PhD Thesis
of Denis R. Akhmetov

The classical solvability of the Cauchy problem for second-order linear parabolic equation, in general form, is considered.

It is well known that the Cauchy problem may fail to possess a classical solution if the coefficients and right-hand side of the equation are continuous but unqualified additionally. The coefficients and right-hand side of the equation are usually supposed to satisfy some Holder condition. We may treat the latter as uniform continuity with the continuity modulus a power function. The following question is natural: what properties of the continuity moduli of the coefficients and right-hand side of the equation guarantee existence of classical solutions and exact coercive estimates for their norms? By exact coercive estimates for the norms of solutions we mean estimates in which the higher-order derivatives of a solution to the problem belong to the same function class as the right-hand side of the equation. In other words, the continuity modulus of the higher-order derivatives of a solution does not exceed (up to a constant factor) the continuity modulus of the right-hand side.

The following results are established in the PhD thesis:

• We introduce function spaces with qualified continuity moduli such that the membership of the coefficients and right-hand side of the equation in these classes guarantees classical solvability of the Cauchy problem. We find necessary and sufficient conditions on the continuity moduli under which the operator of the Cauchy problem establishes an isomorphism between the corresponding spaces of solutions and right-hand sides.
  It is interesting that these conditions coincide with the Zygmund conditions well known in the theory
  of Fourier series, although their appearance here is not stipulated by Fourier series expansions.

• In the second part of the thesis, we establish a necessary and sufficient condition ensuring classical solvability of the Cauchy problem with zero initial data in the class of uniformly parabolic equations whose coefficients are Holder continuous and whose right-hand sides possess a local continuity modulus.
  The condition is stated in terms of the continuity modulus. Interestingly, this necessary and sufficient condition coincides with the well-known Dini condition of the theory of Fourier series.

• In the last part, we find a representation for a classical solution provided that the latter exists. Herewith,
  the growth of the right-hand side of the equation is arbitrary as the temporary variable, t, tends to zero
  and preassigned as the Euclidean norm of the spatial variables, ||x||, tends to infinity.
  Here we do not suppose any additional restrictions on the continuity modulus of the right-hand side.

These results have been published in several papers.