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

Mohamed Jleli; Bessem Samet

doi:10.3934/cpaa.2022036

Article Contents

2022, Volume 21, Issue 6: 2065-2078. Doi: 10.3934/cpaa.2022036

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

Mohamed Jleli and
Bessem Samet^,

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

^* Corresponding author
^* Corresponding author

Received: May 2021

Revised: December 2021

Early access: April 2022

Published: June 2022

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 35B44, 35R01; Secondary: 35B33.

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References

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