Large time behavior of solutions for a complex-valued quadratic heat equation

Abstract

In this paper we study the existence and the asymptotic behavior of global solutions for a parabolic system related to the complex-valued heat equation with quadratic nonlinearity: \({\partial_t z=\Delta z+z^2}\), \({t > 0, x \in \mathbb{R}^{N},}\) with initial data z0 = u0 + iv0. We show that if \({u_{0}(x)\sim c|x|^{-2\alpha_1}}\) and \({v_{0}(x)\sim c|x|^{-2\alpha_{1}'},}\) as \({|x|\rightarrow\infty}\) with \({\alpha_{1} \geq1,\,2\alpha_{1}'-\alpha_{1} \geq1,\,\frac{N}{2\alpha_{1}} > 1,\,\frac{N}{2\alpha_{1}'} > 1}\) (|c| is sufficiently small), then the solution is global and converges to a self-similar solution. We also establish the existence of four different self-similar behaviors. These behaviors depend on the values of α1 and \({\alpha_{1}'}\). In particular, the real and the imaginary parts of the constructed solutions may have different behaviors in the L∞-norm for large time. Also, the real part may have different behaviors from those known for the real-valued quadratic heat equation.