method is adopted in the investigation for the establishment of the well-posedness of the resulting closed-loop system. Constructing an appropriate energy-like function, the exponential decay rate of energy of the closed-loop system is demonstrated by a generalized Gronwall-type integral inequality and the integral multiplier technique.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104143"},"PeriodicalIF":2.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141244510","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-31DOI: 10.1016/j.nonrwa.2024.104144
Inkyung Ahn , Wonhyung Choi , Jong-Shenq Guo
We study the disease-spreading dynamics of the West Nile virus (WNv) epidemic model under shifting climatic conditions. A WNv epidemic model is developed incorporating a shifting net growth term to depict the evolving mosquito habitat. First, we comprehensively characterize the spreading dynamics of mosquitoes for any given climate change speed compared with the intrinsic spreading speed of mosquitoes. Utilizing the results from mosquito dynamics, we determine the spreading dynamics of infected birds and mosquitoes, taking into account relationships among the shifting speed and the spreading speeds of mosquito and WNv. Ultimately, we find that infected mosquitoes and birds propagate, and their population densities converge to a stable positive endemic state. This paper provides crucial insights into the impact of climate change on the spread of vector-borne diseases such as WNv.
{"title":"Spreading dynamics for an epidemic model of West-Nile virus with shifting environment","authors":"Inkyung Ahn , Wonhyung Choi , Jong-Shenq Guo","doi":"10.1016/j.nonrwa.2024.104144","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104144","url":null,"abstract":"<div><p>We study the disease-spreading dynamics of the West Nile virus (WNv) epidemic model under shifting climatic conditions. A WNv epidemic model is developed incorporating a shifting net growth term to depict the evolving mosquito habitat. First, we comprehensively characterize the spreading dynamics of mosquitoes for any given climate change speed compared with the intrinsic spreading speed of mosquitoes. Utilizing the results from mosquito dynamics, we determine the spreading dynamics of infected birds and mosquitoes, taking into account relationships among the shifting speed and the spreading speeds of mosquito and WNv. Ultimately, we find that infected mosquitoes and birds propagate, and their population densities converge to a stable positive endemic state. This paper provides crucial insights into the impact of climate change on the spread of vector-borne diseases such as WNv.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104144"},"PeriodicalIF":2.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141244500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-25DOI: 10.1016/j.nonrwa.2024.104139
Piotr B. Mucha , Šárka Nečasová , Maja Szlenk
We investigate the existence of weak solutions to a multi-component system, consisting of compressible chemically reacting components, coupled with the compressible Stokes equation for the velocity. Specifically, we consider the case of irreversible chemical reactions and assume a nonlinear relation between the pressure and the particular densities. These assumptions cause the additional difficulties in the mathematical analysis, due to the possible presence of vacuum.
It is shown that there exists a global weak solution, satisfying the bounds for all the components. We obtain strong compactness of the sequence of densities in spaces, under the assumption that all components are strictly positive. The applied method captures the properties of models of high generality, which admit an arbitrary number of components. Furthermore, the framework that we develop can handle models that contain both diffusing and non-diffusing elements.
{"title":"A multifluid model with chemically reacting components — Construction of weak solutions","authors":"Piotr B. Mucha , Šárka Nečasová , Maja Szlenk","doi":"10.1016/j.nonrwa.2024.104139","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104139","url":null,"abstract":"<div><p>We investigate the existence of weak solutions to a multi-component system, consisting of compressible chemically reacting components, coupled with the compressible Stokes equation for the velocity. Specifically, we consider the case of irreversible chemical reactions and assume a nonlinear relation between the pressure and the particular densities. These assumptions cause the additional difficulties in the mathematical analysis, due to the possible presence of vacuum.</p><p>It is shown that there exists a global weak solution, satisfying the <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup></math></span> bounds for all the components. We obtain strong compactness of the sequence of densities in <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span> spaces, under the assumption that all components are strictly positive. The applied method captures the properties of models of high generality, which admit an arbitrary number of components. Furthermore, the framework that we develop can handle models that contain both diffusing and non-diffusing elements.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"80 ","pages":"Article 104139"},"PeriodicalIF":2.0,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141097740","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-21DOI: 10.1016/j.nonrwa.2024.104138
Liyan Zhong , Jianhe Shen
In general, critical manifold loses normal hyperbolicity at folded, transcritical and pitchfork singularities. There is another situation where normal hyperbolicity of critical manifold fails, namely, the alignment of the tangent and normal bundles at the unbounded part of critical manifold. In this case, how to reveal the attracting or repelling natures of unbounded critical manifold is essential to detect the birth of relaxation oscillations. In this article, after the compactification of the unbounded critical curve and then blowing-up the resulting degenerate line, we find that return mechanism exists at the -region of the critical curve in a slow–fast excitable Brusselator oscillator. By so doing the birth of relaxation oscillation near the unbounded critical curve in this model is demonstrated. In addition, we reveal the continuation process from Hopf small-amplitude cycle to large relaxation oscillation of size in the blown-up space. This may be the counterpart of canard explosion in unbounded situation. All the theoretical predictions are verified by numerical simulations.
