On Necessary and Sufficient Conditions for Classical Solvability of the~Cauchy Problem for Linear Parabolic Equations

D. R. Akhmetov

Matematičeskie trudy, Tome 1 (1998) no. 1, pp. 3-28.

Résumé

In the first part of the article, we establish a necessary and sufficient condition ensuring classical solvability of the Cauchy problem with zero initial data for uniformly parabolic equations whose coefficients are Hölder continuous and whose right-hand sides possess a local continuity modulus. In the second part, we find a representation for a classical solution provided that the latter exists. Herewith, the growth of the right-hand side of an equation is arbitrary as $t\to 0$ and preassigned as $|x|\to \mathrm{\infty }$. In the last part, we obtain necessary and sufficient conditions for classical solvability of the Cauchy problem with zero initial data for parabolic equations with constant coefficients and right-hand sides infinitely differentiable for $t>0$.