Communications on Pure & Applied Analysis最新文献

{"title":"On an exponentially decaying diffusive chemotaxis system with indirect signals","authors":"Pan Zheng, Jie Xing","doi":"10.3934/cpaa.2022044","DOIUrl":"https://doi.org/10.3934/cpaa.2022044","url":null,"abstract":"<p style='text-indent:20px;'>This paper deals with an exponentially decaying diffusive chemotaxis system with indirect signal production or consumption</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ begin{eqnarray*} label{1a} left{ begin{split}{} &u_t = nablacdot(D(u)nabla u)-nablacdot(S(u)nabla v), &(x,t)in Omegatimes (0,infty), &v_t = Delta v+h(v,w), &(x,t)in Omegatimes (0,infty), &w_t = Delta w- w+u, &(x,t)in Omegatimes (0,infty), end{split} right. end{eqnarray*} $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>under homogeneous Neumann boundary conditions in a smoothly bounded domain <inline-formula><tex-math id=\"M1\">begin{document}$ Omegasubset mathbb{R}^{n} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M2\">begin{document}$ ngeq2 $end{document}</tex-math></inline-formula>, where the nonlinear diffusivity <inline-formula><tex-math id=\"M3\">begin{document}$ D $end{document}</tex-math></inline-formula> and chemosensitivity <inline-formula><tex-math id=\"M4\">begin{document}$ S $end{document}</tex-math></inline-formula> are supposed to satisfy</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE2\"> begin{document}$ K_{1}e^{-beta^{-}s}leq D(s) leq K_{2}e^{-beta^{+}s} ;;;{rm{and}};;;frac{D(s)}{S(s)}geq K_{3}s^{-alpha}+gamma, $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with the constants <inline-formula><tex-math id=\"M5\">begin{document}$ beta^{-}geq beta^{+}>0 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M6\">begin{document}$ K_{1},K_{2},K_{3}>0 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M7\">begin{document}$ alpha,gammageq0 $end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id=\"M8\">begin{document}$ h(v,w) = -v+w $end{document}</tex-math></inline-formula>, we study the global existence and boundedness of solutions for the above system provided that <inline-formula><tex-math id=\"M9\">begin{document}$ alphain[0,frac{2}{n}) $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M10\">begin{document}$ beta^{-}geq beta^{+}>frac{n}{2} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M11\">begin{document}$ gamma>1 $end{document}</tex-math></inline-formula> and the initial mass of <inline-formula><tex-math id=\"M12\">begin{document}$ u_{0} $end{document}</tex-math></inline-formula> is small enough. Moreover, it is proved that the global bounded solution <inline-formula><tex-math id=\"M13\">begin{document}$ (u,v,w) $end{document}</tex-math></inline-formula> converges to <inline-formula><tex-math id=\"M14\">begin{document}$ (overline{u_{0}},overline{u_{0}},overline{u_{0}}) $end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id=\"M15\">begin{document}$ L^{infty} $end{document}</tex-math></inline-formula>-norm as <inline-formula><tex-math id=\"M16\">begin{document}$ trightarrow","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"33 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":"115796811","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":"Formation of singularities of solutions to the Cauchy problem for semilinear Moore-Gibson-Thompson equations","authors":"Sen Ming, Han Yang, Xiongmei Fan","doi":"10.3934/cpaa.2022046","DOIUrl":"https://doi.org/10.3934/cpaa.2022046","url":null,"abstract":"<p style='text-indent:20px;'>This paper is devoted to investigating formation of singularities for solutions to semilinear Moore-Gibson-Thompson equations with power type nonlinearity <inline-formula><tex-math id=\"M1\">begin{document}$ |u|^{p} $end{document}</tex-math></inline-formula>, derivative type nonlinearity <inline-formula><tex-math id=\"M2\">begin{document}$ |u_{t}|^{p} $end{document}</tex-math></inline-formula> and combined type nonlinearities <inline-formula><tex-math id=\"M3\">begin{document}$ |u_{t}|^{p}+|u|^{q} $end{document}</tex-math></inline-formula> in the case of single equation, combined type nonlinearities <inline-formula><tex-math id=\"M4\">begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M5\">begin{document}$ |u_{t}|^{p_{2}}+|u|^{q_{2}} $end{document}</tex-math></inline-formula>, combined and power type nonlinearities <inline-formula><tex-math id=\"M6\">begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M7\">begin{document}$ |u|^{q_{2}} $end{document}</tex-math></inline-formula>, combined and derivative type nonlinearities <inline-formula><tex-math id=\"M8\">begin{document}$ |v_{t}|^{p_{1}}+|v|^{q_{1}} $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=\"M9\">begin{document}$ |u_{t}|^{p_{2}} $end{document}</tex-math></inline-formula> in the case of coupled system, respectively. More precisely, blow-up results of solutions to problems in the sub-critical and critical cases are derived by applying test function technique. Moreover, upper bound lifespan estimates of solutions to the coupled systems are investigated. