Global existence for reaction-diffusion evolution equations driven by the $ {\text{p}} $-Laplacian on manifolds

Gabriele Grillo; Giulia Meglioli; Fabio Punzo; Gabriele Grillo; Giulia Meglioli; Fabio Punzo

doi:10.3934/mine.2023070

Mathematics in Engineering

2023, Volume 5, Issue 3: 1-38. doi: 10.3934/mine.2023070

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Global existence for reaction-diffusion evolution equations driven by the $ {\text{p}} $-Laplacian on manifolds

Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy

Received: 17 September 2022 Revised: 07 December 2022 Accepted: 09 December 2022 Published: 27 December 2022

We consider reaction-diffusion equations driven by the $ p $-Laplacian on noncompact, infinite volume manifolds assumed to support the Sobolev inequality and, in some cases, to have $ L^2 $ spectrum bounded away from zero, the main example we have in mind being the hyperbolic space of any dimension. It is shown that, under appropriate conditions on the parameters involved and smallness conditions on the initial data, global in time solutions exist and suitable smoothing effects, namely explicit bounds on the $ L^\infty $ norm of solutions at all positive times, in terms of $ L^q $ norms of the data. The geometric setting discussed here requires significant modifications w.r.t. the Euclidean strategies.
- nonlinear reaction-diffusion equation,
- Riemannian manifolds,
- $ p- $Laplace equation,
- smoothing estimates,
- global existence,
- Sobolev inequality,
- Poincaré inequality,
- porous medium equation
Citation: Gabriele Grillo, Giulia Meglioli, Fabio Punzo. Global existence for reaction-diffusion evolution equations driven by the $ {\text{p}} $-Laplacian on manifolds[J]. Mathematics in Engineering, 2023, 5(3): 1-38. doi: 10.3934/mine.2023070

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Abstract

We consider reaction-diffusion equations driven by the $ p $-Laplacian on noncompact, infinite volume manifolds assumed to support the Sobolev inequality and, in some cases, to have $ L^2 $ spectrum bounded away from zero, the main example we have in mind being the hyperbolic space of any dimension. It is shown that, under appropriate conditions on the parameters involved and smallness conditions on the initial data, global in time solutions exist and suitable smoothing effects, namely explicit bounds on the $ L^\infty $ norm of solutions at all positive times, in terms of $ L^q $ norms of the data. The geometric setting discussed here requires significant modifications w.r.t. the Euclidean strategies.

