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A Liouville-type theorem for the 3-dimensional parabolic Gross–Pitaevskii and related systems
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We prove a Liouville-type theorem for semilinear parabolic systems of the form $$\begin{aligned} {\partial _t u_i}-\Delta u_i =\sum _{j=1}^{m}\beta _{ij} u_i^ru_j^{r+1}, \quad i=1,2,\ldots ,m \end{aligned}$$in the whole space \({\mathbb R}^N\times {\mathbb R}\). Very recently, Quittner (Math Ann. 364, 269–292, 2016) has established an optimal result for \(m=2\) in dimension \(N\le 2\), and partial results in higher dimensions in the range \(p< N/(N-2)\). By nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron, we partially improve the results of Quittner in dimensions \(N\ge 3\). In particular, our results solve the important case of the parabolic Gross–Pitaevskii system—i.e. the cubic case \(r=1\)—in space dimension \(N=3\), for any symmetric (m, m)-matrix \((\beta _{ij})\) with nonnegative entries, positive on the diagonal. By moving plane and monotonicity arguments, that we actually develop for more general cooperative systems, we then deduce a Liouville-type theorem in the half-space \({\mathbb R}^N_+\times {\mathbb R}\). As applications, we give results on universal singularity estimates, universal bounds for global solutions, and blow-up rate estimates for the corresponding initial value problem.Mathematics Subject ClassificationPrimary 35B5335K58Secondary 35B3335B44References1.Akhmediev, N., Ankiewicz, A.: Partially coherent solitons on a finite background. Phys. Rev. Lett. 82, 2661–2664 (1999)CrossRefGoogle Scholar2.Amann, H.: Global existence for semilinear parabolic systems. J. Reine Angew. Math. 360, 47–83 (1985)MathSciNetMATHGoogle Scholar3.Bartsch, Th, Dancer, N., Wang, Z.-Q.: A Liouville theorem, a-priori bounds, and bifurcating branches of positive solutions for a nonlinear elliptic system. Calc. Var. Part. Differ. Equ. 37(3–4), 345–361 (2010)MathSciNetCrossRefMATHGoogle Scholar4.Bebernes, J., Eberly, D.: Mathematical problems from combustion theory. In: Applied Mathematical Sciences, vol. 83. Springer, New York (1989)5.Bidaut-Véron, M.-F.: Initial blow-up for the solutions of a semilinear parabolic equation with source term. In: Équations aux dérivées partielles et applications. Gauthier-Villars, Éd. Sci. Méd. Elsevier, Paris, pp. 189-198 (1998)6.Bidaut-Véron, M.-F., Raoux, Th: Asymptotics of solutions of some nonlinear elliptic systems. Commun. Part. Differ. Equ. 21(7–8), 1035–1086 (1996)MathSciNetMATHGoogle Scholar7.Dancer, N., Wang, K., Zhang, Z.: Uniform Hölder estimate for singularly perturbed parabolic systems of Bose-Einstein condensates and competing species. J. Differ. Equ. 251(10), 2737-2769 (2011)8.Dancer, N.: Some notes on the method of moving planes. Bull. Aust. Math. Soc. 46(3), 425–434 (1992)MathSciNetCrossRefMATHGoogle Scholar9.Dancer, N., Wei, J., Weth, T.: A priori bounds versus multiple existence of positive solutions for a nonlinear Schrödinger system. Ann. Inst. H. Poincaré Anal. Non Linéaire 27(3), 953-969 (2010)10.Dancer, N., Weth, T.: Liouville-type results for non-cooperative elliptic systems in a half-space. J. Lond. Math. Soc. (2) 86(1), 111-128 (2012)11.Földes, J., Poláčik, P.: On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete Contin. Dyn. Syst. 25(1), 133–157 (2009)MathSciNetCrossRefMATHGoogle Scholar12.Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34(4), 525–598 (1981)MathSciNetCrossRefMATHGoogle Scholar13.Guo, Y., Li, B., Wei, J.: Entire nonradial solutions for non-cooperative coupled elliptic system with critical exponents in \(\mathbb{R}^3\). J. Differ. Equ. 256(10), 3463–3495 (2014)MathSciNetCrossRefMATHGoogle Scholar14.Guo, Y., Liu, J.: Liouville type theorems for positive solutions of elliptic system in \(\mathbb{R}^N\). Commun. Part. Differ. Equ. 33(1–3), 263–284 (2008)MathSciNetCrossRefMATHGoogle Scholar15.Hioe, F.T.: Solitary waves for \(\mathit{N}\) coupled nonlinear Schrödinger equations. Phys. Rev. Lett. 82, 1152–1155 (1999)MathSciNetCrossRefGoogle Scholar16.Liu, J., Liu, X., Wang, Z.