Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds. (English) Zbl 1365.35191
Summary: We are concerned with nonexistence results of nonnegative weak solutions for a class of quasilinear parabolic problems with a potential on complete noncompact Riemannian manifolds. In particular, we highlight the interplay between the geometry of the underlying manifold, the power nonlinearity and the behavior of the potential at infinity.
MSC:
| 35R01 | PDEs on manifolds |
| 35K59 | Quasilinear parabolic equations |
| 35K92 | Quasilinear parabolic equations with \(p\)-Laplacian |
| 53C20 | Global Riemannian geometry, including pinching |
| 35R45 | Partial differential inequalities and systems of partial differential inequalities |
References:
| [1] | Bandle, C., Pozio, M.A., Tesei, A.: The Fujita exponent for the Cauchy problem in the hyperbolic space. J. Differ. Equ. 251, 2143-2163 (2011) · Zbl 1227.35052 · doi:10.1016/j.jde.2011.06.001 |
| [2] | Brooks, R.: A relation between growth and the spectrum of the Laplacian. Math. Z. 178, 501-508 (1981) · Zbl 0458.58024 · doi:10.1007/BF01174771 |
| [3] | D’Ambrosio, L., Mitidieri, V.: A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities. Adv. Math. 224, 967-1020 (2010) · Zbl 1206.35265 · doi:10.1016/j.aim.2009.12.017 |
| [4] | D’Ambrosio, L., Lucente, S.: Nonlinear Liouville theorems for Grushin and Tricomi operators. J. Differ. Equ. 193, 511-541 (2003) · Zbl 1040.35012 · doi:10.1016/S0022-0396(03)00138-4 |
| [5] | Davies, E.B.: Heat Kernel and Spectral Theory. Cambridge University Press, Cambridge (1989) · Zbl 0699.35006 · doi:10.1017/CBO9780511566158 |
| [6] | Fujita, H.: On the blowing up of solutions of the Cauchy problem for \(u_t = \Delta u + u^{1+\alpha }\). J. Fac. Sci. Univ. Tokyo Sect. I(13), 109-124 (1966) · Zbl 0163.34002 |
| [7] | Fujita, H.: On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations. In: Proc. Symp. Pure Math., vol. 18, pp. 138-161. Am. Math. Soc., Providence (1968) · Zbl 1319.35312 |
| [8] | Galaktionov, V.A.: Conditions for the absence of global solutions for a class of quasilinear parabolic equations. Zh. Vychisl. Mat. i Mat. Fiz. 22, 322-338 (1982) · Zbl 0501.35043 |
| [9] | Galaktionov, V.A.: Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proc. R. Soc. Edinb. Sect. A 124, 517-525 (1994) · Zbl 0808.35053 · doi:10.1017/S0308210500028766 |
| [10] | Galaktionov, V.A., Levine, H.A.: A general approach to critical Fujita exponents in nonlinear parabolic problems. Nonlinear Anal. 34, 1005-1027 (1998) · Zbl 1139.35317 · doi:10.1016/S0362-546X(97)00716-5 |
| [11] | Grigor’yan, A.: Analytic and geometric background of recurrence and nonexplosion of the Brownian motion on Riemannian manifolds. Bull. Am. Math. Soc. 36, 135-249 (1999) · Zbl 0927.58019 · doi:10.1090/S0273-0979-99-00776-4 |
| [12] | Grigor’yan, A., Kondratiev, V.A.: On the existence of positive solutions of semilinear elliptic inequalities on Riemannian manifolds. In: Around the Research of Vladimir Maz’ya. II. Int. Math. Ser. (N. Y.), vol. 12, pp. 203-218. Springer, New York (2010) · Zbl 1185.35344 |
| [13] | Grigor’yan, A., Sun, Y.: On non-negative solutions of the inequality \(\Delta u + u^\sigma \le 0\) on Riemannian manifolds. Commun. Pure Appl. Math. 67, 1336-1352 (2014) · Zbl 1296.58011 · doi:10.1002/cpa.21493 |
| [14] | Hayakawa, K.: On nonexistence of global solutions of some semilinear parabolic differential equations. Proc. Jpn. Acad. 49, 503-505 (1973) · Zbl 0281.35039 · doi:10.3792/pja/1195519254 |
