一类半线性波动方程弱耦合系统解的破裂

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Breakdown of Solutions to a Weakly Coupled System of Semilinear Wave Equations

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Abstract:

This paper addresses the Cauchy problem for a weakly coupled system of semilinear wave equations with scale-invariant dampings, mass, and general nonlinear memory terms. Firstly, a local (in time) existence result for this problem is established using Banach's fixed point theorem, subject to suitable assumptions on the exponents $p, q$ and coefficients $\mu_1, \mu_2$ .Here, $p$ and $q$ represent the powers of the nonlinear memory terms, while $\mu_1$ and $\mu_2$ denote the coefficients of the dampings and mass terms, respectively. It is noteworthy that Palmieri's decay estimates for the solution to the corresponding linear homogeneous equation play a crucial role in proving the local well-posedness result. Subsequently, employing an iteration technique in conjunction with the test function method, we obtain a blowup result for energy solutions.

Key words: semilinear wave equation, weakly coupled system, blowup, test function method, iteration skill

CLC Number:

O175.27

Zhendong Feng, Fei Guo, Yuequn Li. Breakdown of Solutions to a Weakly Coupled System of Semilinear Wave Equations[J].Acta mathematica scientia,Series A, 2025, 45(3): 726-747.

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