Pub Date : 2024-12-06DOI: 10.1016/j.nonrwa.2024.104272
Donal O’Regan
In this paper we present collectively fixed point theory for lower semicontinuous maps. In addition we present coincidence results between type maps and lower semicontinuous maps. Our arguments are based on the Schauder-Tychonoff fixed point theorem and a fixed point result based on self maps on an admissible convex set in a Hausdorff topological vector space. As an application we present a new (Nash) equilibrium result for economies.
{"title":"A note on collectively coincidence theory for lower semicontinuous maps","authors":"Donal O’Regan","doi":"10.1016/j.nonrwa.2024.104272","DOIUrl":"10.1016/j.nonrwa.2024.104272","url":null,"abstract":"<div><div>In this paper we present collectively fixed point theory for lower semicontinuous maps. In addition we present coincidence results between <span><math><mrow><mi>K</mi><mi>K</mi><mi>M</mi></mrow></math></span> type maps and lower semicontinuous maps. Our arguments are based on the Schauder-Tychonoff fixed point theorem and a fixed point result based on <span><math><mrow><mi>K</mi><mi>K</mi><mi>M</mi></mrow></math></span> self maps on an admissible convex set in a Hausdorff topological vector space. As an application we present a new (Nash) equilibrium result for economies.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104272"},"PeriodicalIF":1.8,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179286","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 : 2024-12-06DOI: 10.1016/j.nonrwa.2024.104283
Pavel Drábek , Michaela Zahradníková
This paper concerns the existence and properties of traveling wave solutions to reaction–diffusion–convection equations on the real line. We consider a general diffusion term involving the -Laplacian and combustion-type reaction term. We extend and generalize results established for to the case of singular and degenerate diffusion. Our approach allows for non-Lipschitz reaction as well. We also discuss the shape of the traveling wave profile near equilibria, assuming power-type behavior of the reaction and diffusion terms.
本文涉及实线上反应-扩散-对流方程的行波解的存在性和性质。我们考虑了涉及 p 拉普拉斯和燃烧型反应项的一般扩散项。我们将 p=2 时建立的结果扩展和推广到奇异和退化扩散的情况。我们的方法还允许非 Lipschitz 反应。我们还讨论了平衡点附近的行波剖面形状，假定反应和扩散项的幂型行为。
{"title":"Traveling waves in reaction–diffusion–convection equations with combustion nonlinearity","authors":"Pavel Drábek , Michaela Zahradníková","doi":"10.1016/j.nonrwa.2024.104283","DOIUrl":"10.1016/j.nonrwa.2024.104283","url":null,"abstract":"<div><div>This paper concerns the existence and properties of traveling wave solutions to reaction–diffusion–convection equations on the real line. We consider a general diffusion term involving the <span><math><mi>p</mi></math></span>-Laplacian and combustion-type reaction term. We extend and generalize results established for <span><math><mrow><mi>p</mi><mo>=</mo><mn>2</mn></mrow></math></span> to the case of singular and degenerate diffusion. Our approach allows for non-Lipschitz reaction as well. We also discuss the shape of the traveling wave profile near equilibria, assuming power-type behavior of the reaction and diffusion terms.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104283"},"PeriodicalIF":1.8,"publicationDate":"2024-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143178798","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 : 2024-12-04DOI: 10.1016/j.nonrwa.2024.104269
Luan Hoang , Thinh Kieu
We study the anisotropic Forchheimer-typed flows for compressible fluids in porous media. The first half of the paper is devoted to understanding the nonlinear structure of the anisotropic momentum equations. Unlike the isotropic flows, the important monotonicity properties are not automatically satisfied in this case. Therefore, various sufficient conditions for them are derived and applied to the experimental data. In the second half of the paper, we prove the existence and uniqueness of the steady state flows subject to a nonhomogeneous Dirichlet boundary condition. It is also established that these steady states, in appropriate functional spaces, have local Hölder continuous dependence on the forcing function and the boundary data.