| [15] | Kurta, V.V.: On the absence of positive solutions to semilinear elliptic equations. Tr. Mat. Inst. Steklova 227, 162-169 (1999) · Zbl 1018.35030 |
| [16] | Lee, T.Y., Ni, W.-M.: Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem. Trans. Am. Math. Soc. 333, 365-378 (1992) · Zbl 0785.35011 · doi:10.1090/S0002-9947-1992-1057781-6 |
| [17] | Levine, H.A.: The role of critical exponents in blowup theorems. SIAM Rev. 32, 262-288 (1990) · Zbl 0706.35008 · doi:10.1137/1032046 |
| [18] | Mastrolia, P., Monticelli, D.D., Punzo, F.: Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds. Calc. Var. PDE 54, 1345-1372 (2015) · Zbl 1329.58020 · doi:10.1007/s00526-015-0827-0 |
| [19] | Mastrolia, P., Rigoli, M., Setti, A.G.: Yamabe-type equations on complete, noncompact manifolds. In: Progress in Mathematics 302. Birkhäuser, Basel (2012) · Zbl 1323.53004 |
| [20] | Mitidieri, V., Pohozev, S.I.: Absence of global positive solutions of quasilinear elliptic inequalities. Dokl. Akad. Nauk 359, 456-460 (1998) · Zbl 0976.35100 |
| [21] | Mitidieri, V., Pohozaev, S.I.: Nonexistence of positive solutions for quasilinear elliptic problems in \({\bb R\it }^N\). Tr. Mat. Inst. Steklova 227, 192-222 (1999) · Zbl 1056.35507 |
| [22] | Mitidieri, V., Pohozaev, S.I.: A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities. Tr. Mat. Inst. Steklova 234, 1-384 (2001) · Zbl 1074.35500 |
| [23] | Mitidieri, V., Pohozaev, S.I.: Towards a unified approach to nonexistence of solutions for a class of differential inequalities. Milan J. Math. 72, 129-162 (2004) · Zbl 1115.35157 · doi:10.1007/s00032-004-0032-7 |
| [24] | Monticelli, D.D.: Maximum principles and the method of moving planes for a class of degenerate elliptic linear operators. J. Eur. Math. Soc. 12, 611-654 (2010) · Zbl 1208.35068 · doi:10.4171/JEMS/210 |
| [25] | Pohozaev, S.I., Tesei, A.: Nonexistence of local solutions to semilinear partial differential inequalities. Ann. Inst. H. Poincare Anal. Non Linear 21, 487-502 (2004) · Zbl 1064.35220 |
| [26] | Punzo, F.: Blow-up of solutions to semilinear parabolic equations on Riemannian manifolds with negative sectional curvature. J. Math. Anal. Appl. 387, 815-827 (2012) · Zbl 1233.35052 · doi:10.1016/j.jmaa.2011.09.043 |
| [27] | Punzo, F.: Global existence of solutions to the semilinear heat equation on Riemannian manifolds with negative sectional curvature. Riv. Mat. Univ. Parma 5, 113-138 (2014) · Zbl 1321.35094 |
| [28] | Punzo, F., Tesei, A.: On a semilinear parabolic equation with inverse-square potential. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 21, 359-396 (2010) · Zbl 1207.35193 |
| [29] | Sun, Y.: Uniqueness results for nonnegative solutions of semilinear inequalities on Riemannian manifolds. J. Math. Anal. Appl. 419, 646-661 (2014) · Zbl 1297.35297 · doi:10.1016/j.jmaa.2014.05.011 |
| [30] | Sun, Y.: On nonexistence of positive solutions of quasi-linear inequality on Riemannian manifolds. Proc. Am. Math. Soc. 143(7), 2969-2984 (2015) · Zbl 1319.35312 · doi:10.1090/S0002-9939-2015-12705-0 |
| [31] | Weissler, F.B.: Local existence and nonexistence for semilinear parabolic equations in \(L^p\). Indiana Univ. Math. J. 29, 79-102 (1980) · Zbl 0443.35034 · doi:10.1512/iumj.1980.29.29007 |
| [32] | Weissler, F.B.: Existence and nonexistence of global solutions for a semilinear heat equation. Isr. J. Math. 38, 29-40 (1981) · Zbl 0476.35043 · doi:10.1007/BF02761845 |
| [33] | Zhang, Q.S.: Blow-up results for nonlinear parabolic equations on manifolds. Duke Math. J. 97, 515-539 (1999) · Zbl 0954.35029 · doi:10.1215/S0012-7094-99-09719-3 |
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