ABSTRACT

In this paper, we study the Cauchy problems for weakly coupled systems of semi-linear structurally damped σ-evolution models with different power nonlinearities. By assuming additional $L^m$ regularity on the initial data, with $m\in [1,2)$, we use $(L^m\cap L^2)-L^2$ and $L^2-L^2$ estimates for solutions to the corresponding linear Cauchy problems to prove the global (in time) existence of small data Sobolev solutions to the weakly coupled systems of semi-linear models from suitable function spaces for any $\sigma \ge 1$ and $\delta \in (0,\sigma )$. Moreover, not only the nonexistence of global solution is indicated by using test function method, but also a lifespan estimate is discussed when σ and $\delta \in (0,\sigma )$ are integers.

摘要

在本文中，我们研究了具有不同功率非线性的半线性结构阻尼σ演化模型的弱耦合系统的柯西问题。通过假设额外${\mathrm{大号}}^{米}$初始数据的规律性，与$米\in [1,2)$， 我们用$({\mathrm{大号}}^{米}\cap {\mathrm{大号}}^2)-{\mathrm{大号}}^2$和${\mathrm{大号}}^2-{\mathrm{大号}}^2$估计对应的线性柯西问题的解，以证明小数据的全局（及时）存在 Sobolev 从合适的函数空间对任何半线性模型的弱耦合系统的解$\sigma \ge 1$和$\delta \in (0,\sigma )$. 此外，不仅用检验函数法表明不存在全局解，而且讨论了当σ和$\delta \in (0,\sigma )$是整数。