Journal of Differential Equations最新文献

{"title":"Random dynamics of the stochastic Landau-Lifshitz-Bloch equation with colored noise in the real line","authors":"Daiwen Huang ,&nbsp;Zhaoyang Qiu","doi":"10.1016/j.jde.2025.113314","DOIUrl":"10.1016/j.jde.2025.113314","url":null,"abstract":"<div><div>In this paper, we are concerned the stochastic Landau-Lifshitz-Bloch equation driven by the colored noise, evolving in the entire real line. First, the well-posedness of strong solution is established using a domain expansion method. Then, we consider the existence and uniqueness of the pullback random attractors in regularity space <span><math><msup><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>. Finally, we prove the upper semi-continuity of the attractors as the noise coefficient <em>α</em> tending to zero. The uniform tail-ends estimates of solutions for overcoming the non-compactness difficulty of Sobolev embedding in unbounded domains and the energy method due to Ball are invoked to establish the asymptotic compactness of solutions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113314"},"PeriodicalIF":2.4,"publicationDate":"2025-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143847909","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Global existence and uniqueness of strong solutions to the 2D nonhomogeneous primitive equations with density-dependent viscosity","authors":"Quansen Jiu ,&nbsp;Lin Ma ,&nbsp;Fengchao Wang","doi":"10.1016/j.jde.2025.113321","DOIUrl":"10.1016/j.jde.2025.113321","url":null,"abstract":"<div><div>This paper is concerned with an initial-boundary value problem of the two-dimensional inhomogeneous primitive equations with density-dependent viscosity. The global well-posedness of strong solutions is established, provided the initial horizontal velocity is suitably small, that is, <span><math><msub><mrow><mo>‖</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>≤</mo><msub><mrow><mi>η</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> for suitably small <span><math><msub><mrow><mi>η</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>&gt;</mo><mn>0</mn></math></span>. The initial data may contain vacuum. The proof is based on the local well-posedness and the blow-up criterion proved in <span><span>[15]</span></span>, which states that if <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup></math></span> is the maximal existence time of the local strong solutions <span><math><mo>(</mo><mi>ρ</mi><mo>,</mo><mi>u</mi><mo>,</mo><mi>w</mi><mo>,</mo><mi>P</mi><mo>)</mo></math></span> and <span><math><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>&lt;</mo><mo>∞</mo></math></span>, then<span><span><span><math><munder><mi>sup</mi><mrow><mn>0</mn><mo>≤</mo><mi>t</mi><mo>&lt;</mo><msup><mrow><mi>T</mi></mrow><mrow><mo>⁎</mo></mrow></msup></mrow></munder><mo>⁡</mo><mo>(</mo><msub><mrow><mo>‖</mo><mi>∇</mi><mi>ρ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><msup><mrow><mi>∇</mi></mrow><mrow><mn>2</mn></mrow></msup><mi>ρ</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>+</mo><msub><mrow><mo>‖</mo><mi>∇</mi><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>‖</mo></mrow><mrow><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></msub><mo>)</mo><mo>=</mo><mo>∞</mo><mo>.</mo></math></span></span></span> To complete the proof, it is required to make an estimate on a key term <span><math><msub><mrow><mo>‖</mo><mi>∇</mi><msub><mrow><mi>u</mi></mrow><mrow><mi>t</mi></mrow></msub><mo>‖</mo></mrow><mrow><msubsup><mrow><mi>L</mi></mrow><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msubsup><msubsup><mrow><mi>L</mi></mrow><mrow><mi>Ω</mi></mrow><mrow><mn>2</mn></mrow></msubsup></mrow></msub></math></span>. We prove that it is bounded and could be as small as desired under certain smallness conditions, by making use of the regularity result of hydrostatic Stokes equations and some careful time weighted estimates.