Riv. Mat. Univ. Parma, Vol. 5, No. 1, 2014

Fabio Punzo [1]

Global existence of solutions to the semilinear heat equation on Riemannian manifolds with negative sectional curvature

Pages: 113-138
Received: 30 April 2013   
Accepted in revised form: 30 July 2013
Mathematics Subject Classification (2010): 35B51, 35B44, 35K08, 35K58, 35R01.

Keywords: Local existence, finite time blow-up, global existence, mild solutions, Laplace-Beltrami operator, heat kernel.
Author addresse:
[1] : Dipartimento di Matematica "F. Enriques", UniversitÓ degli Studi di Milano, via C. Saldini 50, 20133 Milano, Italy.

Abstract: We address local existence, blow-up and global existence of mild solutions to the semilinear heat equation on Riemannian manifolds with negative sectional curvature. We deal with a power nonlinearity multiplied by a time-dependent positive function \(h(t)\), and initial conditions \(u_0\in L^p(M)\). We show that depending on the behavior at infinity of \(h\), either every solution blows up in finite time, or a global solution exists, if the initial datum is small enough. In particular, for any power nonlinearity, if \(h\equiv 1\) we have global existence for small initial data, whereas if \(h(t)=e^{\alpha t}\) a Fujita type phenomenon prevails varying the parameter \(\alpha>0\).

References

[1] C. Bandle, M. A. Pozio and A. Tesei, The Fujita exponent for the Cauchy problem in the hyperbolic space, J. Differential Equations 251 (2011), 2143-2163.
[2] P. Baras, Non-unicité des solutions d'une equations d'evolutions non-linéaire, Ann. Fac. Sci. Toulouse Math. 5 (1983), 287-302.
[3] I. Chavel and L. Karp, Large time behavior of the heat kernel: the parabolic \(\lambda\)-potential alternative, Comment. Math. Helv. 66 (1991), 541- 556.
[4] E. B. Davies, Heat kernel and spectral theory, Cambridge Tracts in Mathematics, 92, Cambridge University Press, Cambridge 1989.
[5] 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.
[6] H. Fujita, On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations, Nonlin. Functional Analysis, (Proc. Sympos. Pure Math., 18, part 1, Chicago 1968), Amer. Math. Soc., Providence, R.I. 1970, 105-113.
[7] S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, Berlin 1990.
[8] A. Grigoryan, Analytic and geometric background of recurrence and nonexplosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. 36 (1999), 135-249.
[9] A. Grigoryan, Heat kernels on weighted manifolds and applications, Contemp. Math. 398 (2006), 93-191.
[10] A. Grigoryan, Heat kernel and analysis on manifolds, Amer. Math. Soc., International Press, Boston, MA 2009.
[11] H. A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), 262-288.
[12] L. Ji and A. Weber, \(L^p\) spectral theory and heat dynamics of symmetric spaces, J. Funct. Anal. 258 (2010), 1121-1139.
[13] H. P. McKean, An upper bound to the spectrum of \(\Delta\) on manifold of negative curvature, J. Differential Geometry 4 (1970), 359-366.
[14] 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.
[15] P. Quittner and P. Souplet, Superlinear parabolic problems. Blow-up, global existence and steady states, Birkhäuser Verlag, Basel 2007.
[16] F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in \(L^p\), Indiana Univ. Math. J. 29 (1980), 79-102.
[17] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math. 38 (1981), 29-40.
[18] F. B. Weissler, \(L^p\)-energy and blow-up for a semilinear heat equation, Proc. Sympos. Pure Math., 45, part 2, Amer. Math. Soc., Providence, RI 1986, 545-551.
[19] Q. S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J. 97 (1999), 515-539.


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