Well posedness and regularity for heat equation with the initial condition in weighted Orlicz–Slobodetskii space subordinated to Orlicz space like \(\lambda (\mathrm{log} \lambda )^\alpha \) and the logarithmic weight

We consider the initial-value problem \(\tilde{u}_t=\Delta _x \tilde{u}(x,t)\), \(\tilde{u}(x,0)=u(x)\), where \(x\in \mathbb {R}^{n-1},t\in (0,T)\) and \(u\) belongs to certain weighted Orlicz–Slobodetskii space \(Y^{ \Phi ,\Phi }_{log}(\mathbb {R}^{n-1})\) subordinated to the logarithmic weight. We prove that under certain assumptions on Orlicz function \({\Phi }\), the solution \(\tilde{u}\) belongs to Orlicz–Sobolev space \(W^{1,{\Psi }}(\Omega \times (0,T))\) for certain function \(\Psi \) which in general dominates \(\Phi \). The typical representants are \(\Phi (\lambda )= \lambda (\mathrm{log} (2+\lambda ))^\alpha \), \(\Psi (\lambda )= \lambda (\mathrm{log} (2+\lambda ))^{\alpha +1}\) where \(\alpha > 0\).