-Q.: Multiple mixed states of nodal solutions for nonlinear Schrödinger systems. Calc. Var. Part. Differ. Equ. 52(3–4), 565–586 (2015)CrossRefMATHGoogle Scholar17.Ma, L., Zhao, L.: Uniqueness of ground states of some coupled nonlinear Schrödinger systems and their application. J. Differ. Equ. 245(9), 2551–2565 (2008)MathSciNetCrossRefMATHGoogle Scholar18.Merle, F., Zaag, H.: A Liouville theorem for vector-valued nonlinear heat equations and applications. Math. Ann. 316(1), 103–137 (2000)MathSciNetCrossRefMATHGoogle Scholar19.Montaru, A., Sirakov, B., Souplet, Ph: Proportionality of components, Liouville theorems and a priori estimates for noncooperative elliptic systems. Arch. Rat. Mech. Anal. 213(1), 129–169 (2014)MathSciNetCrossRefMATHGoogle Scholar20.Morgan, J.: Global existence for semilinear parabolic systems. SIAM J. Math. Anal. 20(5), 1128–1144 (1989)MathSciNetCrossRefMATHGoogle Scholar21.Phan, Q.H.: Optimal Liouville-type theorem for a parabolic system. Discrete Contin. Dyn. Syst. 35(1), 399–409 (2015)MathSciNetCrossRefMATHGoogle Scholar22.Poláčik, P.: Symmetry properties of positive solutions of parabolic equations: a survey. In: Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions. World Sci. Publ., Hackensack, pp. 170-208 (2009)23.Poláčik, P., Quittner, P.: A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation. Nonlinear Anal. 64, 1679–1689 (2006)MathSciNetCrossRefMATHGoogle Scholar24.Poláčik, P., Quittner, P., Souplet, Ph: Singularity and decay estimates in superlinear problems via Liouville-type theorems. I. Elliptic equations and systems. Duke Math. J. 139(3), 555–579 (2007)MathSciNetCrossRefMATHGoogle Scholar25.Poláčik, P., Quittner, P., Souplet, Ph: Singularity and decay estimates in superlinear problems via Liouville-type theorems. II. Parabolic equations. Indiana Univ. Math. J. 56(2), 879–908 (2007)MathSciNetCrossRefMATHGoogle Scholar26.Protter, M., Weinberger, H.: Maximum Principles in Differential Equations. Prentice Hall, Englewood Cliffs (1967)MATHGoogle Scholar27.Quittner, P.: Liouville theorems for scaling invariant superlinear parabolic problems with gradient structure. Math. Ann. 364, 269–292 (2016)MathSciNetCrossRefMATHGoogle Scholar28.Quittner, P.: Liouville theorems, universal estimates and periodic solutions for cooperative parabolic Lotka–Volterra systems. J. Differ. Equ. 260, 3524–3537 (2016)MathSciNetCrossRefMATHGoogle Scholar29.Quittner, P., Souplet, Ph.: Superlinear parabolic problems. In: Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Blow-up, Global Existence and Steady States. Birkhäuser Verlag, Basel (2007)30.Quittner, P., Souplet, Ph: Optimal Liouville-type theorems for noncooperative elliptic Schrödinger systems and applications. Commun. Math. Phys. 311(1), 1–19 (2012)MathSciNetCrossRefMATHGoogle Scholar31.Quittner, P., Souplet, Ph: Symmetry of components for semilinear elliptic systems. SIAM J. Math. Anal. 44(4), 2545–2559 (2012)MathSciNetCrossRefMATHGoogle Scholar32.Reichel, W., Zou, H.: Non-existence results for semilinear cooperative elliptic systems via moving spheres. J. Differ. Equ. 161(1), 219–243 (2000)MathSciNetCrossRefMATHGoogle Scholar33.Tavares, H., Terracini, S., Verzini, G., Weth, T.: Existence and nonexistence of entire solutions for non-cooperative cubic elliptic systems. Commun. Part. Differ. Equ. 36(11), 1988–2010 (2011)MathSciNetCrossRefMATHGoogle Scholar34.Wei, J., Weth, T.: Radial solutions and phase separation in a system of two coupled Schrödinger equations. Arch. Rat. Mech. Anal. 190(1), 83–106 (2008)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer-Verlag Berlin Heidelberg 2016Authors and AffiliationsQuoc Hung Phan1Philippe Souplet2Email author1.Institute of Research and DevelopmentDuy Tan UniversityDa NangVietnam2.Sorbonne Paris Cité, CNRS UMR 7539, Laboratoire Analyse, Géométrie et ApplicationsUniversité Paris 13VilletaneuseFrance About this article CrossMark Print ISSN 0025-5831 Online ISSN 1432-1807 Publisher Name Springer Berlin Heidelberg About this journal Reprints and Permissions Article actions function trackAddToCart() { var buyBoxPixel = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox", product: "10.1007/s00208-016-1368-3_A Liouville-type theorem for the 3", productStatus: "add", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); buyBoxPixel.sendinfo(); } function trackSubscription() { var subscription = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "springer_com.buybox" }); subscription.sendinfo({linkId: "inst. subscription info"}); } window.addEventListener("load", function(event) { var viewPage = new webtrekkV3({ trackDomain: "springergmbh01.webtrekk.net", trackId: "196033507532344", domain: "link.springer.com", contentId: "SL-article", product: "10.1007/s00208-016-1368-3_A Liouville-type theorem for the 3", productStatus: "view", productCategory : { 1 : "ppv" }, customEcommerceParameter : { 9 : "link.springer.com" } }); viewPage.sendinfo(); }); Log in to check your access to this article Buy (PDF)EUR 34,95 Unlimited access to full article Instant download (PDF) Price includes local sales tax if applicable Find out about institutional subscriptions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. 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