Asymptotically self-similar global solutions for a complex-valued quadratic heat equation with a generalized kernel

We study the global existence and the large time behavior of solutions for the complex-valued quadratic heat equation: $$\partial _t z=\mathcal {L}\,z+z^2,\,t>0,\,x\in \mathbb {R}^{N},$$ with initial data $$z_0=u_0+i\,v_0$$ . $$\mathcal {L}$$ is a linear operator with $$e^{t\mathcal {L}}$$ its semigroup having a generalized heat kernel G satisfying in particular $$G(t,x)= t^{-\frac{N}{d}} G(1,xt^{-1/d}),\,d>0,\, t>0$$ and $$x\in \mathbb {R}^N.$$ For $$\sigma \ge 1,\, 2\rho -\sigma \ge 1$$ , $$\frac{N}{{\text {d}}\sigma }>1,$$ $$\frac{N}{{\text {d}}\rho }>1$$ and if $$u_{0}(x)\sim c|x|^{-{\text {d}}\sigma }$$ and $$v_{0}(x)\sim c|x|^{-{\text {d}}\rho },$$ as $$|x|\rightarrow \infty $$ (|c| is small) we prove the global existence and we establish the existence of four different asymptotically self-similar behaviors. Our results apply for the fractional and higher order complex-valued quadratic heat equation and are new even in these cases.