Journal of Differential Equations最新文献

{"title":"Asymptotic blow-up behavior for the semilinear heat equation with non scale invariant nonlinearity","authors":"Loth Damagui Chabi","doi":"10.1016/j.jde.2025.113303","DOIUrl":"10.1016/j.jde.2025.113303","url":null,"abstract":"<div><div>We characterize the asymptotic behavior near blowup points for positive solutions of the semilinear heat equation<span><span><span><math><msub><mrow><mo>∂</mo></mrow><mrow><mi>t</mi></mrow></msub><mi>u</mi><mo>−</mo><mi>Δ</mi><mi>u</mi><mo>=</mo><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo></math></span></span></span> for nonlinearities which are genuinely non scale invariant, unlike in the standard case <span><math><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>. Indeed, our results apply to a large class of nonlinearities of the form <span><math><mi>f</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>u</mi></mrow><mrow><mi>p</mi></mrow></msup><mi>L</mi><mo>(</mo><mi>u</mi><mo>)</mo></math></span>, where <span><math><mi>p</mi><mo>&gt;</mo><mn>1</mn></math></span> is Sobolev subcritical and <em>L</em> is a slowly varying function at infinity (which includes for instance logarithms and their powers and iterates, as well as some strongly oscillating functions).</div><div>More precisely, denoting by <em>ψ</em> the unique positive solution of the corresponding ODE <span><math><msup><mrow><mi>y</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>)</mo></math></span> which blows up at the same time <em>T</em>, we show that if <span><math><mi>a</mi><mo>∈</mo><mi>Ω</mi></math></span> is a blowup point of <em>u</em>, then<span><span><span><math><munder><mi>lim</mi><mrow><mi>t</mi><mo>→</mo><mi>T</mi></mrow></munder><mo>⁡</mo><mfrac><mrow><mi>u</mi><mo>(</mo><mi>a</mi><mo>+</mo><mi>y</mi><msqrt><mrow><mi>T</mi><mo>−</mo><mi>t</mi></mrow></msqrt><mo>,</mo><mi>t</mi><mo>)</mo></mrow><mrow><mi>ψ</mi><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mfrac><mo>=</mo><mn>1</mn><mo>,</mo><mspace></mspace><mrow><mtext>uniformly for </mtext><mi>y</mi><mtext> bounded.</mtext></mrow></math></span></span></span> Additional blow-up properties are obtained, including the compactness of the blow-up set for the Cauchy problem with decaying initial data.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113303"},"PeriodicalIF":2.4,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826454","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":"Some geometric relations for equipotential curves","authors":"Yajun Zhou","doi":"10.1016/j.jde.2025.113296","DOIUrl":"10.1016/j.jde.2025.113296","url":null,"abstract":"<div><div>Let <span><math><mi>U</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>,</mo><mi>r</mi><mo>∈</mo><mi>Ω</mi><mo>⊂</mo><msup><mrow><mi>R</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span> be a harmonic function that solves an exterior Dirichlet problem. If all the level sets of <span><math><mi>U</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>,</mo><mi>r</mi><mo>∈</mo><mi>Ω</mi></math></span> are smooth Jordan curves, then there are several geometric inequalities that correlate the curvature <span><math><mi>κ</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> with the magnitude of gradient <span><math><mo>|</mo><mi>∇</mi><mi>U</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>|</mo></math></span> on each level set (“equipotential curve”). One of such inequalities is <span><math><mo>〈</mo><mo>[</mo><mi>κ</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>−</mo><mo>〈</mo><mi>κ</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>〉</mo><mo>]</mo><mo>[</mo><mo>|</mo><mi>∇</mi><mi>U</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>|</mo><mo>−</mo><mo>〈</mo><mo>|</mo><mi>∇</mi><mi>U</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>|</mo><mo>〉</mo><mo>]</mo><mo>〉</mo><mo>≥</mo><mn>0</mn></math></span>, where <span><math><mo>〈</mo><mo>⋅</mo><mo>〉</mo></math></span> denotes average over a level set, weighted by the arc length of the Jordan curve. We prove such a geometric inequality by constructing an entropy for each level set <span><math><mi>U</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>=</mo><mi>φ</mi></math></span>, and showing that such an entropy is convex in <em>φ</em>. The geometric inequality for <span><math><mi>κ</mi><mo>(</mo><mi>r</mi><mo>)</mo></math></span> and <span><math><mo>|</mo><mi>∇</mi><mi>U</mi><mo>(</mo><mi>r</mi><mo>)</mo><mo>|</mo></math></span> then follows from convexity and monotonicity of our entropy formula. A few other geometric relations for equipotential curves are also built on a convexity argument.