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}
Pub Date : 2024-11-22DOI: 10.1016/j.nonrwa.2024.104264
A. Nachaoui , A. Laghrib , A. Hadri , M. Nachaoui
Within the realm of image processing, image denoising holds significant importance. This study focuses on tackling denoising challenges posed by Speckle noise. We introduce a novel variable -PDE-based denoising model, offering a fresh perspective. Our approach involves a unique class of PDEs, wherein the nonlinear structure relies on spatially nonlocal exponent dependent factors linked to the target solution and also its gradient. This innovation incorporates grayscale information by introducing the variable exponent , which controls much better the diffusion and incorporates information from wide regions. The existence and uniqueness of the proposed PDE are established through Galerkin’s approximation. Furthermore, a series of experiments are conducted for denoising, including comparisons with other models, in order to validate the selection of the variable exponent parameter. This research contributes to the advancement of image denoising methods with high theoretical foundations and potential implications for other applications.
{"title":"A novel variable exponent PDE with dependency on γ(u,|∇u0,σ|) for image despeckling application","authors":"A. Nachaoui , A. Laghrib , A. Hadri , M. Nachaoui","doi":"10.1016/j.nonrwa.2024.104264","DOIUrl":"10.1016/j.nonrwa.2024.104264","url":null,"abstract":"<div><div>Within the realm of image processing, image denoising holds significant importance. This study focuses on tackling denoising challenges posed by Speckle noise. We introduce a novel variable <span><math><mrow><mi>γ</mi><mrow><mo>(</mo><mi>u</mi><mo>,</mo><mrow><mo>|</mo><mo>∇</mo><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn><mo>,</mo><mi>σ</mi></mrow></msub><mo>|</mo></mrow><mo>)</mo></mrow></mrow></math></span>-PDE-based denoising model, offering a fresh perspective. Our approach involves a unique class of PDEs, wherein the nonlinear structure relies on spatially nonlocal exponent dependent factors linked to the target solution and also its gradient. This innovation incorporates grayscale information by introducing the variable exponent <span><math><mi>γ</mi></math></span>, which controls much better the diffusion and incorporates information from wide regions. The existence and uniqueness of the proposed PDE are established through Galerkin’s approximation. Furthermore, a series of experiments are conducted for denoising, including comparisons with other models, in order to validate the selection of the variable exponent parameter. This research contributes to the advancement of image denoising methods with high theoretical foundations and potential implications for other applications.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"83 ","pages":"Article 104264"},"PeriodicalIF":1.8,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704857","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 : 2024-11-20DOI: 10.1016/j.nonrwa.2024.104261
Shengfeng Zhao, Li Xie
<div><div>This paper is focused on the zero-flux initial–boundary value problem for a forager-exploiter model of the form <span><span><span><math><mfenced><mrow><mtable><mtr><mtd></mtd><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>u</mi><mo>∇</mo><mi>w</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo></mtd><mtd></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>v</mi><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>−</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo></mtd><mtd></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mi>w</mi><mo>−</mo><mi>g</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mi>w</mi><mo>−</mo><mi>μ</mi><mi>w</mi><mo>+</mo><mi>r</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mtd><mtd></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>in a smoothly bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, where <span><math><mi>μ</mi></math></span>, <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><mi>m</mi></math></span>, <span><math><mi>l</mi></math></span> are positive constants, <span><math><mrow><mi>r</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>¯</mo></mover><mo>×</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>∩</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is a given nonnegative function, the functions <span><math><mrow><mi>f</mi><mo>,</mo><mspace></mspace><mi>g</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>
本文主要研究觅食者-开发者模型的零流量初始边界值问题，其形式为 ut=Δu-∇⋅(u∇w)+μ1(u-um),x∈Ω,t>；0,vt=Δv-∇⋅(v∇u)+μ2(v-vl),x∈Ω,t>0,wt=Δw-f(u)w-g(v)w-μw+r(x,t),x∈Ω,t>；0,in a smooth bounded domain Ω⊂R2, where μ, μ1, μ2, m, l are positive constants, r(x,t)∈C1(Ω¯×[0,∞))∩L∞(Ω×(0,∞)) is a given nonnegative function、假设函数 f，g∈C1[0,∞]的性质分别类似于 uα，vβ，并有一些正常数 α 和 β。结果表明，只要 m≥1 , l≥1, α≤m2 和 β<l2, 初界值问题就具有全局有界经典解。
