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

In this work, we propose new contributions to a complex system of quadratic heat equations with a generalized kernel of the form: \(\partial _t z=\mathfrak {L}\,z+ \widetilde{z}^{2},\;\partial _t \widetilde{z}=\mathfrak {L}\,\widetilde{z}+ z^2,\;t>0,\) ${\partial }_{t}z=\mathfrak{L}z+{\tilde{z}}^2,{\partial }_t\tilde{z}=\mathfrak{L}\tilde{z}+z^2,t>0,$ with initial conditions \(z_{0}=u_0+v_0,\;\widetilde{z}_{0}=\widetilde{u}_0+\widetilde{v}_0\) $z_0=u_0+v_0,{\tilde{z}}_0={\tilde{u}}_0+{\tilde{v}}_0$ , and \(\mathfrak {L}\) $\mathfrak{L}$ is a linear operator with \(e^{t\mathcal {L}}\) $e^{t\mathcal{L}}$ its semigroup having a generalized heat kernel G satisfying in particular \(G(t,x)= t^{-\frac{N}{d}} G(1,xt^{-1/d}),\,d>0,\, t>0\) $G(t,x)=t^{-\frac{N}{d}}G(1,x{t}^{-1/d}),d>0,t>0$ and \(x\in \mathbb {R}^N.\) $x\in {\mathbb{R}}^N.$ Under conditions on the parameters \(\sigma _{1},\,\widetilde{\sigma }_{1},\,\rho _{1},\,\) ${\sigma }_1,{\tilde{\sigma }}_1,{\rho }_1,$ and \(\widetilde{\rho _{1}}\) $\widetilde{{\rho }_1}$ we show results on global-in time solution for small data \(u_{0}(x)\sim c|x|^{-d\sigma _{1}},\,v_{0}(x)\sim c|x|^{-d\rho _{1}},\,\widetilde{u}_{0}(x)\sim c|x|^{-d\widetilde{\sigma }_{1}}\) $u_0(x)\sim {c|x|}^{-d{\sigma }_1},v_0(x)\sim {c|x|}^{-d{\rho }_1},{\tilde{u}}_0(x)\sim c{|x|}^{-d{\tilde{\sigma }}_1}$ and \(\widetilde{v}_{0}(x)\sim c|x|^{-d\widetilde{\rho }_{1}}\) ${\tilde{v}}_0(x)\sim c{|x|}^{-d{\tilde{\rho }}_1}$ as \(|x|\rightarrow \infty \) $|x|\to \infty$ , ( |c| is sufficiently small ). We investigate the global existence of solutions to the given system.