{"title":"Anisotropic flows of Forchheimer-type in porous media and their steady states","authors":"Luan Hoang , Thinh Kieu","doi":"10.1016/j.nonrwa.2024.104269","DOIUrl":"10.1016/j.nonrwa.2024.104269","url":null,"abstract":"<div><div>We study the anisotropic Forchheimer-typed flows for compressible fluids in porous media. The first half of the paper is devoted to understanding the nonlinear structure of the anisotropic momentum equations. Unlike the isotropic flows, the important monotonicity properties are not automatically satisfied in this case. Therefore, various sufficient conditions for them are derived and applied to the experimental data. In the second half of the paper, we prove the existence and uniqueness of the steady state flows subject to a nonhomogeneous Dirichlet boundary condition. It is also established that these steady states, in appropriate functional spaces, have local Hölder continuous dependence on the forcing function and the boundary data.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104269"},"PeriodicalIF":1.8,"publicationDate":"2024-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179287","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 : 2024-12-03DOI: 10.1016/j.nonrwa.2024.104268
Paolo Secchi
We consider the initial–boundary value problem in the quarter space for the system of equations of ideal Magneto-Hydrodynamics for compressible fluids with perfectly conducting wall boundary conditions. On the two parts of the boundary the solution satisfies different boundary conditions, which make the problem an initial–boundary value problem with non-uniformly characteristic boundary.
We identify a subspace of the Sobolev space , obtained by addition of suitable boundary conditions on one portion of the boundary, such that for initial data in there exists a solution in the same space , for all times in a small time interval. This yields the well-posedness of the problem combined with a persistence property of full -regularity, although in general we expect a loss of normal regularity near the boundary. Thanks to the special geometry of the quarter space the proof easily follows by the “reflection technique”.
{"title":"A note on ideal Magneto-Hydrodynamics with perfectly conducting boundary conditions in the quarter space","authors":"Paolo Secchi","doi":"10.1016/j.nonrwa.2024.104268","DOIUrl":"10.1016/j.nonrwa.2024.104268","url":null,"abstract":"<div><div>We consider the initial–boundary value problem in the quarter space for the system of equations of ideal Magneto-Hydrodynamics for compressible fluids with perfectly conducting wall boundary conditions. On the two parts of the boundary the solution satisfies different boundary conditions, which make the problem an initial–boundary value problem with non-uniformly characteristic boundary.</div><div>We identify a subspace <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> of the Sobolev space <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, obtained by addition of suitable boundary conditions on one portion of the boundary, such that for initial data in <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span> there exists a solution in the same space <span><math><mrow><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow></math></span>, for all times in a small time interval. This yields the well-posedness of the problem combined with a persistence property of full <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span>-regularity, although in general we expect a loss of normal regularity near the boundary. Thanks to the special geometry of the quarter space the proof easily follows by the “reflection technique”.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104268"},"PeriodicalIF":1.8,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179284","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 : 2024-12-03DOI: 10.1016/j.nonrwa.2024.104266
Junying Chen, Ruixiang Xing
In this paper, we study a 2-dimensional free boundary problem modeling the tumor growth with a necrotic core. This model has three parameters: is a threshold value of nutrient concentration for distinguishing whether tumor cells are alive or not, is the death rate of proliferating cells and is the removal rate of necrotic cells. With the assumption of , we first give a complete classification of and under which the necrotic problem either has the unique radially symmetric stationary solution or no solutions. Furthermore, we derive the existence of symmetry-breaking solutions bifurcating from the radially symmetric solution for every .
