Communications on Pure & Applied Analysis最新文献

{"title":"Particle paths in equatorial flows","authors":"Tony Lyons","doi":"10.3934/cpaa.2022041","DOIUrl":"https://doi.org/10.3934/cpaa.2022041","url":null,"abstract":"We investigate particle trajectories in equatorial flows with geophysical corrections caused by the earth's rotation. Particle trajectories in the flows are constructed using pairs of analytic functions defined over the labelling space used in the Lagrangian formalism. Several classes of flow are investigated, and the physical regime in which each is valid is determined using the pressure distribution function of the flow, while the vorticity distribution of each flow is also calculated and found to be effected by earth's rotation.","PeriodicalId":435074,"journal":{"name":"Communications on Pure & Applied Analysis","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124010021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Gagliardo-Nirenberg-Sobolev inequalities on planar graphs","authors":"M. Esteban","doi":"10.3934/cpaa.2022051","DOIUrl":"https://doi.org/10.3934/cpaa.2022051","url":null,"abstract":"In this paper we study a family of the interpolation Gagliardo-Nirenberg-Sobolev inequalities on planar graphs. We are interested in knowing when the best constants in the inequalities are achieved. The inequalities being equivalent to some minimization problems, we also analyse the set of solutions of the Euler-Lagrange equations satisfied by extremal functions, or equivalently, by minimizers.","PeriodicalId":435074,"journal":{"name":"Communications on Pure & Applied Analysis","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114078021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiplicity results for nonhomogeneous elliptic equations with singular nonlinearities","authors":"R. Arora","doi":"10.3934/cpaa.2022056","DOIUrl":"https://doi.org/10.3934/cpaa.2022056","url":null,"abstract":"<p style='text-indent:20px;'>This paper is concerned with the study of multiple positive solutions to the following elliptic problem involving a nonhomogeneous operator with nonstandard growth of <inline-formula><tex-math id=\"M1\">begin{document}$ p $end{document}</tex-math></inline-formula>-<inline-formula><tex-math id=\"M2\">begin{document}$ q $end{document}</tex-math></inline-formula> type and singular nonlinearities</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ left{ begin{alignedat}{2} {} - mathcal{L}_{p,q} u & {} = lambda frac{f(u)}{u^gamma}, u>0 && quadmbox{ in } , Omega, u & {} = 0 && quadmbox{ on } partialOmega, end{alignedat} right. $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M3\">begin{document}$ Omega $end{document}</tex-math></inline-formula> is a bounded domain in <inline-formula><tex-math id=\"M4\">begin{document}$ mathbb{R}^N $end{document}</tex-math></inline-formula> with <inline-formula><tex-math id=\"M5\">begin{document}$ C^2 $end{document}</tex-math></inline-formula> boundary, <inline-formula><tex-math id=\"M6\">begin{document}$ N geq 1 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M7\">begin{document}$ lambda >0 $end{document}</tex-math></inline-formula> is a real parameter,</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE2\"> begin{document}$ mathcal{L}_{p,q} u : = {rm{div}}(|nabla u|^{p-2} nabla u + |nabla u|^{q-2} nabla u), $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'><inline-formula><tex-math id=\"M8\">begin{document}$ 1<p<q< infty $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M9\">begin{document}$ gamma in (0,1) $end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id=\"M10\">begin{document}$ f $end{document}</tex-math></inline-formula> is a continuous nondecreasing map satisfying suitable conditions. By constructing two distinctive pairs of strict sub and super solution, and using fixed point theorems by Amann [<xref ref-type=\"bibr\" rid=\"b1\">1</xref>], we prove existence of three positive solutions in the positive cone of <inline-formula><tex-math id=\"M11\">begin{document}$ C_delta(overline{Omega}) $end{document}</tex-math></inline-formula> and in a certain range of <inline-formula><tex-math id=\"M12\">begin{document}$ lambda $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124945134","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Curvature-driven front propagation through planar lattices in oblique directions","authors":"Mia Juki'c, H. Hupkes","doi":"10.3934/cpaa.2022055","DOIUrl":"https://doi.org/10.3934/cpaa.2022055","url":null,"abstract":"<p style='text-indent:20px;'>In this paper we investigate the long-term behaviour of solutions to the discrete Allen-Cahn equation posed on a two-dimensional lattice. We show that front-like initial conditions evolve towards a planar travelling wave modulated by a phaseshift <inline-formula><tex-math id=\"M1\">begin{document}$ gamma_l(t) $end{document}</tex-math></inline-formula> that depends on the coordinate <inline-formula><tex-math id=\"M2\">begin{document}$ l $end{document}</tex-math></inline-formula> transverse to the primary direction of propagation. This direction is allowed to be general, but rational, generalizing earlier known results for the horizontal direction. We show that the behaviour of <inline-formula><tex-math id=\"M3\">begin{document}$ gamma $end{document}</tex-math></inline-formula> can be asymptotically linked to the behaviour of a suitably discretized mean curvature flow. This allows us to show that travelling waves propagating in rational directions are nonlinearly stable with respect to perturbations that are asymptotically periodic in the transverse direction.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"61 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131692172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two-sided estimates of total bandwidth for Schrödinger operators on periodic graphs","authors":"E. Korotyaev, N. Saburova","doi":"10.3934/cpaa.2022042","DOIUrl":"https://doi.org/10.3934/cpaa.2022042","url":null,"abstract":"We consider Schrödinger operators with periodic potentials on periodic discrete graphs. Their spectrum consists of a finite number of bands. We obtain two-sided estimates of the total bandwidth for the Schrödinger operators in terms of geometric parameters of the graph and the potentials. In particular, we show that these estimates are sharp. It means that these estimates become identities for specific graphs and potentials. The proof is based on the Floquet theory and trace formulas for fiber operators. The traces are expressed as finite Fourier series of the quasimomentum with coefficients depending on the potentials and cycles of the quotient graph from some specific cycle sets. In order to obtain our results we estimate these Fourier coefficients in terms of geometric parameters of the graph and the potentials.