A blow-up result for a nonlinear wave equation on manifolds: the critical case

Abstract

We consider a inhomogeneous semilinear wave equation on a noncompact complete Riemannian manifold $(\mathbb{M},g)$ of dimension $N\ge 3$, without boundary. The reaction exhibits the combined effects of a critical term and of a forcing term. Using a rescaled test function argument together with appropriate estimates, we show that the equation admits no global solution. Moreover, in the special case when $\mathbb{M}={\mathbb{R}}^3$, our result improves the existing literature. Namely, our main result is valid without assuming that the initial values are compactly supported.

Keywords:

• Blow-up
• inhomogeneous semilinear wave equation
• critical case
• manifold

2010 Mathematical subject classifications::

• 35B44
• 35B33

Acknowledgments

The authors wish to thank the knowledgeable referees for the constructive remarks.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The first author was supported by Researchers Supporting Project number RSP-2021/57, King Saud University, Riyadh, Saudi Arabia.