Global existence and asymptotic behavior of solutions for the complex-valued nonlinear heat equation
Volume 121 / 2018
Abstract
We are concerned with a parabolic system derived from the complex-valued nonlinear heat equation $\partial _t z=\varDelta z+z^p,$ $t \gt 0$, $x\in \mathbb {R}^{N},$ with initial data $z_{0}=u_{0}+{\rm i}v_{0},$ where $p \gt 1$ is an integer. We study the global existence and the large time behavior of solutions for data $u_{0}(x)\sim c|x|^{-2\alpha _1}$, $v_{0}(x)\sim c|x|^{-2\alpha _1’},$ as $|x|\rightarrow \infty $ ($|c|$ is sufficiently small) such that $\max(\alpha _1,\alpha _1’) \lt N/2,$ $\alpha _1\geq 1/(p-1)$ and $\alpha _1’\geq 1/(p-1)$ if $p$ is odd, $\alpha _1’\geq (1+\alpha _1)/p$ if $p$ is even. Since we may take different decay rates for the real part and imaginary part of the initial data, we obtain asymptotic behaviors which cannot occur for the real-valued nonlinear heat equation. Also, these asymptotic behaviors depend on the parity of the power of the nonlinearity.
