Abstract:
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 $|x|\to\infty$.
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 $t>0$.
Key words:parabolic equations, Cauchy problem, classical solution, necessary and sufficient conditions, Tikhonov function class, Hölder condition, continuity modulus, Dini condition, Duhamel integral.