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"22 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131432544","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":"Multi-dimensional degenerate operators in $L^p$-spaces","authors":"S. Fornaro, G. Metafune, D. Pallara, R. Schnaubelt","doi":"10.3934/cpaa.2022052","DOIUrl":"https://doi.org/10.3934/cpaa.2022052","url":null,"abstract":"<p style='text-indent:20px;'>This paper is concerned with second-order elliptic operators whose diffusion coefficients degenerate at the boundary in first order. In this borderline case, the behavior strongly depends on the size and direction of the drift term. Mildly inward (or outward) pointing and strongly outward pointing drift terms were studied before. Here we treat the intermediate case equipped with Dirichlet boundary conditions, and show generation of an analytic positive <inline-formula><tex-math id=\"M2\">begin{document}$ C_0 $end{document}</tex-math></inline-formula>-semigroup. The main result is a precise description of the domain of the generator, which is more involved than in the other cases and exhibits reduced regularity compared to them.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"279 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":"131988851","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 ocean and the atmosphere: An applied mathematician's view","authors":"R. S. Johnson","doi":"10.3934/cpaa.2022040","DOIUrl":"https://doi.org/10.3934/cpaa.2022040","url":null,"abstract":"In this survey article, we provide a mathematical description of oceanic and atmospheric flows, based on the incompressible Navier–Stokes equation (for the ocean), and the compressible version with an equation of state and the first law of thermodynamics for the atmosphere. We show that, in both cases, the only fundamental assumption that we need to make is that of a thin shell on a (nearly) spherical Earth, so that the main elements of spherical geometry are included, with all other attributes of the fluid motion retained at leading order. (The small geometrical correction that is needed to represent the Earth's geoid as an oblate spheroid is briefly described.) We argue that this is the only reliable theoretical approach to these types of fluid problem. A generic formulation is presented for the ocean, and for the steady and unsteady atmosphere, these latter two differing slightly in the details. Based on these governing equations, a number of examples are presented (in outline only), some of which provide new insights into familiar flows. The examples include the Ekman flow and large gyres in the ocean; and in the atmosphere: Ekman flow, geostrophic balance, Brunt–Väisälä frequency, Hadley–Ferrel–polar cells, harmonic waves, equatorially trapped waves.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"5 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":"115587831","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":"Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator","authors":"Dinh Nguyen Duy Hai","doi":"10.3934/cpaa.2022043","DOIUrl":"https://doi.org/10.3934/cpaa.2022043","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the <inline-formula><tex-math id=\"M1\">begin{document}$ n $end{document}</tex-math></inline-formula>-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., <b>33</b> (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space <inline-formula><tex-math id=\"M2\">begin{document}$ H^q(mathbb{R}^n) $end{document}</tex-math></inline-formula>.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"5 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":"116086747","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":"Monotonicity and nonexistence of positive solutions for pseudo-relativistic equation with indefinite nonlinearity","authors":"Yuxia Guo, Shaolong Peng","doi":"10.3934/cpaa.2022037","DOIUrl":"https://doi.org/10.3934/cpaa.2022037","url":null,"abstract":"<p style='text-indent:20px;'>In this paper, we consider the following general pseudo-relativistic Schrödinger equation with indefinite nonlinearities:</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=\"FE1\"> begin{document}$ (-Delta+m^{2})^{s}u = a(x_1)fleft(u,nabla uright),quad {rm{in}} ,,mathbb R^{N}, $end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">begin{document}$ sin(0,1) $end{document}</tex-math></inline-formula>, mass <inline-formula><tex-math id=\"M2\">begin{document}$ m>0 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=\"M3\">begin{document}$ a $end{document}</tex-math></inline-formula> is a non-decreasing functions. We prove the nonexistence and the monotonicity of the positive bounded solution for the above equation via the direct method of moving planes.