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"436 ","pages":"Article 113321"},"PeriodicalIF":2.4,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844318","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fixed point indices of iterates of orientation-reversing homeomorphisms","authors":"Grzegorz Graff ,&nbsp;Patryk Topór","doi":"10.1016/j.jde.2025.113322","DOIUrl":"10.1016/j.jde.2025.113322","url":null,"abstract":"<div><div>We show that any sequence of integers satisfying necessary Dold's congruences is realized as the sequence of fixed point indices of the iterates of an orientation-reversing homeomorphism of <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>m</mi></mrow></msup></math></span> for <span><math><mi>m</mi><mo>≥</mo><mn>3</mn></math></span>. As an element of the construction of the above homeomorphism, we consider the class of boundary-preserving homeomorphisms of <span><math><msubsup><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> and give the answer to the problem posed by Barge and Wójcik (2017) <span><span>[2]</span></span> providing a complete description of the forms of fixed point indices for this class of maps.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"436 ","pages":"Article 113322"},"PeriodicalIF":2.4,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The non-homogeneous Dirichlet problem for the p(x)-Laplacian with unbounded p(x) on a smooth domain","authors":"Mohamed A. Khamsi ,&nbsp;Jan Lang ,&nbsp;Osvaldo Méndez ,&nbsp;Aleš Nekvinda","doi":"10.1016/j.jde.2025.113316","DOIUrl":"10.1016/j.jde.2025.113316","url":null,"abstract":"<div><div>This paper examines the solvability of the Dirichlet problem for the variable exponent <em>p</em>-Laplacian in the case of unbounded <span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. For a bounded domain <span><math><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> with a smooth boundary and <em>p</em> satisfying <span><math><msub><mrow><mi>p</mi></mrow><mrow><mo>−</mo></mrow></msub><mo>=</mo><munder><mrow><mi>essinf</mi></mrow><mrow><mi>Ω</mi></mrow></munder><mspace></mspace><mi>p</mi><mo>&gt;</mo><mi>n</mi></math></span> and <em>φ</em> in the Sobolev space <span><math><msup><mrow><mi>W</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>, we investigate the problem<span><span><span><math><msub><mrow><mi>Δ</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><mn>0</mn><mspace></mspace><mtext>in</mtext><mspace></mspace><mi>Ω</mi><mo>,</mo><mspace></mspace><mi>u</mi><msub><mrow><mo>|</mo></mrow><mrow><mo>∂</mo><mi>Ω</mi></mrow></msub><mo>=</mo><mi>φ</mi><mo>.</mo></math></span></span></span> We introduce the space <span><math><msubsup><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span>, which is the natural solution space for the minimization of the Dirichlet integral given the unbounded nature of <span><math><mi>p</mi><mo>(</mo><mi>x</mi><mo>)</mo></math></span>. Our main results establish the existence and uniqueness of solutions within this space. Since <span><math><msubsup><mrow><mi>V</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mi>p</mi></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> is not defined via a TVS topology, the paper includes the description of the necessary modular topological framework and discusses Clarkson-type inequalities for unbounded variable exponents, which are interesting in their own right.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113316"},"PeriodicalIF":2.4,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Linear inviscid damping for monotonic shear flow in unbounded domain","authors":"Siqi Ren","doi":"10.1016/j.jde.2025.113287","DOIUrl":"10.1016/j.jde.2025.113287","url":null,"abstract":"<div><div>In this paper, we study the 2-D incompressible Euler equation in unbounded domain <span><math><mi>T</mi><mo>×</mo><mi>R</mi></math></span>, linearized around a class of monotonic shear flow whose derivatives degenerate with same exponentially rate at infinity. We prove the linear inviscid damping with exponential weighted Sobolev initial data.</div><div>Our proof includes four parts: limiting absorption principle for Rayleigh equation, space-time estimate, vector field method and semigroup estimate. To seize the degeneracy of the derivatives of the flow, all of our estimates are weighted with the widest range. To handle the lack of compactness for the non-local term, we use blow-up analysis in the proof of limiting absorption principle.