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August  2022, 42(8): 3733-3746. doi: 10.3934/dcds.2022030

Global solutions of semilinear parabolic equations with drift term on Riemannian manifolds

Dipartimento di Matematica, Politecnico di Milano, via Bonardi 9 - Milano, 20133, Italia

*Corresponding author: Fabio Punzo

Received  July 2021 Revised  January 2022 Published  August 2022 Early access  March 2022

We study existence and non-existence of global solutions to the semilinear heat equation with a drift term and a power-like source term $ u^p $, on Cartan-Hadamard manifolds. Under suitable assumptions on Ricci and sectional curvatures, we show that, for any $ p>1 $, global solutions cannot exists if the initial datum is large enough. Furthermore, under appropriate conditions on the drift term, global existence is obtained for any $ p>1 $, if the initial datum is sufficiently small. We also deal with Riemannian manifolds whose Ricci curvature tends to zero at infinity sufficiently fast. We show that for any non trivial initial datum, for certain $ p $ depending on the Ricci curvature bound, global solutions cannot exist. On the other hand, for certain values of $ p $, depending on the vector field $ b $, global solutions exist, for sufficiently small initial data.

Citation: Fabio Punzo. Global solutions of semilinear parabolic equations with drift term on Riemannian manifolds. Discrete and Continuous Dynamical Systems, 2022, 42 (8) : 3733-3746. doi: 10.3934/dcds.2022030
References:
[1]

J. Aguirre and M. Escobedo, On the blow-up of solutions of a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 433-460.  doi: 10.1017/S0308210500025828.

[2]

L. J. Alias, P. Mastrolia and M. Rigoli, Maximum Principles and Geometric Applications, Springer Monographs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-24337-5.

[3]

C. Bandle and H. Levine, Fujita phenomena for reaction-diffusion equations with convection like terms, Diff. Integral Eq., 7 (1994), 1169-1193. 

[4]

C. BandleM. A. Pozio and A. Tesei, The fujita exponent for the Cauchy problem in the Hyperbolic space, J. Diff. Eq., 251 (2011), 2143-2163.  doi: 10.1016/j.jde.2011.06.001.

[5]

L. BrandoliniM. Rigoli and A. G. Setti, Positive solutions of Yamabe type equationson complete manifolds and applications, J. Functional Anal., 160 (1998), 176-222.  doi: 10.1006/jfan.1998.3313.

[6]

A. Friedman, Partial Differential Equations of Parabolic Type, Dover Publications, New York, 1992.

[7]

H. Fujita, 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 (1966), 109-124. 

[8]

A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 135-249.  doi: 10.1090/S0273-0979-99-00776-4.

[9]

A. Grigor'yan, Heat Kernel and Analysis on Manifold, AMS/IP Studies in Advanced Mathematics, 47. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. doi: 10.1090/amsip/047.

[10]

G. GrilloG. Meglioli and F. Punzo, Smoothing effects and infinite time blow-up for reaction diffusion equation: An approach based on Sobolev and Poincaré inequalities, J. Math. Pures Appl., 151 (2021), 99-131.  doi: 10.1016/j.matpur.2021.04.011.

[11]

G. GrilloG. Meglioli and F. Punzo, Global existence of solutions and smoothing effect for classes of reaction-diffusion equations on manifolds, J. Evol. Eq., 21 (2021), 2339-2375.  doi: 10.1007/s00028-021-00685-3.

[12]

G. GrilloM. Muratori and F. Punzo, Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds, J. Diff. Eq., 266 (2019), 4305-4336.  doi: 10.1016/j.jde.2018.09.037.

[13]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505. 

[14]

S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.

[15]

K. KobayashiT. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424.  doi: 10.2969/jmsj/02930407.

[16]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. A. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.

[17]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.  doi: 10.1137/1032046.

[18]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995.

[19]

P. MastroliaD. D. Monticelli and F. Punzo, Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds, Math. Ann., 367 (2017), 929-963.  doi: 10.1007/s00208-016-1393-2.

[20]

S. PigolaM. Rigoli and A. G. Setti, A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds, J. Functional Anal., 219 (2005), 400-432.  doi: 10.1016/j.jfa.2004.05.009.

[21]

F. Punzo, Blow-up of solutions to semilinear parabolic equations on Riemannian manifolds with negative sectional curvature, J. Math. Anal. Appl., 387 (2012), 815-827.  doi: 10.1016/j.jmaa.2011.09.043.

[22]

F. Punzo, Global solutions of semilinear parabolic equations on negatively curved Riemannian manifolds, J. Geom. Anal., 31 (2021), 543-559.  doi: 10.1007/s12220-019-00289-x.