{"title":"Large relaxation oscillation in slow–fast excitable Brusselator oscillator","authors":"Liyan Zhong , Jianhe Shen","doi":"10.1016/j.nonrwa.2024.104138","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104138","url":null,"abstract":"<div><p>In general, critical manifold loses normal hyperbolicity at folded, transcritical and pitchfork singularities. There is another situation where normal hyperbolicity of critical manifold fails, namely, the alignment of the tangent and normal bundles at the unbounded part of critical manifold. In this case, how to reveal the attracting or repelling natures of unbounded critical manifold is essential to detect the birth of relaxation oscillations. In this article, after the compactification of the unbounded critical curve and then blowing-up the resulting degenerate line, we find that return mechanism exists at the <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>ɛ</mi><mo>)</mo></mrow></mrow></math></span>-region of the critical curve in a slow–fast excitable Brusselator oscillator. By so doing the birth of relaxation oscillation near the unbounded critical curve in this model is demonstrated. In addition, we reveal the continuation process from Hopf small-amplitude cycle to large relaxation oscillation of size <span><math><mrow><mi>O</mi><mrow><mo>(</mo><mn>1</mn><mo>/</mo><mi>ɛ</mi><mo>)</mo></mrow></mrow></math></span> in the blown-up space. This may be the counterpart of canard explosion in unbounded situation. All the theoretical predictions are verified by numerical simulations.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"79 ","pages":"Article 104138"},"PeriodicalIF":2.0,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141072615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-16DOI: 10.1016/j.nonrwa.2024.104137
Martina Magliocca
We study existence results for a fourth order problem describing single-component film models assuming initial data in Wiener spaces.
我们研究了描述单组分薄膜模型的四阶问题的存在性结果，假设初始数据在维纳空间中。
{"title":"On a fourth order equation describing single-component film models","authors":"Martina Magliocca","doi":"10.1016/j.nonrwa.2024.104137","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104137","url":null,"abstract":"<div><p>We study existence results for a fourth order problem describing single-component film models assuming initial data in Wiener spaces.</p></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"79 ","pages":"Article 104137"},"PeriodicalIF":2.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1468121824000774/pdfft?md5=fa090be4457e4225e16eb90fa56fba0e&pid=1-s2.0-S1468121824000774-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140951343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-15DOI: 10.1016/j.nonrwa.2024.104135
Alessandro Columbu, Rafael Díaz Fuentes, Silvia Frassu
<div><p>The following fully nonlinear attraction–repulsion and zero-flux chemotaxis model is studied: <span><span><span>(<span><math><mo>♢</mo></math></span>)</span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mi>⋅</mi><mfenced><mrow><msup><mrow><mrow><mo>(</mo><mi>u</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>−</mo><mi>χ</mi><mi>u</mi><msup><mrow><mrow><mo>(</mo><mi>u</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>∇</mo><mi>v</mi></mrow></mfenced><mspace></mspace></mtd></mtr><mtr><mtd><mfenced><mrow><mspace></mspace><mo>+</mo><mi>ξ</mi><mi>u</mi><msup><mrow><mrow><mo>(</mo><mi>u</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>∇</mo><mi>w</mi></mrow></mfenced><mo>+</mo><mi>λ</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>r</mi></mrow></msup><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>τ</mi><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>τ</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>ψ</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>Herein, <span><math><mi>Ω</mi></math></span> is a bounded and smooth domain of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, for <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>χ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>r</mi></mrow></math></span> proper positive numbers, <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></
{"title":"Uniform-in-time boundedness in a class of local and nonlocal nonlinear attraction–repulsion chemotaxis models with logistics","authors":"Alessandro Columbu, Rafael Díaz Fuentes, Silvia Frassu","doi":"10.1016/j.nonrwa.2024.104135","DOIUrl":"https://doi.org/10.1016/j.nonrwa.2024.104135","url":null,"abstract":"<div><p>The following fully nonlinear attraction–repulsion and zero-flux chemotaxis model is studied: <span><span><span>(<span><math><mo>♢</mo></math></span>)</span><span><math><mfenced><mrow><mtable><mtr><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mo>∇</mo><mi>⋅</mi><mfenced><mrow><msup><mrow><mrow><mo>(</mo><mi>u</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>∇</mo><mi>u</mi><mo>−</mo><mi>χ</mi><mi>u</mi><msup><mrow><mrow><mo>(</mo><mi>u</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>∇</mo><mi>v</mi></mrow></mfenced><mspace></mspace></mtd></mtr><mtr><mtd><mfenced><mrow><mspace></mspace><mo>+</mo><mi>ξ</mi><mi>u</mi><msup><mrow><mrow><mo>(</mo><mi>u</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>−</mo><mn>1</mn></mrow></msup><mo>∇</mo><mi>w</mi></mrow></mfenced><mo>+</mo><mi>λ</mi><mi>u</mi><mo>−</mo><mi>μ</mi><msup><mrow><mi>u</mi></mrow><mrow><mi>r</mi></mrow></msup><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>τ</mi><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>v</mi><mo>)</mo></mrow><mo>+</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>)</mo></mrow><mo>,</mo></mtd></mtr><mtr><mtd><mi>τ</mi><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>ψ</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>w</mi><mo>)</mo></mrow><mo>+</mo><mi>g</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mspace></mspace></mtd><mtd><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>)</mo></mrow><mo>.</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>Herein, <span><math><mi>Ω</mi></math></span> is a bounded and smooth domain of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, for <span><math><mrow><mi>n</mi><mo>∈</mo><mi>N</mi></mrow></math></span>, <span><math><mrow><mi>χ</mi><mo>,</mo><mi>ξ</mi><mo>,</mo><mi>λ</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>r</mi></mrow></math></span> proper positive numbers, <span><math><mrow><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"79 ","pages":"Article 104135"},"PeriodicalIF":2.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S1468121824000750/pdfft?md5=7ea2ce86ba1b3e1921a481bb478cddb6&pid=1-s2.0-S1468121824000750-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140951342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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