On the isomorphism generated by the heat equation (Q1972639)

The author studies the existence problem for a classical solution to the Cauchy problem for the heat equation \[ u_t = u_{xx}+f(x,t) \quad \text{for}\quad t>0, \qquad\qquad u(x,0)= \varphi(x) . \] A classical solution is a function \(u(x,t)\) that is continuous for \(t\geq 0\), has continuous derivatives \(u_t\) and \(u_{xx}\) for \(t>0\), and satisfies the above equation and initial condition. In a standard way, the author reformulates the above problem as the problem of characterizing the range of the linear operator \(L\: u \mapsto (f,\varphi)\), where \(f(x,t)=u_t-u_{xx}\) for \(t>0\) and \(\varphi (x)= u(x,0)\). The domain of \(L\) is the set \(C(\overline{\mathbb R^2_+})\cup C^{2,1}_{x,t}(\mathbb R^2_+)\). Obviously, the range of \(L\) consists of pairs of functions \(f\) and \(\varphi\) for which there exists a classical solution to the original problem. In the article under review, the author uses an arbitrary continuity modulus \(\omega\) for constructing weighted norm linear spaces \(\mathcal D_\omega\) and \(\mathcal R_\omega\) such that \(L\) is an isomorphism between \(\mathcal D_\omega\) and \(\mathcal R_\omega\). The results obtained generalize results by \textit{V.~S.~Belonosov} [Mat. Sb., Nov. Ser. 110(152), 163-188 (1979; Zbl 0434.35056)] who has used weighted Hölder norms for describing subspaces that are isomorphic under \(L\).