标题： 关于带有双耗散的弱耦合半线性$σ$-演化方程组的柯西问题 显示英文标题

标题： On the Cauchy problem for a weakly coupled system of semi-linear $σ$-evolution equations with double dissipation

摘要： 在本文中，我们考虑多组分弱耦合的半线性$\sigma$-演化方程组的柯西问题，该方程组具有双重耗散，并且适用于任何$\sigma\ge 1$。 第一个主要目的是通过假设初始数据具有额外的$L^1$正则性，并巧妙地使用多损失衰减，来获得超临界条件下的小数据解的整体（时间上）存在性。 对于第二个主要目的，我们感兴趣的是在亚临界情况下建立爆破结果以及解的寿命的精确估计。 证明基于通过修改测试函数的帮助进行矛盾论证以导出上界估计。 最后，我们成功地通过构造带有时间依赖多项式权重函数的Sobolev空间，在其相应范数中获得了下界估计。 显示英文摘要

摘要： In this paper, we would like to consider the Cauchy problem for a multi-component weakly coupled system of semi-linear $\sigma$-evolution equations with double dissipation for any $\sigma\ge 1$. The first main purpose is to obtain the global (in time) existence of small data solutions in the supercritical condition by assuming additional $L^1$ regularity for the initial data and using multi-loss of decay wisely. For the second main one, we are interested in establishing the blow-up results together with sharp estimates for lifespan of solutions in the subcritical case. The proof is based on a contradiction argument with the help of modified test functions to derive the upper bound estimates. Finally, we succeed in catching the lower bound estimate by constructing Sobolev spaces with the time-dependent weighted functions of polynomial type in their corresponding norms.