{"title":"Global existence and boundedness of solutions to a two-dimensional forager-exploiter model with/without logistic source","authors":"Shengfeng Zhao, Li Xie","doi":"10.1016/j.nonrwa.2024.104261","DOIUrl":"10.1016/j.nonrwa.2024.104261","url":null,"abstract":"<div><div>This paper is focused on the zero-flux initial–boundary value problem for a forager-exploiter model of the form <span><span><span><math><mfenced><mrow><mtable><mtr><mtd></mtd><mtd><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>u</mi><mo>−</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>u</mi><mo>∇</mo><mi>w</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>u</mi><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo></mtd><mtd></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>v</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>v</mi><mo>−</mo><mo>∇</mo><mi>⋅</mi><mrow><mo>(</mo><mi>v</mi><mo>∇</mo><mi>u</mi><mo>)</mo></mrow><mo>+</mo><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>v</mi><mo>−</mo><msup><mrow><mi>v</mi></mrow><mrow><mi>l</mi></mrow></msup><mo>)</mo></mrow><mo>,</mo></mtd><mtd></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr><mtr><mtd></mtd><mtd><msub><mrow><mi>w</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>=</mo><mi>Δ</mi><mi>w</mi><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mi>w</mi><mo>−</mo><mi>g</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mi>w</mi><mo>−</mo><mi>μ</mi><mi>w</mi><mo>+</mo><mi>r</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo></mtd><mtd></mtd><mtd><mi>x</mi><mo>∈</mo><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>></mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></mfenced></math></span></span></span>in a smoothly bounded domain <span><math><mrow><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span>, where <span><math><mi>μ</mi></math></span>, <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>μ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><mi>m</mi></math></span>, <span><math><mi>l</mi></math></span> are positive constants, <span><math><mrow><mi>r</mi><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>(</mo><mover><mrow><mi>Ω</mi></mrow><mo>¯</mo></mover><mo>×</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>∩</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>∞</mi></mrow></msup><mrow><mo>(</mo><mi>Ω</mi><mo>×</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mi>∞</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> is a given nonnegative function, the functions <span><math><mrow><mi>f</mi><mo>,</mo><mspace></mspace><mi>g</mi><mo>∈</mo><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn></mrow></msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"83 ","pages":"Article 104261"},"PeriodicalIF":1.8,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704853","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 : 2024-11-20DOI: 10.1016/j.nonrwa.2024.104259
Abderrahim Charkaoui , Anouar Ben-Loghfyry
This research paper proposes a novel parabolic model driven by a nonlinear operator with a variable exponent applied to multi-frame image super-resolution. Our idea is based essentially on enhancing the classical image super-resolution models by considering novel regularized terms involving -growth structure. This regularization leads to deriving a new nonlinear parabolic PDE with nonstandard growth conditions. We start initially by examining the theoretical solvability of our model. We employ the so-called variable exponents Lebesgue-Sobolev spaces to establish an appropriate functional framework for the theoretical investigation of our proposed model. We then apply the Faedo–Galerkin method to establish both the existence and uniqueness of a weak solution for the proposed model. To validate the effectiveness of our model in the multi-frame super resolution (SR) context, we conduct numerical experiments on Magnetic Resonance Images (MRI) featuring diverse characteristics, including corners and edges, while applying different warping, decimation and blurring matrices with noises on the low-resolution (LR) images. We initiate the evaluation by introducing an adaptive discrete scheme of the proposed model. To prove the robustness of our approach, we subject our images to varying levels of noise while conducting many behavior tests on some parameters with major contributions. Additionally, we perform simulations on real data (videos) to show the superiority of the proposed model. The obtained high resolution (HR) results demonstrate notable efficiency and robustness against noise, outperforming the competitive models visually and quantitatively.