{"title":"Symmetry-breaking bifurcation for necrotic tumor model with two free boundaries","authors":"Junying Chen, Ruixiang Xing","doi":"10.1016/j.nonrwa.2024.104266","DOIUrl":"10.1016/j.nonrwa.2024.104266","url":null,"abstract":"<div><div>In this paper, we study a 2-dimensional free boundary problem modeling the tumor growth with a necrotic core. This model has three parameters: <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub></math></span> is a threshold value of nutrient concentration for distinguishing whether tumor cells are alive or not, <span><math><mover><mrow><mi>σ</mi></mrow><mrow><mo>̃</mo></mrow></mover></math></span> is the death rate of proliferating cells and <span><math><mi>ν</mi></math></span> is the removal rate of necrotic cells. With the assumption of <span><math><mrow><mover><mrow><mi>σ</mi></mrow><mrow><mo>̃</mo></mrow></mover><mo>−</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub><mo>−</mo><mi>ν</mi><mo>≤</mo><mn>0</mn></mrow></math></span>, we first give a complete classification of <span><math><msub><mrow><mi>σ</mi></mrow><mrow><mi>D</mi></mrow></msub></math></span> and <span><math><mover><mrow><mi>σ</mi></mrow><mrow><mo>̃</mo></mrow></mover></math></span> under which the necrotic problem either has the unique radially symmetric stationary solution <span><math><mfenced><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></mfenced></math></span> or no solutions. Furthermore, we derive the existence of symmetry-breaking solutions bifurcating from the radially symmetric solution <span><math><mfenced><mrow><msub><mrow><mi>σ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>p</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>s</mi></mrow></msub></mrow></mfenced></math></span> for every <span><math><msub><mrow><mi>γ</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span> <span><math><mrow><mo>(</mo><mi>l</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>)</mo></mrow></math></span>.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104266"},"PeriodicalIF":1.8,"publicationDate":"2024-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179289","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 : 2024-12-02DOI: 10.1016/j.nonrwa.2024.104252
Sheng-Jie Li , Shugen Chai , Irena Lasiecka
The stabilization of a weak viscoelastic wave equation with variable coefficients in the principal part of elliptic operator and an interior delay is considered. The dynamics is subject to a nonlinear boundary dissipation. This leads to a non-dissipative dynamics. The existence of solution is demonstrated by means of Faedo–Galerkin method combined with monotone operator theory in handling nonlinear boundary conditions. The main result pertains to exponential decay rates for energy, which depend on the geometry of the spatial domain, viscoelastic effects, the strength of delay and the strength of mechanical boundary damping. An important feature of the model is the fact that the delay term and stabilizing mechanism are not collocated geometrically — in contrast with many other works on the subject. This aspect of the problem requires the appropriate tools in order to exhibit propagation of the dissipation from one location to another. The precise ranges of admissible parameters characterizing the model and ensuring the stability are provided. The methods of proofs are routed in Riemannian geometry.
{"title":"Stabilization of a weak viscoelastic wave equation in Riemannian geometric setting with an interior delay under nonlinear boundary dissipation","authors":"Sheng-Jie Li , Shugen Chai , Irena Lasiecka","doi":"10.1016/j.nonrwa.2024.104252","DOIUrl":"10.1016/j.nonrwa.2024.104252","url":null,"abstract":"<div><div>The stabilization of a <em>weak</em> viscoelastic wave equation with variable coefficients in the principal part of elliptic operator and an interior delay is considered. The dynamics is subject to a nonlinear boundary dissipation. This leads to a non-dissipative dynamics. The existence of solution is demonstrated by means of Faedo–Galerkin method combined with monotone operator theory in handling nonlinear boundary conditions. The main result pertains to exponential decay rates for energy, which depend on the geometry of the spatial domain, viscoelastic effects, the strength of delay and the strength of mechanical boundary damping. An important feature of the model is the fact that the delay term and stabilizing mechanism are not collocated geometrically — in contrast with many other works on the subject. This aspect of the problem requires the appropriate tools in order to exhibit propagation of the dissipation from one location to another. The precise ranges of admissible parameters characterizing the model and ensuring the stability are provided. The methods of proofs are routed in Riemannian geometry.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104252"},"PeriodicalIF":1.8,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143179288","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 : 2024-11-30DOI: 10.1016/j.nonrwa.2024.104270
Lijing Zhao, Shenzhou Zheng
