Sobolev hyperbola for periodic Lane-Emden heat flow system in N spatial dimension. (English) Zbl 1480.35012
Summary: The well-known Lane-Emden conjecture indicates that, for the elliptic Lane-Emden system \(- \Delta u = v^p, - \Delta v = u^q\) in \(\mathbb{R}^N\), the Sobolev hyperbola \(\frac{1}{p + 1} + \frac{1}{ q + 1} = \frac{N - 2}{ N}\) is expected as the critical curve for the existence and nonexistence of entire solutions. In this paper, we study the periodic Lane-Emden heat flow system \(u_t - \Delta u = a (t) v^p, v_t - \Delta v = b (t) u^q\) in a bounded domain \(\Omega\) of \(\mathbb{R}^N\), subject to homogeneous Dirichlet boundary condition. We will show that the Sobolev hyperbola is also a critical curve for the existence and nonexistence of periodic solutions. Moreover, if \(pq = 1\), the nontrivial periodic solutions may exist or not exist.
MSC:
| 35B10 | Periodic solutions to PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K58 | Semilinear parabolic equations |
References:
| [1] | Lane, J. H., On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known to terrestrial experiment, Am. J. Sci., 50, 57-74 (1870), 2nd Ser. |
| [2] | Emden, R., Gaskugeln: Anwendungen der Mechanischen Warmetheorie auf Kosmologische und Meteorologische Probleme (1907), Teubner: Teubner Berlin · JFM 38.0952.02 |
| [3] | Serrin, J.; Zou, H., Non-existence of positive solutions of Lane-Emden systems, Differential Integral Equations, 9, 635-653 (1996) · Zbl 0868.35032 |
| [4] | de Figueiredo, D. G., Semilinear elliptic systems: Existence, multiplicity, symmetry of solutions, (Chipot, M., Handbook of Differential Equations: Stationary Partial Differential Equations, Vol. 5 (2008), Elsevier/North-Holland: Elsevier/North-Holland Amsterdam), 1-48 · Zbl 1187.35056 |
| [5] | 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, 555-579 (2007) · Zbl 1146.35038 |
| [6] | Souplet, Ph., The proof of the Lane-Emden conjecture in four space dimensions, Adv. Math., 221, 1409-1427 (2009) · Zbl 1171.35035 |
| [7] | Clement, Ph.; de Figueiredo, D. G.; Mitidier, E., Positive solutions of semilinear elliptic systems, Comm. Partial Differential Equations, 17, 923-940 (1992) · Zbl 0818.35027 |
| [8] | Peletier, L.; van der Vorst, R. C.A. M., Existence and non-existence of positive solutions of nonlinear elliptic systems and the biharmonic equation, Differential Integral Equations, 5, 747-767 (1992) · Zbl 0758.35029 |
| [9] | van der Vorst, R. C.A. M., Variational identities and applications to differential systems, Arch. Ration. Mech. Anal., 116, 375-398 (1992) · Zbl 0796.35059 |
| [10] | Quittner, P.; Souplet, Ph., (Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States. Superlinear Parabolic Problems: Blow-Up, Global Existence and Steady States, Birkhäuser Advanced Texts (2019), Birkhäuser: Birkhäuser Basel) · Zbl 1423.35004 |
| [11] | Esteban, M. J.; Herrero, M. A., Boundedness and blow up for a semilinear reaction-diffusion system, J. Differential Equations, 89, 176-202 (1991) · Zbl 0735.35013 |
| [12] | Escobedo, M.; Herrero, M. A., A semilinear parabolic system in a bounded domain, Ann. Mat. Pura Appl., 165, 315-336 (1993) · Zbl 0806.35088 |
| [13] | Huang, H.; Huang, R.; Yin, J.; Cao, Y., Sobolev hyperbola for the periodic parabolic Lane-Emden system, Nonlinearity, 33, 762-789 (2020) · Zbl 1427.35101 |
| [14] | Huang, H.; Huang, R.; Yin, J., Singular curve and critical curve for doubly nonlinear Lane-Emden type equations, Math. Methods Appl. Sci., 1-17 (2021) |
| [15] | Mitidieri, E., Non-existence of positive solutions of semilinear elliptic systems in \(\mathbb{R}^N\), Differential Integral Equations, 9, 465-479 (1996) · Zbl 0848.35034 |
| [16] | Serrin, J.; Zou, H., Non-existence of positive solutions of semilinear elliptic systems, (Discourses in Mathematics and its Applications 3 (1994), Department of Mathematics, Texas A & M University: Department of Mathematics, Texas A & M University College Station, Texas), 55-68 · Zbl 0900.35121 |
| [17] | Guo, Y.; Liu, J., Liouville type theorems for positive solutions of elliptic system in \(\mathbb{R}^N\), Comm. Partial Differential Equations, 33, 263-284 (2008) · Zbl 1139.35305 |
| [18] | Li, K.; Zhang, Z., Proof of the Hénon-Lane-Emden conjecture in \(\mathbb{R}^3\), J. Differential Equations, 266, 202-226 (2019) · Zbl 1410.35040 |
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