Convergence and blow-up of solutions for a complex-valued heat equation with a quadratic nonlinearity
Convergence and blow-up of solutions for a complex-valued heat equation with a quadratic nonlinearity
复制标题
DOI：
10.1090/s0002-9947-2012-05797-7
复制
发表时间：
2012-10
影响因子：
1.3
通讯作者：
Jong-Shenq Guo;H. Ninomiya;Masahiko Shimojo;E. Yanagida
中科院分区：
文献类型：
--
作者：
Jong-Shenq Guo;H. Ninomiya;Masahiko Shimojo;E. Yanagida
本文研究的是抛物型方程组的柯西问题，该方程是从具有二次非线性的复值方程导出的。首先我们证明，如果初始数据图像的凸包不与正实轴相交，则解在时间上全局存在并收敛到平凡稳态。接下来，在一维空间上，我们提供了一些具有同时爆炸的非平凡虚部的解决方案。最后，我们考虑渐近恒定初始数据的情况，并表明，根据极限，解在空间无穷大处非同时爆炸或在时间上全局存在并收敛到平凡稳态。
This paper is concerned with the Cauchy problem for a system of parabolic equations which is derived from a complex-valued equation with a quadratic nonlinearity. First we show that if the convex hull of the image of initial data does not intersect the positive real axis, then the solution exists globally in time and converges to the trivial steady state. Next, on the onedimensional space, we provide some solutions with nontrivial imaginary parts that blow up simultaneously. Finally, we consider the case of asymptotically constant initial data and show that, depending on the limit, the solution blows up nonsimultaneously at space infinity or exists globally in time and converges to the trivial steady state.