<div><div>We devote this paper to a higher fractional differentiability of solutions for a class of parabolic double-phase equations <span><span><span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mtext>div</mtext><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>D</mi><mi>u</mi><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>D</mi><mi>u</mi></mrow></mfenced><mo>=</mo><mo>−</mo><mtext>div</mtext><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>F</mi><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>F</mi></mrow></mfenced><mspace></mspace><mtext>in</mtext><mspace></mspace><mspace></mspace><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>.</mo></mrow></math></span></span></span>A higher fractional differentiability of spatial gradients is established by way of the finite difference quotient, under assumptions that <span><math><mrow><mn>0</mn><mo>≤</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi><mo>,</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>, the exponents <span><math><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></math></span> satisfies <span><math><mrow><mn>2</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>q</mi><mo>≤</mo><mi>p</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>α</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span>, and <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> belongs to <span><math><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mi>ϑ</mi></mrow></msubsup><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>Φ</mi><mo>,</mo><mi>∞</mi><mo>;</mo><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mspace></mspace><mi>β</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mn>0</mn><mo><</mo><mi>β</mi><mo><</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>ϑ</mi><mo>≔</mo><mo>max</mo><mrow><mo>{</mo><mfrac><mrow><mi>q</mi><mrow><mo>(</mo><mn>2</mn><mi>q</mi><mo>−</mo><mi>p</mi><mo>)</mo></mro
{"title":"Higher fractional differentiability for solutions to parabolic equations with double-phase growth","authors":"Lijing Zhao, Shenzhou Zheng","doi":"10.1016/j.nonrwa.2024.104270","DOIUrl":"10.1016/j.nonrwa.2024.104270","url":null,"abstract":"<div><div>We devote this paper to a higher fractional differentiability of solutions for a class of parabolic double-phase equations <span><span><span><math><mrow><msub><mrow><mi>∂</mi></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mtext>div</mtext><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>D</mi><mi>u</mi><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>D</mi><mi>u</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>D</mi><mi>u</mi></mrow></mfenced><mo>=</mo><mo>−</mo><mtext>div</mtext><mfenced><mrow><msup><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow></mrow><mrow><mi>p</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>F</mi><mo>+</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><msup><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow></mrow><mrow><mi>q</mi><mo>−</mo><mn>2</mn></mrow></msup><mi>F</mi></mrow></mfenced><mspace></mspace><mtext>in</mtext><mspace></mspace><mspace></mspace><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>.</mo></mrow></math></span></span></span>A higher fractional differentiability of spatial gradients is established by way of the finite difference quotient, under assumptions that <span><math><mrow><mn>0</mn><mo>≤</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mi>α</mi><mo>,</mo><mfrac><mrow><mi>α</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mrow><mo>(</mo><msub><mrow><mi>Ω</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mi>α</mi><mo>∈</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow></math></span>, the exponents <span><math><mrow><mi>p</mi><mo>,</mo><mi>q</mi></mrow></math></span> satisfies <span><math><mrow><mn>2</mn><mo>≤</mo><mi>p</mi><mo>≤</mo><mi>q</mi><mo>≤</mo><mi>p</mi><mo>+</mo><mfrac><mrow><mn>2</mn><mi>α</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>2</mn></mrow></mfrac></mrow></math></span>, and <span><math><mrow><mi>F</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></mrow></math></span> belongs to <span><math><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mi>ϑ</mi></mrow></msubsup><mrow><mo>(</mo><mrow><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>Φ</mi><mo>,</mo><mi>∞</mi><mo>;</mo><mi>l</mi><mi>o</mi><mi>c</mi></mrow><mrow><mspace></mspace><mi>β</mi></mrow></msubsup><mrow><mo>(</mo><mi>Ω</mi><mo>)</mo></mrow></mrow><mo>)</mo></mrow></mrow></math></span> for <span><math><mrow><mn>0</mn><mo><</mo><mi>β</mi><mo><</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>ϑ</mi><mo>≔</mo><mo>max</mo><mrow><mo>{</mo><mfrac><mrow><mi>q</mi><mrow><mo>(</mo><mn>2</mn><mi>q</mi><mo>−</mo><mi>p</mi><mo>)</mo></mro","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104270"},"PeriodicalIF":1.8,"publicationDate":"2024-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142757322","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 : 2024-11-29DOI: 10.1016/j.nonrwa.2024.104265
Hiroshi Watanabe
In this paper, we consider qualitative properties for entropy solutions to one-dimensional Cauchy problems (CP) for scalar parabolic–hyperbolic conservation laws. Since the equations have both properties of hyperbolic equations and those of parabolic equations, it is difficult to investigate the behavior of solutions to (CP). In our previous works, we focused on the traveling wave structure instead of the self-similar structure. In fact, we succeeded in constructing shock wave type traveling waves with multiple discontinuity. Moreover, we constructed rarefaction wave type sub-, super-solutions to (CP) and investigated their properties.