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113287"},"PeriodicalIF":2.4,"publicationDate":"2025-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143844502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Cauchy problem in function spaces with asymptotic expansions with respect to time variable t","authors":"Sunao Ōuchi","doi":"10.1016/j.jde.2025.113335","DOIUrl":"10.1016/j.jde.2025.113335","url":null,"abstract":"<div><div>A system of nonlinear Cauchy problem <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>U</mi><mo>,</mo><msub><mrow><mi>∇</mi></mrow><mrow><mi>x</mi></mrow></msub><mi>U</mi><mo>)</mo></math></span> <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mn>0</mn><mo>,</mo><mi>x</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mo>,</mo><mn>0</mn></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> is studied in function spaces with asymptotic expansion with respect to <em>t</em>. To be specific, it is discussed in Borel summable or multisummable function space. It is recognized that these functions are important classes in asymptotic analysis. We study equations under the condition <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>f</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>U</mi><mo>,</mo><mi>P</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> are in these function spaces with respect to <em>t</em> and show <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>)</mo><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>m</mi></mrow></msubsup></math></span> have also the same summability.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"437 ","pages":"Article 113335"},"PeriodicalIF":2.4,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143837940","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On a class of superlinear obstacle problems","authors":"Cong Wang","doi":"10.1016/j.jde.2025.113318","DOIUrl":"10.1016/j.jde.2025.113318","url":null,"abstract":"<div><div>In this paper, we focus on the superlinear obstacle problem,<span><span><span><math><mrow><mrow><mo>{</mo><mtable><mtr><mtd><mi>Δ</mi><mi>u</mi><mo>=</mo><mo>(</mo><mn>1</mn><mo>−</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo><msub><mrow><mi>χ</mi></mrow><mrow><mo>{</mo><mi>u</mi><mo>&gt;</mo><mn>0</mn><mo>}</mo></mrow></msub><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mi>in</mi></mrow><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>≥</mo><mn>0</mn><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mi>in</mi></mrow><mspace></mspace><mi>Ω</mi><mo>,</mo></mtd></mtr><mtr><mtd><mi>u</mi><mo>∈</mo><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>,</mo><mspace></mspace></mtd></mtr></mtable></mrow></mrow></math></span></span></span> where Ω is a smooth open bounded domain in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>,</mo><mi>n</mi><mo>≥</mo><mn>2</mn></math></span>, and the exponent <span><math><mi>p</mi><mo>&gt;</mo><mn>2</mn></math></span> satisfies the condition<span><span><span><math><mrow><mn>2</mn><mo>&lt;</mo><mi>p</mi><mo>&lt;</mo><mrow><mo>{</mo><mtable><mtr><mtd><msup><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msup><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mi>as</mi></mrow><mspace></mspace><mi>n</mi><mo>≥</mo><mn>3</mn><mo>,</mo></mtd></mtr><mtr><mtd><mo>+</mo><mo>∞</mo><mo>,</mo><mspace></mspace><mspace></mspace></mtd><mtd><mrow><mi>as</mi></mrow><mspace></mspace><mi>n</mi><mo>=</mo><mn>2</mn><mo>,</mo></mtd></mtr></mtable></mrow></mrow></math></span></span></span> with <span><math><msup><mrow><mn>2</mn></mrow><mrow><mo>⁎</mo></mrow></msup><mo>=</mo><mfrac><mrow><mn>2</mn><mi>n</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></math></span> is the Sobolev critical exponent of embedding <span><math><msubsup><mrow><mi>W</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msubsup><mo>(</mo><mi>Ω</mi><mo>)</mo><mo>↪</mo><msup><mrow><mi>L</mi></mrow><mrow><mi>q</mi></mrow></msup><mo>(</mo><mi>Ω</mi><mo>)</mo></math></span> when <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>. We prove the existence of a non-minimizing solution and establish the optimal regularity of solutions to the aforementioned equation. Furthermore, utilizing blowup analysis, we derive the regularity of the free boundary at regular points and characterize the structure of the singular set.