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arXiv · 2506.16126

Critical curve for the weakly coupled system of damped wave equations with mixed nonlinearities

Dinh Van Duong·Tuan Anh Dao·Masahiro Ikeda
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Abstract

In this paper, we would like to consider the Cauchy problem for a weakly coupled system of semi-linear damped wave equations with mixed nonlinear terms. Our main objective is to draw conclusions about the critical curve of this problem using tools from Harmonic Analysis. Precisely, we obtain a new critical curve $pq = 1+ \frac{2}{n}$ for $n =1,2$ by proving global (in time) existence of small data Sobolev solutions when $pq > 1 +\frac{2}{n}$ and blow-up of weak solutions in finite time even for small data when $pq < 1+ \frac{2}{n}$ for $n \geq 1$. From this, we infer the impact of the nonlinearities of time derivative-type on the critical curve associated with the system.

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Dinh Van Duong, Tuan Anh Dao, Masahiro Ikeda. 2025-06-19. Critical curve for the weakly coupled system of damped wave equations with mixed nonlinearities. https://arxiv.org/abs/2506.16126

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