{"title":"A class of nonlinear parabolic PDEs with variable growth structure applied to multi-frame MRI super-resolution","authors":"Abderrahim Charkaoui , Anouar Ben-Loghfyry","doi":"10.1016/j.nonrwa.2024.104259","DOIUrl":"10.1016/j.nonrwa.2024.104259","url":null,"abstract":"<div><div>This research paper proposes a novel parabolic model driven by a nonlinear operator with a variable exponent applied to multi-frame image super-resolution. Our idea is based essentially on enhancing the classical image super-resolution models by considering novel regularized terms involving <span><math><mrow><mi>p</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></math></span>-growth structure. This regularization leads to deriving a new nonlinear parabolic PDE with nonstandard growth conditions. We start initially by examining the theoretical solvability of our model. We employ the so-called variable exponents Lebesgue-Sobolev spaces to establish an appropriate functional framework for the theoretical investigation of our proposed model. We then apply the <em>Faedo–Galerkin</em> method to establish both the existence and uniqueness of a weak solution for the proposed model. To validate the effectiveness of our model in the multi-frame super resolution (SR) context, we conduct numerical experiments on Magnetic Resonance Images (MRI) featuring diverse characteristics, including corners and edges, while applying different warping, decimation and blurring matrices with noises on the low-resolution (LR) images. We initiate the evaluation by introducing an adaptive discrete scheme of the proposed model. To prove the robustness of our approach, we subject our images to varying levels of noise while conducting many behavior tests on some parameters with major contributions. Additionally, we perform simulations on real data (videos) to show the superiority of the proposed model. The obtained high resolution (HR) results demonstrate notable efficiency and robustness against noise, outperforming the competitive models visually and quantitatively.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"83 ","pages":"Article 104259"},"PeriodicalIF":1.8,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142704858","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 : 2024-11-16DOI: 10.1016/j.nonrwa.2024.104260
Duong Xuan Vinh , Mai Duc Thanh , Nguyen Huu Hiep
Undercompressive phase transitions violating Lax shock inequalities in a model of fluid flows in a nozzle with discontinuous cross-section area are studied. The Riemann problem involving phase transitions is considered. Depending on the choice of admissibility criteria suitable for a specific application, one can obtain a Riemann solver, which may involve nonclassical shock wave. The resonance phenomenon is also observed as multiple shocks waves of the same speed can apparently appear in a single solution. The Riemann problem may admit a unique solution in some region, but may have up to three distinct solutions in other regions.
{"title":"Undercompressive phase transitions for the model of fluid flows in a nozzle with discontinuous cross-sectional area","authors":"Duong Xuan Vinh , Mai Duc Thanh , Nguyen Huu Hiep","doi":"10.1016/j.nonrwa.2024.104260","DOIUrl":"10.1016/j.nonrwa.2024.104260","url":null,"abstract":"<div><div>Undercompressive phase transitions violating Lax shock inequalities in a model of fluid flows in a nozzle with discontinuous cross-section area are studied. The Riemann problem involving phase transitions is considered. Depending on the choice of admissibility criteria suitable for a specific application, one can obtain a Riemann solver, which may involve nonclassical shock wave. The resonance phenomenon is also observed as multiple shocks waves of the same speed can apparently appear in a single solution. The Riemann problem may admit a unique solution in some region, but may have up to three distinct solutions in other regions.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"82 ","pages":"Article 104260"},"PeriodicalIF":1.8,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659195","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 : 2024-11-16DOI: 10.1016/j.nonrwa.2024.104258
Braulio B.V. Maia , Alânnio B. Nóbrega
In this paper, we study a class of elliptic problems with a interior interface condition, which arise in population dynamics. In these problems, each population lives in a subdomain and they interact in a common border, which acts as a geographical barrier. The main novelty in our work is the presence of a nonlocal reaction terms. To obtain our results we employ mainly bifurcation methods.
