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A Liouville-type theorem and the decay of radial solutions of a semilinear heat equation. (English) Zbl 1092.35045

Summary: We consider the semilinear parabolic equation \(u_t= \Delta u+u^p\) on \(\mathbb R^N\), where the power nonlinearity is subcritical. We first address the question of existence of entire solutions, that is, solutions defined for all \(x\in \mathbb R^N\) and \(t\in \mathbb R\). Our main result asserts that there are no positive radially symmetric bounded entire solutions. Then we consider radial solutions of the Cauchy problem. We show that if such a solution is global, that is, defined for all \(t\geq 0\), then it necessarily converges to 0, as \(t\to\infty\), uniformly with respect to \(x\in\mathbb R^N\).

MSC:

35K55 Nonlinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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[1] Angenent, S., The zero set of a solution of a parabolic equation, J. Reine Angew. Math., 390, 79-96 (1988) · Zbl 0644.35050
[2] Bidaut-Véron, M.-F., Initial blow-up for the solutions of a semilinear parabolic equation with source term, (Equations aux dérivées partielles et applications, articles dédiés à Jacques-Louis Lions (1998), Gauthier- Villars: Gauthier- Villars Paris), 189-198 · Zbl 0914.35055
[3] Cazenave, T.; Lions, P.-L., Solutions globales d’équations de la chaleur semi linéaires, Comm. Part. Differential Equations, 9, 955-978 (1984) · Zbl 0555.35067
[4] Chen, W.; Li, C., Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63, 615-622 (1991) · Zbl 0768.35025
[5] Chen, X.-Y.; Poláčik, P., Asymptotic periodicity of positive solutions of reaction-diffusion equations on a ball, J. Reine Angew. Math., 472, 17-51 (1996) · Zbl 0839.35059
[6] Y. Du, S. Li, Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations 10 (2005) 814-860.; Y. Du, S. Li, Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations, Adv. Differential Equations 10 (2005) 814-860. · Zbl 1161.35388
[7] Galaktionov, V. A.; Lacey, A. A., Monotonicity in time of large solutions to a nonlinear heat equation, Rocky Mountain J. Math., 28, 1279-1301 (1998) · Zbl 0933.35092
[8] Gidas, B.; Spruck, J., Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34, 525-598 (1981) · Zbl 0465.35003
[9] Ladyzhenskaya, O. A.; Solonnikov, V. A.; Urall’ceva, N. N., Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, vol. 23 (1968), American Mathematical Society: American Mathematical Society Providence, RI, (Russian Original: “Nauka”, Moscow 1967) · Zbl 0174.15403
[10] Lions, P.-L., Asymptotic behavior of some nonlinear heat equations, Physica D, 5, 293-306 (1982) · Zbl 1194.35459
[11] Matos, J.; Souplet, Ph., Universal blow-up rates for a semilinear heat equation and applications, Adv. Differential Equations, 8, 615-639 (2003) · Zbl 1028.35065
[12] Merle, F.; Zaag, H., Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math., 51, 139-196 (1998) · Zbl 0899.35044
[13] P. Poláčik, P. Quittner, Liouville type theorems and complete blow-up for indefinite superlinear parabolic equations, Progress in nonlinear diffrential equations and their applications, Vol. 64, Birkhäuser, Basel, 2005, to appear.; P. Poláčik, P. Quittner, Liouville type theorems and complete blow-up for indefinite superlinear parabolic equations, Progress in nonlinear diffrential equations and their applications, Vol. 64, Birkhäuser, Basel, 2005, to appear. · Zbl 1093.35037
[14] Quittner, P.; Simondon, F., A priori bounds and complete blow-up of positive solutions of indefinite superlinear parabolic problems, J. Math. Anal. Appl., 304, 614-631 (2005) · Zbl 1071.35026
[15] Quittner, P.; Souplet, Ph., A priori estimates of global solutions of superlinear parabolic problems without variational structure, Discrete Contin. Dyn. Systems, 9, 1277-1292 (2003) · Zbl 1029.35049
[16] Souplet, Ph., Sur l’asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domaines non bornés, C. R. Acad. Sci. Paris, 323, 877-882 (1996) · Zbl 0860.35051
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