caa a22 4500 60619441X CHVBK 20210128100914.0 cr unu---uuuuu 210128e20151001xx s 000 0 eng 10.1007/s00030-015-0312-7 doi (NATIONALLICENCE)springer-10.1007/s00030-015-0312-7 Large time behavior of solutions for a complex-valued quadratic heat equation [Elektronische Daten] [Amel Chouichi, Sarah Otsmane, Slim Tayachi] 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 z = Δ z + z 2 , $${t > 0, x \in \mathbb{R}^{N},}$$ t > 0 , x ∈ R N , with initial data z 0=u 0+iv 0. We show that if $${u_{0}(x)\sim c|x|^{-2\alpha_1}}$$ u 0 ( x ) ∼ c | x | - 2 α 1 and $${v_{0}(x)\sim c|x|^{-2\alpha_{1}'},}$$ v 0 ( x ) ∼ c | x | - 2 α 1 ′ , as $${|x|\rightarrow\infty}$$ | x | → ∞ with $${\alpha_{1} \geq1,\,2\alpha_{1}'-\alpha_{1} \geq1,\,\frac{N}{2\alpha_{1}} > 1,\,\frac{N}{2\alpha_{1}'} > 1}$$ α 1 ≥ 1 , 2 α 1 ′ - α 1 ≥ 1 , N 2 α 1 > 1 , N 2 α 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 $${\alpha_{1}'}$$ α 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. Springer Basel, 2015 Parabolic system nationallicence Semi-linear parabolic equations nationallicence Global solutions nationallicence Large time behavior nationallicence Self-similar solutions nationallicence Nonlinear heat equation nationallicence Chouichi Amel Laboratoire équations aux dérivées partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia aut Otsmane Sarah Laboratoire équations aux dérivées partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia aut Tayachi Slim Laboratoire équations aux dérivées partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia aut Nonlinear Differential Equations and Applications NoDEA Springer Basel 22/5(2015-10-01), 1005-1045 1021-9722 22:5<1005 2015 22 30 https://doi.org/10.1007/s00030-015-0312-7 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00030-015-0312-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Chouichi Amel Laboratoire équations aux dérivées partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia aut NATIONALLICENCE 700 1- Otsmane Sarah Laboratoire équations aux dérivées partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia aut NATIONALLICENCE 700 1- Tayachi Slim Laboratoire équations aux dérivées partielles LR03ES04, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El Manar, 2092, Tunis, Tunisia aut NATIONALLICENCE 773 0- Nonlinear Differential Equations and Applications NoDEA Springer Basel 22/5(2015-10-01), 1005-1045 1021-9722 22:5<1005 2015 22 30