References

[1]	E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285–320. https://doi.org/10.1215/S0012-7094-07-13623-8 doi: 10.1215/S0012-7094-07-13623-8
[2]	N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Commun. Part. Diff. Eq., 4 (1979), 827–868. https://doi.org/10.1080/03605307908820113 doi: 10.1080/03605307908820113
[3]	C. Bandle, M. A. Pozio, A. Tesei, The Fujita exponent for the Cauchy problem in the hyperbolic space, J. Differ. Equations, 251 (2011), 2143–2163. https://doi.org/10.1016/j.jde.2011.06.001 doi: 10.1016/j.jde.2011.06.001
[4]	P. Bénilan, Opérateurs accrétifs et semi-groupes dans les espaces $L^p$ ($1 \le p \le +\infty$), Japan-France seminar, Japan Society for the Advancement of Science, 1978.
[5]	P. Bénilan, M. G. Crandall, Completely accretive operators, semigroup theory and evolution equations (Delft, 1989), In: Lecture Notes in Pure and Applied Mathematics, Volume 135, Dekker, 1991, 41–75.
[6]	V. Bögelein, F. Duzaar, G. Mingione, The regularity of general parabolic systems with degenerate diffusion, Memoirs of the American Mathematical Society, 2013. https://doi.org/10.1090/S0065-9266-2012-00664-2
[7]	X. Chen, M. Fila, J. S. Guo, Boundedness of global solutions of a supercritical parabolic equation, Nonlinear Anal., 68 (2008), 621–628. https://doi.org/10.1016/j.na.2006.11.023 doi: 10.1016/j.na.2006.11.023
[8]	T. Coulhon, D. Hauer, Regularisation effects of nonlinear semigroups, SMAI - Mathématiques et Applications, Springer, to appear.
[9]	K. Deng, H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., 243 (2000), 85–126. https://doi.org/10.1006/jmaa.1999.6663 doi: 10.1006/jmaa.1999.6663
[10]	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.
[11]	Y. Fujishima, K. Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 231–247. https://doi.org/10.1016/j.anihpc.2013.03.001 doi: 10.1016/j.anihpc.2013.03.001
[12]	V. A. Galaktionov, The conditions for there to be no global solutions of a class of quasilinear parabolic equations, USSR Computational Mathematics and Mathematical Physics, 22 (1982), 73–90. https://doi.org/10.1016/0041-5553(82)90037-4 doi: 10.1016/0041-5553(82)90037-4
[13]	V. A. Galaktionov, Blow-up for quasilinear heat equations with critical Fujita's exponents, Proc. Roy. Soc. Edinb. A, 124 (1994), 517–525. https://doi.org/10.1017/S0308210500028766 doi: 10.1017/S0308210500028766
[14]	V. A. Galaktionov, H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34 (1998), 1005–1027. https://doi.org/10.1016/S0362-546X(97)00716-5 doi: 10.1016/S0362-546X(97)00716-5
[15]	A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., 36 (1999), 135–249. https://doi.org/10.1090/s0273-0979-99-00776-4 doi: 10.1090/s0273-0979-99-00776-4
[16]	A. Grigor'yan, Heat kernel and analysis on manifolds, Providence, RI: American Mathematical Society, 2009.
[17]	G. Grillo, G. Meglioli, F. Punzo, Global existence of solutions and smoothing effects for classes of reaction-diffusion equations on manifolds, J. Evol. Equ., 21 (2021), 2339–2375. https://doi.org/10.1007/s00028-021-00685-3 doi: 10.1007/s00028-021-00685-3
[18]	G. Grillo, G. Meglioli, F. Punzo, Smoothing effects and infinite time blowup for reaction-diffusion equations: an approach via Sobolev and Poincaré inequalities, J. Math. Pure. Appl., 151 (2021), 99–131. https://doi.org/10.1016/j.matpur.2021.04.011 doi: 10.1016/j.matpur.2021.04.011
[19]	G. Grillo, G. Meglioli, F. Punzo, Blow-up versus global existence of solutions for reaction-diffusion equations on classes of Riemannian manifolds, Annali di Matematica Pura e Applicata, in press. https://doi.org/10.1007/s10231-022-01279-7
[20]	Q. Gu, Y. Sun, J. Xiao, F. Xu, Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold, Calc. Var., 59 (2020), 170. https://doi.org/10.1007/s00526-020-01837-y doi: 10.1007/s00526-020-01837-y
[21]	D. Hauer, Regularizing effect of homogeneous evolution equations with perturbation, Nonlinear Anal., 206 (2021), 112245. https://doi.org/10.1016/j.na.2021.112245 doi: 10.1016/j.na.2021.112245
[22]	K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503–505. https://doi.org/10.3792/pja/1195519254 doi: 10.3792/pja/1195519254
[23]	K. Kobayashi, T. Sirao, H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407–424. https://doi.org/10.2969/jmsj/02930407 doi: 10.2969/jmsj/02930407
[24]	T. Kuusi, G. Mingione, Gradient regularity for nonlinear parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 755–822. https://doi.org/10.2422/2036-2145.201103_006 doi: 10.2422/2036-2145.201103_006
[25]	O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Providence, RI: American Mathematical Society, 1968.
[26]	H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32 (1990), 262–288. https://doi.org/10.1137/1032046 doi: 10.1137/1032046
[27]	A. V. Martynenko, A. F. Tedeev, On the behavior of solutions of the Cauchy problem for a degenerate parabolic equation with nonhomogeneous density and a source, Comput. Math. Math. Phys., 48 (2008), 1145–1160. https://doi.org/10.1134/S0965542508070087 doi: 10.1134/S0965542508070087
[28]	P. Mastrolia, D. D. Monticelli, F. Punzo, Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds, Math. Ann., 367 (2017), 929–963. https://doi.org/10.1007/s00208-016-1393-2 doi: 10.1007/s00208-016-1393-2
[29]	G. Meglioli, D. D. Monticelli, F. Punzo, Nonexistence of solutions to quasilinear parabolic equations with a potential in bounded domains, Calc. Var., 61 (2022), 23. https://doi.org/10.1007/s00526-021-02132-0 doi: 10.1007/s00526-021-02132-0
[30]	E. Mitidieri, S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1–362.
[31]	E. Mitidieri, S. I. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129–162. https://doi.org/10.1007/s00032-004-0032-7 doi: 10.1007/s00032-004-0032-7
[32]	S. I. Pohozaev, A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 487–502. https://doi.org/10.1016/j.anihpc.2003.06.002 doi: 10.1016/j.anihpc.2003.06.002
[33]	F. Punzo, A. Tesei, On a semilinear parabolic equation with inverse-square potential, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., 21 (2010), 359–396. https://doi.org/10.4171/RLM/578 doi: 10.4171/RLM/578
[34]	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. https://doi.org/10.1016/j.jmaa.2011.09.043 doi: 10.1016/j.jmaa.2011.09.043
[35]	P. Quittner, The decay of global solutions of a semilinear heat equation, Discrete Contin. Dyn. Syst., 21 (2008), 307–318. https://doi.org/10.3934/dcds.2008.21.307 doi: 10.3934/dcds.2008.21.307
[36]	P. Souplet, Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in $\mathbb R^N$, J. Funct. Anal., 272 (2017), 2005–2037. https://doi.org/10.1016/j.jfa.2016.09.002 doi: 10.1016/j.jfa.2016.09.002
[37]	J. L. Vázquez, The porous medium equation: mathematical theory, Oxford: Oxford University Press, 2007. https://doi.org/10.1093/acprof:oso/9780198569039.001.0001
[38]	L. Véron, Effets régularisants de semi-groupes non linéaires dans des espaces de Banach, Ann. Fac. Sci. Toulouse Math. (5), 1 (1979), 171–200.
[39]	Z. Wang, J. Yin, A note on semilinear heat equation in hyperbolic space, J. Differ. Equations, 256 (2014), 1151–1156. https://doi.org/10.1016/j.jde.2013.10.011 doi: 10.1016/j.jde.2013.10.011
[40]	Z. Wang, J. Yin, Asymptotic behaviour of the lifespan of solutions for a semilinear heat equation in hyperbolic space, Proc. Roy. Soc. Edinb. A, 146 (2016), 1091–1114. https://doi.org/10.1017/S0308210515000785 doi: 10.1017/S0308210515000785
[41]	F. B. Weissler, $L^p$-energy and blow-up for a semilinear heat equation, Proc. Sympos. Pure Math., 45 (1986), 545–551.
[42]	E. Yanagida, Behavior of global solutions of the Fujita equation, Sugaku Expositions, 26 (2013), 129–147.
[43]	Q. S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., 97 (1999), 515–539. https://doi.org/10.1215/S0012-7094-99-09719-3 doi: 10.1215/S0012-7094-99-09719-3

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