Finite Time Blow-Up for an Inhomogeneous Wave Equation with Riemann--Liouville Fractional Integral

changyong Xiang; Jisong Duan; Qianqian Luo

doi:10.61383/ejam.202532105

Authors

Changyong Xiang School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China Corresponding Author https://orcid.org/0009-0003-8568-1000
Jisong Duan School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China https://orcid.org/0009-0008-8662-2578
Qianqian Luo School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550025, China https://orcid.org/0009-0007-5758-4130

DOI:

https://doi.org/10.61383/ejam.202532105

Keywords:

Finite time blow-up, Inhomogeneous wave equation, Riemann-Liouville fractional integral, Strauss exponent

Abstract

This paper is devoted to the study of the effect of an inhomogeneity \( \omega(x) \) on the blow-up phenomena for the wave equation
\[
u_{tt} - \triangle u = I_{0^{+}}^{\alpha}(|u|^{p}) + \omega(x), \quad
(t,x) \in (0,T) \times \mathbb{R}^{N},
\]where \( N \ge 1 \), \( p > 1 \), \( 0 \le \alpha < 1 \), and \( \omega(x) \) satisfies \( \int_{\mathbb{R}^{N}} \omega(x)\,dx > 0 .\) Specifically, by the method of cut-functions, this paper first proves that any non-trivial solution blows up in finite time under \( 0 < \alpha < 1 \), and later proves that any non-trivial solution blows up in finite time under \( \alpha = 0 \), \( 2 \le N \le 4 \), and \( p \) being the Strauss exponent.

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