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113318"},"PeriodicalIF":2.4,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143838627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Invariant manifolds of 3D piecewise vector fields","authors":"Bruno R. Freitas,&nbsp;Samuel C.S. Ferreira,&nbsp;João C.R. Medrado","doi":"10.1016/j.jde.2025.113313","DOIUrl":"10.1016/j.jde.2025.113313","url":null,"abstract":"<div><div>We analyze a 3D piecewise linear dynamical system <span><math><mi>Z</mi><mo>=</mo><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></math></span> with a plane Σ as its switching manifold containing two-fold parallel straight lines. The eigenvalues associated with <em>X</em> and <em>Y</em> are composed of two complex eigenvalues and one non-zero real eigenvalue. Using a suitable canonical form and exponential matrices theory, we generate two closing equations, from which we derive two half-return Poincaré maps. By defining the displacement map as the difference between the two half-return Poincaré maps from the same point, we prove using the Weierstrass preparation theorem that there exists a 3D piecewise linear dynamical system that admits three invariant cylinders of big amplitude, with exactly one limit cycle in each cylinder, a surface cone-like cylinder, and a cylinder filled with closed orbits. Lastly, we provide examples of 3D piecewise linear dynamical systems that present three limit cycles, a cone-like surface, and a cylinder filled with closed orbits, respectively.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113313"},"PeriodicalIF":2.4,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Zero dissipation and time decay rates to the planar rarefaction wave for 3-D compressible Euler equation in the whole space","authors":"Guiqin Qiu ,&nbsp;Lingda Xu","doi":"10.1016/j.jde.2025.113324","DOIUrl":"10.1016/j.jde.2025.113324","url":null,"abstract":"<div><div>We investigate the large-time behavior and inviscid limit of solutions to a class of three-dimensional (3D) viscous hyperbolic systems, focusing on their convergence toward a planar rarefaction wave of the 3D isentropic compressible Euler equation. We prove that the planar rarefaction wave is time-asymptotically stable and the solutions converge to the Euler planar rarefaction wave. Notably, decay rates that depends explicitly on both time and the viscosity coefficient are obtained. Moreover, the strength of the rarefaction wave can be large, and the initial perturbation is in the whole space. To the best of our knowledge, this is the first result to derive both the time-dependent and viscosity-dependent dissipation rates for rarefaction waves. Further, it is also the first result which successfully obtained the time decay rate around the planar rarefaction wave in the case of systems.</div><div>To address the error terms arising from approximating the viscous hyperbolic system with the inviscid planar rarefaction wave, we construct a new smooth viscous profile using Riemann invariants. This profile satisfies a one-dimensional (1D) parabolic system and serves as an effective approximation to the planar rarefaction wave. By applying a scaling transformation and deriving delicate time-weighted <span><math><msup><mrow><mi>L</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>-energy estimates, we establish a uniform convergence rate to the planar rarefaction wave. Moreover, the result is in the whole space, which means we don't need any periodic assumptions on any spatial directions.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113324"},"PeriodicalIF":2.4,"publicationDate":"2025-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143839068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Regularity of weak solutions to generalized quasilinear elliptic obstacle problems","authors":"J. Vanterler da C. Sousa ,&nbsp;Kishor D. Kucche ,&nbsp;Adrian R.G. Plata","doi":"10.1016/j.jde.2025.113317","DOIUrl":"10.1016/j.jde.2025.113317","url":null,"abstract":"<div><div>In this paper, the work is divided into two parts. First, we are interested in presenting some estimates involving the generalized <em>ψ</em>-Hilfer operator in the domain of the ball <span><math><mi>B</mi><mo>(</mo><mi>R</mi><mo>)</mo></math></span> by means of classical inequalities such as Young's, Holder's, and Harnack's. On the other hand, motivated by such results, we will investigate the <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>α</mi></mrow></msup></math></span>-regularity of weak solutions for a new class of <em>m</em>-Laplacian obstacle problems with <span><math><msup><mrow><mi>C</mi></mrow><mrow><mn>1</mn><mo>,</mo><mover><mrow><mi>β</mi></mrow><mo>‾</mo></mover></mrow></msup></math></span>-obstacle function under controllable growth conditions in the <em>ψ</em>-fractional space.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113317"},"PeriodicalIF":2.4,"publicationDate":"2025-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143833782","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}