[23]

Z. Wang and J. Yin, A note on semilinear heat equation in hyperbolic space, J. Diff. Eq., 256 (2014), 1151-1156.  doi: 10.1016/j.jde.2013.10.011.

[24]

Q. S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., 97 (1999), 515-539.  doi: 10.1215/S0012-7094-99-09719-3.

show all references

References:
[1]

J. Aguirre and M. Escobedo, On the blow-up of solutions of a convective reaction diffusion equation, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 433-460.  doi: 10.1017/S0308210500025828.

[2]

L. J. Alias, P. Mastrolia and M. Rigoli, Maximum Principles and Geometric Applications, Springer Monographs in Mathematics, Springer, Cham, 2016. doi: 10.1007/978-3-319-24337-5.

[3]

C. Bandle and H. Levine, Fujita phenomena for reaction-diffusion equations with convection like terms, Diff. Integral Eq., 7 (1994), 1169-1193. 

[4]

C. BandleM. A. Pozio and A. Tesei, The fujita exponent for the Cauchy problem in the Hyperbolic space, J. Diff. Eq., 251 (2011), 2143-2163.  doi: 10.1016/j.jde.2011.06.001.

[5]

L. BrandoliniM. Rigoli and A. G. Setti, Positive solutions of Yamabe type equationson complete manifolds and applications, J. Functional Anal., 160 (1998), 176-222.  doi: 10.1006/jfan.1998.3313.

[6]

A. Friedman, Partial Differential Equations of Parabolic Type, Dover Publications, New York, 1992.

[7]

H. Fujita, 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 (1966), 109-124. 

[8]

A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 135-249.  doi: 10.1090/S0273-0979-99-00776-4.

[9]

A. Grigor'yan, Heat Kernel and Analysis on Manifold, AMS/IP Studies in Advanced Mathematics, 47. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2009. doi: 10.1090/amsip/047.

[10]

G. GrilloG. Meglioli and F. Punzo, Smoothing effects and infinite time blow-up for reaction diffusion equation: An approach based on Sobolev and Poincaré inequalities, J. Math. Pures Appl., 151 (2021), 99-131.  doi: 10.1016/j.matpur.2021.04.011.

[11]

G. GrilloG. Meglioli and F. Punzo, Global existence of solutions and smoothing effect for classes of reaction-diffusion equations on manifolds, J. Evol. Eq., 21 (2021), 2339-2375.  doi: 10.1007/s00028-021-00685-3.

[12]

G. GrilloM. Muratori and F. Punzo, Blow-up and global existence for the porous medium equation with reaction on a class of Cartan-Hadamard manifolds, J. Diff. Eq., 266 (2019), 4305-4336.  doi: 10.1016/j.jde.2018.09.037.

[13]

K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503-505. 

[14]

S. Kaplan, On the growth of solutions of quasilinear parabolic equations, Comm. Pure Appl. Math., 16 (1963), 305-330.  doi: 10.1002/cpa.3160160307.

[15]

K. KobayashiT. Sirao and H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424.  doi: 10.2969/jmsj/02930407.

[16]

O. A. Ladyzhenskaya, V. A. Solonnikov and N. A. Uraltseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, Vol. 23 American Mathematical Society, Providence, R.I. 1968.

[17]

H. A. Levine, The role of critical exponents in blowup theorems, SIAM Review, 32 (1990), 262-288.  doi: 10.1137/1032046.

[18]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser/Springer Basel AG, Basel, 1995.

[19]

P. MastroliaD. D. Monticelli and F. Punzo, Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds, Math. Ann., 367 (2017), 929-963.  doi: 10.1007/s00208-016-1393-2.

[20]

S. PigolaM. Rigoli and A. G. Setti, A Liouville-type result for quasi-linear elliptic equations on complete Riemannian manifolds, J. Functional Anal., 219 (2005), 400-432.  doi: 10.1016/j.jfa.2004.05.009.

[21]

F. Punzo, Blow-up of solutions to semilinear parabolic equations on Riemannian manifolds with negative sectional curvature, J. Math. Anal. Appl., 387 (2012), 815-827.  doi: 10.1016/j.jmaa.2011.09.043.

[22]

F. Punzo, Global solutions of semilinear parabolic equations on negatively curved Riemannian manifolds, J. Geom. Anal., 31 (2021), 543-559.  doi: 10.1007/s12220-019-00289-x.

[23]

Z. Wang and J. Yin, A note on semilinear heat equation in hyperbolic space, J. Diff. Eq., 256 (2014), 1151-1156.  doi: 10.1016/j.jde.2013.10.011.

[24]

Q. S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., 97 (1999), 515-539.  doi: 10.1215/S0012-7094-99-09719-3.

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