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116715816","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A potential approach for planning mean-field games in one dimension","authors":"T. Bakaryan, Rita Ferreira, D. Gomes","doi":"10.3934/cpaa.2022054","DOIUrl":"https://doi.org/10.3934/cpaa.2022054","url":null,"abstract":"This manuscript discusses planning problems for first- and second-order one-dimensional mean-field games (MFGs). These games are comprised of a Hamilton–Jacobi equation coupled with a Fokker–Planck equation. Applying Poincaré's Lemma to the Fokker–Planck equation, we deduce the existence of a potential. Rewriting the Hamilton–Jacobi equation in terms of the potential, we obtain a system of Euler–Lagrange equations for certain variational problems. Instead of the mean-field planning problem (MFP), we study this variational problem. By the direct method in the calculus of variations, we prove the existence and uniqueness of solutions to the variational problem. The variational approach has the advantage of eliminating the continuity equation.We also consider a first-order MFP with congestion. We prove that the congestion problem has a weak solution by introducing a potential and relying on the theory of variational inequalities. We end the paper by presenting an application to the one-dimensional Hughes' model.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126278659","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace","authors":"Anton S. Zadorin","doi":"10.3934/cpaa.2022030","DOIUrl":"https://doi.org/10.3934/cpaa.2022030","url":null,"abstract":"A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propagation front problems. This article presents an exact travelling solution for a reaction-diffusion system with a piecewise constant production restricted to a codimension-1 subset. The solution is monotone, propagates with the unique constant velocity, and connects the trivial solution to a nontrivial nonhomogeneous stationary solution of the problem. The properties of the solution closely parallel the properties of monotone travelling solutions in classical bistable reaction-diffusion systems.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"120 17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126313018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Parabolic Systems with measurable coefficients in weighted Sobolev spaces","authors":"Doyoon Kim, Kyeong-Hun Kim, Kijung Lee","doi":"10.3934/cpaa.2022062","DOIUrl":"https://doi.org/10.3934/cpaa.2022062","url":null,"abstract":"<p style='text-indent:20px;'>We present a weighted <inline-formula><tex-math id=\"M1\">begin{document}$ L_p $end{document}</tex-math></inline-formula>-theory of parabolic systems on a half space <inline-formula><tex-math id=\"M2\">begin{document}$ {mathbb{R}}^d_+ $end{document}</tex-math></inline-formula>. The leading coefficients are assumed to be only measurable in time <inline-formula><tex-math id=\"M3\">begin{document}$ t $end{document}</tex-math></inline-formula> and have small bounded mean oscillations (BMO) with respect to the spatial variables <inline-formula><tex-math id=\"M4\">begin{document}$ x $end{document}</tex-math></inline-formula>, and the lower order coefficients are allowed to blow up near the boundary.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132170170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Multiple localized nodal solutions of high topological type for Kirchhoff-type equation with double potentials","authors":"Zhi-Guo Wu, Wen Guan, Da-Bin Wang","doi":"10.3934/cpaa.2022058","DOIUrl":"https://doi.org/10.3934/cpaa.2022058","url":null,"abstract":"<p style='text-indent:20px;'>We are concerned with sign-changing solutions and their concentration behaviors of singularly perturbed Kirchhoff problem</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{equation*} -(varepsilon^{2}a+ varepsilon bint _{mathbb{R}^{3}}|nabla v|^{2}dx)Delta v+V(x)v = P(x)f(v); ; {rm{in}}; mathbb{R}^{3}, end{equation*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">begin{document}$ varepsilon $end{document}</tex-math></inline-formula> is a small positive parameter, <inline-formula><tex-math id=\"M2\">begin{document}$ a, b>0 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M3\">begin{document}$ V, Pin C^{1}(mathbb{R}^{3}, mathbb{R}) $end{document}</tex-math></inline-formula>. Without using any non-degeneracy conditions, we obtain multiple localized sign-changing solutions of higher topological type for this problem. Furthermore, we also determine a concrete set as the concentration position of these sign-changing solutions. The main methods we use are penalization techniques and the method of invariant sets of descending flow. It is notice that, when nonlinear potential <inline-formula><tex-math id=\"M4\">begin{document}$ P $end{document}</tex-math></inline-formula> is a positive constant, our result generalizes the result obtained in [<xref ref-type=\"bibr\" rid=\"b5\">5</xref>] to Kirchhoff problem.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"44 12","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132153962","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The number of limit cycles from the perturbation of a quadratic isochronous system with two switching lines","authors":"Ai Ke, Maoan Han, Wei-Jian Geng","doi":"10.3934/cpaa.2022047","DOIUrl":"https://doi.org/10.3934/cpaa.2022047","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we give an upper bound (for <inline-formula><tex-math id=\"M1\">begin{document}$ ngeq3 $end{document}</tex-math></inline-formula>) and the least upper bound (for <inline-formula><tex-math id=\"M2\">begin{document}$ n = 1,2 $end{document}</tex-math></inline-formula>) of the number of limit cycles bifurcated from period annuli of a quadratic isochronous system under the piecewise polynomial perturbations of degree <inline-formula><tex-math id=\"M3\">begin{document}$ n $end{document}</tex-math></inline-formula>, respectively. The results improve the conclusions in [<xref ref-type=\"bibr\" rid=\"b19\">19</xref>].</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125788984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}