New contributions to a complex system of quadratic heat equations with a generalized kernels: global solutions

Abstract

In this work, we propose new contributions to a complex system of quadratic heat equations with a generalized kernel of the form: $\partial _t z=\mathfrak {L}\,z+ \widetilde{z}^{2},\;\partial _t \widetilde{z}=\mathfrak {L}\,\widetilde{z}+ z^2,\;t>0,$ with initial conditions $z_{0}=u_0+v_0,\;\widetilde{z}_{0}=\widetilde{u}_0+\widetilde{v}_0$, and $\mathfrak {L}$ 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 $x\in \mathbb {R}^N.$ Under conditions on the parameters $\sigma _{1},\,\widetilde{\sigma }_{1},\,\rho _{1},\,$ and $\widetilde{\rho _{1}}$ we show results on global-in time solution for small data $u_{0}(x)\sim c|x|^{-d\sigma _{1}},\,v_{0}(x)\sim c|x|^{-d\rho _{1}},\,\widetilde{u}_{0}(x)\sim c|x|^{-d\widetilde{\sigma }_{1}}$ and $\widetilde{v}_{0}(x)\sim c|x|^{-d\widetilde{\rho }_{1}}$ as $|x|\rightarrow \infty$, ( |c| is sufficiently small ). We investigate the global existence of solutions to the given system.