In the present paper, we investigate “-wave-like properties” for entropy solutions to (CP) while we are not able to construct an analogue of -waves. In particular, we derive generalized one-sided Lipschitz estimates (Oleinik type entropy estimates) and decay estimates for entropy solutions to (CP). Based on the decay estimates, we discuss the asymptotic profiles of entropy solutions to (CP) under some specific setting.
{"title":"N-wave-like properties for entropy solutions to scalar parabolic–hyperbolic conservation laws","authors":"Hiroshi Watanabe","doi":"10.1016/j.nonrwa.2024.104265","DOIUrl":"10.1016/j.nonrwa.2024.104265","url":null,"abstract":"<div><div>In this paper, we consider qualitative properties for entropy solutions to one-dimensional Cauchy problems (CP) for scalar parabolic–hyperbolic conservation laws. Since the equations have both properties of hyperbolic equations and those of parabolic equations, it is difficult to investigate the behavior of solutions to (CP). In our previous works, we focused on the traveling wave structure instead of the self-similar structure. In fact, we succeeded in constructing shock wave type traveling waves with multiple discontinuity. Moreover, we constructed rarefaction wave type sub-, super-solutions to (CP) and investigated their properties.</div><div>In the present paper, we investigate “<span><math><mi>N</mi></math></span>-wave-like properties” for entropy solutions to (CP) while we are not able to construct an analogue of <span><math><mi>N</mi></math></span>-waves. In particular, we derive generalized one-sided Lipschitz estimates (Oleinik type entropy estimates) and decay estimates for entropy solutions to (CP). Based on the decay estimates, we discuss the asymptotic profiles of entropy solutions to (CP) under some specific setting.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"84 ","pages":"Article 104265"},"PeriodicalIF":1.8,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142745191","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 : 2024-11-22DOI: 10.1016/j.nonrwa.2024.104263
Cecilia Cavaterra , Maurizio Grasselli , Muhammed Ali Mehmood , Riccardo Voso
We consider an evolution system modeling a flow of colloidal particles which are suspended in an incompressible fluid and accounts for colloidal crystallization. The system consists of the Navier–Stokes equations for the volume averaged velocity coupled with the so-called Phase-Field Crystal equation for the density deviation. Considering this system in a periodic domain and assuming that the viscosity as well as the mobility depend on the density deviation, we first prove the existence of a weak solution in dimension three. Then, in dimension two, we establish the existence of a (unique) strong solution.
{"title":"Analysis of a Navier–Stokes phase-field crystal system","authors":"Cecilia Cavaterra , Maurizio Grasselli , Muhammed Ali Mehmood , Riccardo Voso","doi":"10.1016/j.nonrwa.2024.104263","DOIUrl":"10.1016/j.nonrwa.2024.104263","url":null,"abstract":"<div><div>We consider an evolution system modeling a flow of colloidal particles which are suspended in an incompressible fluid and accounts for colloidal crystallization. The system consists of the Navier–Stokes equations for the volume averaged velocity coupled with the so-called Phase-Field Crystal equation for the density deviation. Considering this system in a periodic domain and assuming that the viscosity as well as the mobility depend on the density deviation, we first prove the existence of a weak solution in dimension three. Then, in dimension two, we establish the existence of a (unique) strong solution.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"83 ","pages":"Article 104263"},"PeriodicalIF":1.8,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704854","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 : 2024-11-22DOI: 10.1016/j.nonrwa.2024.104262
Tiantian Zhao , Kai Yan
In the present study, we construct a new blow-up of strong solution to show wave breaking for the well-known Degasperis–Procesi equation. Unlike the previous related results for the shallow water wave models, no conservation law is used here.
{"title":"Wave breaking for the Degasperis–Procesi equation","authors":"Tiantian Zhao , Kai Yan","doi":"10.1016/j.nonrwa.2024.104262","DOIUrl":"10.1016/j.nonrwa.2024.104262","url":null,"abstract":"<div><div>In the present study, we construct a new blow-up of strong solution to show wave breaking for the well-known Degasperis–Procesi equation. Unlike the previous related results for the shallow water wave models, no conservation law is used here.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"83 ","pages":"Article 104262"},"PeriodicalIF":1.8,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704855","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}
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