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"12 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":"125348484","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":"Global behavior for the classical solution of compressible viscous micropolar fluid with cylinder symmetry","authors":"L. Huang, Zhiying Sun, Xinguang Yang, A. Miranville","doi":"10.3934/cpaa.2022033","DOIUrl":"https://doi.org/10.3934/cpaa.2022033","url":null,"abstract":"<p style='text-indent:20px;'>This paper is concerned with the global solutions of the 3D compressible micropolar fluid model in the domain to a subset of <inline-formula><tex-math id=\"M1\">begin{document}$ R^3 $end{document}</tex-math></inline-formula> bounded with two coaxial cylinders that present the solid thermo-insulated walls, which is in a thermodynamical sense perfect and polytropic. Compared with the classical Navier-Stokes equations, the angular velocity <inline-formula><tex-math id=\"M2\">begin{document}$ w $end{document}</tex-math></inline-formula> in this model brings benefit that is the damping term -<inline-formula><tex-math id=\"M3\">begin{document}$ uw $end{document}</tex-math></inline-formula> can provide extra regularity of <inline-formula><tex-math id=\"M4\">begin{document}$ w $end{document}</tex-math></inline-formula>. At the same time, the term <inline-formula><tex-math id=\"M5\">begin{document}$ uw^2 $end{document}</tex-math></inline-formula> is bad, it increases the nonlinearity of our system. Moreover, the regularity and exponential stability in <inline-formula><tex-math id=\"M6\">begin{document}$ H^4 $end{document}</tex-math></inline-formula> also are proved.</p>","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"1 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":"130294651","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":"Shock polars for non-polytropic compressible potential flow","authors":"V. Elling","doi":"10.3934/cpaa.2022032","DOIUrl":"https://doi.org/10.3934/cpaa.2022032","url":null,"abstract":"We consider compressible potential flow for general equations of state. Assuming hyperbolicity and convex equation of state, we prove that shock polars have a unique critical point (in each half), as well as a unique sonic point, with critical and strong shocks always on the subsonic side. We also show existence of normal and oblique shocks, as well as monotonicity of density, enthalpy, pressure along each half-polar, with Mach number monotone only along the subsonic part.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"1 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":"130666959","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":"Weakly nonlinear waves in stratified shear flows","authors":"A. Geyer, Ronald Quirchmayr","doi":"10.3934/cpaa.2022061","DOIUrl":"https://doi.org/10.3934/cpaa.2022061","url":null,"abstract":"We develop a Korteweg–De Vries (KdV) theory for weakly nonlinear waves in discontinuously stratified two-layer fluids with a generally prescribed rotational steady current. With the help of a classical asymptotic power series approach, these models are directly derived from the divergence-free incompressible Euler equations for unidirectional free surface and internal waves over a flat bed. Moreover, we derive a Burns condition for the determination of wave propagation speeds. Several examples of currents are given; explicit calculations of the corresponding propagation speeds and KdV coefficients are provided as well.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"281 ","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133720452","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 Łojasiewicz inequality for free energy functionals on a graph","authors":"Kong Li, X. Xue","doi":"10.3934/cpaa.2022066","DOIUrl":"https://doi.org/10.3934/cpaa.2022066","url":null,"abstract":"Rencently Chow, Huang, Li and Zhou proposed discrete forms of the Fokker-Planck equations on a finite graph. As a primary step, they constructed Riemann metrics on the graph by endowing it with some kinds of weight. In this paper, we reveal the relation between these Riemann metrics and the Euclidean metric, by showing that they are locally equivalent. Moreover, various Riemann metrics have this property provided the corresponding weight satisfies a bounded condition. Based on this, we prove that the two-side Łojasiewicz inequality holds near the Gibbs distribution with Łojasiewicz exponent begin{document}$ frac{1}{2} $end{document}. Then we use it to prove the solution of the discrete Fokker-Planck equation converges to the Gibbs distribution with exponential rate. As a corollary of Łojasiewicz inequality, we show that the two-side Talagrand-type inequality holds under different Riemann metrics.","PeriodicalId":435074,"journal":{"name":"Communications on Pure &amp; Applied Analysis","volume":"20 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":"126141298","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}