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"433 ","pages":"Article 113296"},"PeriodicalIF":2.4,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825790","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":"Existence of bounded asymptotic solutions of autonomous differential equations","authors":"Vu Trong Luong ,&nbsp;William Barker ,&nbsp;Nguyen Duc Huy ,&nbsp;Nguyen Van Minh","doi":"10.1016/j.jde.2025.113320","DOIUrl":"10.1016/j.jde.2025.113320","url":null,"abstract":"<div><div>We study the existence of bounded asymptotic mild solutions to evolution equations of the form <span><math><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>=</mo><mi>A</mi><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>,</mo><mi>t</mi><mo>≥</mo><mn>0</mn></math></span> in a Banach space <span><math><mi>X</mi></math></span>, where <em>A</em> generates an (analytic) <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>-semigroup and <em>f</em> is bounded. We find spectral conditions on <em>A</em> and <em>f</em> for the existence and uniqueness of asymptotic mild solutions with the same “profile” as that of <em>f</em>. In the resonance case, a sufficient condition of Massera type theorem is found for the existence of bounded solutions with the same profile as <em>f</em>. The obtained results are stated in terms of spectral properties of <em>A</em> and <em>f</em>, and they are analogs of classical results of Katznelson-Tzafriri and Massera for the evolution equations on the half line. Applications from PDE are given.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113320"},"PeriodicalIF":2.4,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826384","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 viscosity solutions to Lorentzian eikonal equation on globally hyperbolic space-times","authors":"Siyao Zhu ,&nbsp;Hongguang Wu ,&nbsp;Xiaojun Cui","doi":"10.1016/j.jde.2025.113323","DOIUrl":"10.1016/j.jde.2025.113323","url":null,"abstract":"<div><div>In this paper, we show that any globally hyperbolic space-time admits at least one globally defined locally semiconcave function, which is a viscosity solution to the Lorentzian eikonal equation. According to whether the time orientation is changed, we divide the set of viscosity solutions into some subclasses. We show if the time orientation is consistent, then a viscosity solution has a variational representation locally. As a result, such a viscosity solution is locally semiconcave and has some weak KAM properties, as the one in the Riemannian case. On the other hand, if the time orientation of a viscosity solution is non-consistent, it will exhibit some peculiar properties which makes this kind of viscosity solutions totally different from the ones in the Riemannian case.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113323"},"PeriodicalIF":2.4,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143826455","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":"Existence for a second-order differential equation in a Banach space governed by an m-accretive operator","authors":"Parisa Jamshidnezhad ,&nbsp;Shahram Saeidi","doi":"10.1016/j.jde.2025.113310","DOIUrl":"10.1016/j.jde.2025.113310","url":null,"abstract":"<div><div>In the framework of uniformly smooth Banach spaces, we derive the existence and uniqueness of bounded solutions for the general differential equation (inclusion) <span><math><mi>p</mi><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mo>″</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>q</mi><mo>(</mo><mi>t</mi><mo>)</mo><msup><mrow><mi>u</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>(</mo><mi>t</mi><mo>)</mo><mo>∈</mo><mi>A</mi><mi>u</mi><mo>(</mo><mi>t</mi><mo>)</mo><mo>+</mo><mi>f</mi><mo>(</mo><mi>t</mi><mo>)</mo></math></span>, almost everywhere on <span><math><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>=</mo><mo>[</mo><mn>0</mn><mo>,</mo><mo>∞</mo><mo>)</mo></math></span>, with the initial condition <span><math><mi>u</mi><mo>(</mo><mn>0</mn><mo>)</mo><mo>=</mo><mi>x</mi><mo>∈</mo><mover><mrow><mi>D</mi><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>‾</mo></mover></math></span>. Here, <em>A</em> is a nonlinear m-accretive operator with <span><math><mn>0</mn><mo>∈</mo><mi>R</mi><mo>(</mo><mi>A</mi><mo>)</mo></math></span>, <span><math><mi>f</mi><mo>:</mo><msub><mrow><mi>R</mi></mrow><mrow><mo>+</mo></mrow></msub><mo>→</mo><mi>X</mi></math></span> is a given suitable function, and <span><math><mi>p</mi><mo>,</mo><mi>q</mi></math></span> are continuous functions. By developing new methods, we extend several previously known results in the literature, including the works of Poffald-Reich 1986 and Moroşanu 2014, and prove the existence of solutions to the aforementioned differential equation for the first time in Banach spaces. We apply our results to investigate the weak and strong <span><math><msup><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msup></math></span>-valued solutions for certain wave equations on bounded domains. Most of the results are new, even for Hilbert spaces.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"436 ","pages":"Article 113310"},"PeriodicalIF":2.4,"publicationDate":"2025-04-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143825854","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":"Limit cycles and critical periods with non-hyperbolic slow-fast systems","authors":"Peter De Maesschalck ,&nbsp;Joan Torregrosa","doi":"10.1016/j.jde.2025.113307","DOIUrl":"10.1016/j.jde.2025.113307","url":null,"abstract":"<div><div>By considering planar slow-fast systems with a curve of double singular points, we obtain lower bounds on the number of limit cycles of polynomial systems surrounding a single singular point, as well as on the number of critical periods in one annulus of periodic orbits. In some circumstances, orbits of such slow-fast systems do not exhibit the typical slow-fast behavior but instead follow a hit-and-run pattern: they quickly move toward the critical curve, pause briefly there, and then continue their path.