{"title":"Bifurcation results for a class of elliptic equations with a nonlocal reaction term and interior interface boundary conditions","authors":"Braulio B.V. Maia , Alânnio B. Nóbrega","doi":"10.1016/j.nonrwa.2024.104258","DOIUrl":"10.1016/j.nonrwa.2024.104258","url":null,"abstract":"<div><div>In this paper, we study a class of elliptic problems with a interior interface condition, which arise in population dynamics. In these problems, each population lives in a subdomain and they interact in a common border, which acts as a geographical barrier. The main novelty in our work is the presence of a nonlocal reaction terms. To obtain our results we employ mainly bifurcation methods.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"82 ","pages":"Article 104258"},"PeriodicalIF":1.8,"publicationDate":"2024-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659196","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 : 2024-11-09DOI: 10.1016/j.nonrwa.2024.104256
Tiago Carvalho , Jackson Cunha , Rodrigo Euzébio , Marco Florentino
In this paper we study the dynamics of a piecewise smooth vector field modeling an intermittent human immunodeficiency virus treatment where the patient is recurrently submitted and removed from drug administration. In fact, the protocol says that the drugs are administered when the level of CD4 T defense cells is smaller than a fixed number . When the level of CD4 T cells is greater than a fixed number (distinct from ) the drugs are not administered to provide a better recovery from side effects. Moreover, the orbits of the piecewise smooth vector fields are trapped within a compact set, which proves that the protocol controls the disease.
在本文中，我们研究了一个片断平滑矢量场的动力学模型，它模拟了一种间歇性人类免疫缺陷病毒治疗方法，在这种治疗方法中，病人会反复服药和停药。事实上，治疗方案规定，当 CD4+ T 防御细胞的水平小于一个固定的数字 CoffT 时，就会给药。当 CD4+ T 细胞的水平大于一个固定的数字 ConT（与 CoffT 不同）时，则不用药，以便更好地从副作用中恢复过来。此外，片断平滑矢量场的轨道被困在一个紧凑集合内，这证明该方案能控制疾病。
{"title":"Dynamics of an intermittent HIV treatment using piecewise smooth vector fields with two switching manifolds","authors":"Tiago Carvalho , Jackson Cunha , Rodrigo Euzébio , Marco Florentino","doi":"10.1016/j.nonrwa.2024.104256","DOIUrl":"10.1016/j.nonrwa.2024.104256","url":null,"abstract":"<div><div>In this paper we study the dynamics of a piecewise smooth vector field modeling an intermittent human immunodeficiency virus treatment where the patient is recurrently submitted and removed from drug administration. In fact, the protocol says that the drugs are administered when the level of CD4<span><math><msup><mrow></mrow><mrow><mo>+</mo></mrow></msup></math></span> T defense cells is smaller than a fixed number <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>o</mi><mi>f</mi><mi>f</mi></mrow><mrow><mi>T</mi></mrow></msubsup></math></span>. When the level of CD4<span><math><msup><mrow></mrow><mrow><mo>+</mo></mrow></msup></math></span> T cells is greater than a fixed number <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>o</mi><mi>n</mi></mrow><mrow><mi>T</mi></mrow></msubsup></math></span> (distinct from <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mi>o</mi><mi>f</mi><mi>f</mi></mrow><mrow><mi>T</mi></mrow></msubsup></math></span>) the drugs are not administered to provide a better recovery from side effects. Moreover, the orbits of the piecewise smooth vector fields are trapped within a compact set, which proves that the protocol controls the disease.</div></div>","PeriodicalId":49745,"journal":{"name":"Nonlinear Analysis-Real World Applications","volume":"82 ","pages":"Article 104256"},"PeriodicalIF":1.8,"publicationDate":"2024-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142659198","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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