# American Institute of Mathematical Sciences

June  2022, 21(6): 2065-2078. doi: 10.3934/cpaa.2022036

## Instantaneous blow-up for nonlinear Sobolev type equations with potentials on Riemannian manifolds

 Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

* Corresponding author

Received  May 2021 Revised  December 2021 Published  June 2022 Early access  April 2022

Fund Project: The second author is supported by Researchers Supporting Project number (RSP–2021/4), King Saud University, Riyadh, Saudi Arabia

We investigate Cauchy problems for two classes of nonlinear Sobolev type equations with potentials defined on complete noncompact Riemannian manifolds. The first one involves a polynomial nonlinearity and the second one involves a gradient nonlinearity. Namely, we derive sufficient conditions depending on the geometry of the manifold, the power nonlinearity, the behavior of the potential at infinity, and the initial data, for which the considered problems admit no nontrivial local weak solutions, i.e., an instantaneous blow-up occurs.

Citation: Mohamed Jleli, Bessem Samet. Instantaneous blow-up for nonlinear Sobolev type equations with potentials on Riemannian manifolds. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2065-2078. doi: 10.3934/cpaa.2022036
##### References:
 [1] L. J. Alias, P. Mastrolia and M. Rigoli, Maximum Principles and Geometric Applications, Springer, 2016. doi: 10.1007/978-3-319-24337-5. [2] D. Bianchi and A. Setti, Laplacian cut-offs, porous and fast diffusion on manifolds and other applications, Calc. Var. Partial Differ. Equ., 57 (2018), 1-33.  doi: 10.1007/978-3-319-24337-5. [3] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital., 1(B) (1998), 223-262. [4] H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304. [5] E. I. Galakhov, On the nonexistence of local solutions of some evolution equations, Math. Notes., 86 (2009), 314-324. [6] E. I. Galakhov, On the instantaneous blow-up of solutions of some quasilinear evolution equations, Differ. Equ., 46 (2010), 329–338. [7] 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.  doi: 10.1090/S0273-0979-99-00776-4. [8] A. Grigor'yan and Y. Sun, On non-negative solutions of the inequality $\Delta u+ u^{\sigma}\leq 0$ on Riemannian manifolds, Commun. Pure Appl. Math., 67 (2014), 1336-1352.  doi: 10.1002/cpa.21493. [9] M. O. Korpusov, Critical exponents of instantaneous blow-up or local solubility of non-linear equations of Sobolev type, Izv. RAN. Ser. Mat., 79 (2015), 103-162.  doi: 10.4213/im8285. [10] P. Mastrolia, D. Monticelli and F. Punzo, Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds, Calc. Var. Partial Differ. Equ., 54 (2015), 1345-1372.  doi: 10.1007/s00526-015-0827-0. [11] P. Mastrolia, 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. [12] E. Mitidieri and S. I. Pokhozhaev, Nonexistence of positive solutions for quasilinear elliptic problems in $\mathbb{R}^N$, Tr. Mat. Inst. Steklova., 227 (1999), 192-222. [13] D. D. Monticelli, F. Punzo and M. Squassina, Nonexistence for hyperbolic problems on Riemannian manifolds, Asympt. Anal., 120 (2020), 87-101.  doi: 10.3233/ASY-191580. [14] P. Petersen, Riemannian geometry, Graduate Texts in Mathematics, New York, NY: Springer New York, 2006. [15] 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. [16] F. B. Weissler, Local existence and nonexistence for semi-linear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-101. [17] 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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##### References:
 [1] L. J. Alias, P. Mastrolia and M. Rigoli, Maximum Principles and Geometric Applications, Springer, 2016. doi: 10.1007/978-3-319-24337-5. [2] D. Bianchi and A. Setti, Laplacian cut-offs, porous and fast diffusion on manifolds and other applications, Calc. Var. Partial Differ. Equ., 57 (2018), 1-33.  doi: 10.1007/978-3-319-24337-5. [3] H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital., 1(B) (1998), 223-262. [4] H. Brezis and T. Cazenave, A nonlinear heat equation with singular initial data, J. Anal. Math., 68 (1996), 277-304. [5] E. I. Galakhov, On the nonexistence of local solutions of some evolution equations, Math. Notes., 86 (2009), 314-324. [6] E. I. Galakhov, On the instantaneous blow-up of solutions of some quasilinear evolution equations, Differ. Equ., 46 (2010), 329–338. [7] 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.  doi: 10.1090/S0273-0979-99-00776-4. [8] A. Grigor'yan and Y. Sun, On non-negative solutions of the inequality $\Delta u+ u^{\sigma}\leq 0$ on Riemannian manifolds, Commun. Pure Appl. Math., 67 (2014), 1336-1352.  doi: 10.1002/cpa.21493. [9] M. O. Korpusov, Critical exponents of instantaneous blow-up or local solubility of non-linear equations of Sobolev type, Izv. RAN. Ser. Mat., 79 (2015), 103-162.  doi: 10.4213/im8285. [10] P. Mastrolia, D. Monticelli and F. Punzo, Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds, Calc. Var. Partial Differ. Equ., 54 (2015), 1345-1372.  doi: 10.1007/s00526-015-0827-0. [11] P. Mastrolia, 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. [12] E. Mitidieri and S. I. Pokhozhaev, Nonexistence of positive solutions for quasilinear elliptic problems in $\mathbb{R}^N$, Tr. Mat. Inst. Steklova., 227 (1999), 192-222. [13] D. D. Monticelli, F. Punzo and M. Squassina, Nonexistence for hyperbolic problems on Riemannian manifolds, Asympt. Anal., 120 (2020), 87-101.  doi: 10.3233/ASY-191580. [14] P. Petersen, Riemannian geometry, Graduate Texts in Mathematics, New York, NY: Springer New York, 2006. [15] 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. [16] F. B. Weissler, Local existence and nonexistence for semi-linear parabolic equations in $L^p$, Indiana Univ. Math. J., 29 (1980), 79-101. [17] 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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