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"433 ","pages":"Article 113307"},"PeriodicalIF":2.4,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143824174","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":"Existence of nonzero nonnegative solutions of Sturm-Liouville boundary value problems and applications","authors":"Kunquan Lan,&nbsp;Chongming Li","doi":"10.1016/j.jde.2025.113291","DOIUrl":"10.1016/j.jde.2025.113291","url":null,"abstract":"<div><div>Sufficient conditions for the boundary value problems (BVPs) of linear Sturm-Liouville (S-L) homogeneous equations subject to the separated boundary conditions (BCs) to have only zero solution are provided in this paper for the first time. Some previous papers and classical books used the assertion that the BVPs have only zero solution as a hypothesis and did not provide any sufficient conditions to ensure that the assertion holds. The sufficient conditions obtained in this paper are a key toward obtaining both the Green's functions to such BVPs and uniqueness of solutions for the linear S-L nonhomogeneous BVPs including the one-dimensional elliptic BVPs. New results on the existence of nonzero nonnegative or strictly positive solutions for the BVPs of nonlinear S-L equations with the separated BCs are obtained by using the fixed point index theory for nowhere normal-outward maps in Banach spaces. The new results allow the nonlinearities in the S-L BVPs to take negative values and have no lower bounds and are applied to deal with the logistic type population models which contain such nonlinearities.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"434 ","pages":"Article 113291"},"PeriodicalIF":2.4,"publicationDate":"2025-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143823756","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 the critical exponent for the semilinear Euler-Poisson-Darboux-Tricomi equation with power nonlinearity","authors":"Alessandro Palmieri","doi":"10.1016/j.jde.2025.113309","DOIUrl":"10.1016/j.jde.2025.113309","url":null,"abstract":"<div><div>In this note, we derive a blow-up result for a semilinear generalized Tricomi equation with damping and mass terms having time-dependent coefficients. We consider these coefficients with critical decay rates. Due to this threshold nature of the time-dependent coefficients (both for the damping and for the mass), the multiplicative constants appearing in these lower order terms strongly influence the value of the critical exponent, determining a competition between a Fujita-type exponent and a Strauss-type exponent.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"437 ","pages":"Article 113309"},"PeriodicalIF":2.4,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143820790","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 the capillary water waves with constant vorticity","authors":"Lizhe Wan","doi":"10.1016/j.jde.2025.113308","DOIUrl":"10.1016/j.jde.2025.113308","url":null,"abstract":"<div><div>This article is devoted to the study of local well-posedness for deep water waves with constant vorticity in two space dimensions on the real line. The water waves can be paralinearized and written as a quasilinear dispersive system of equations. By using the energy estimate and the Strichartz estimate, we show that for <span><math><mi>s</mi><mo>&gt;</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac></math></span>, the gravity-capillary water wave system with constant vorticity is locally well-posed in <span><math><msup><mrow><mi>H</mi></mrow><mrow><mi>s</mi></mrow></msup><mo>(</mo><mi>R</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113308"},"PeriodicalIF":2.4,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815613","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 solutions of multispeed semilinear Klein-Gordon systems in space dimension two","authors":"Xilu Zhu","doi":"10.1016/j.jde.2025.113279","DOIUrl":"10.1016/j.jde.2025.113279","url":null,"abstract":"<div><div>We consider general semilinear, multispeed Klein-Gordon systems in space dimension two with some non-degeneracy conditions. We prove that with small initial data such solutions are always global and scatter to a linear solution. This result partly extends the previous result obtained by Deng <span><span>[3]</span></span>, who completely proved the 3D quasilinear case. To prove our result, we mainly work on Fourier side and explore the contribution from the vicinity of space-time resonance.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"437 ","pages":"Article 113279"},"PeriodicalIF":2.4,"publicationDate":"2025-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143815818","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}