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Solution of the nonregular problem for the heat equation with the time derivative in the boundary conditions. (Russian. English summary) Zbl 1488.35264

Summary: There is studied one dimensional problem for a parabolic equation with a time derivative in the boundary condition with non fulfillment of the compatibility conditions of all necessary orders. It is shown due to this there are appeared singular solutions, the derivatives of which have the singularities at an initial moment in the vicinity of a boundary. Existence, uniqueness, estimate of the regular solution is proved in the Hölder space.

MSC:

35K05 Heat equation
35K20 Initial-boundary value problems for second-order parabolic equations
35C05 Solutions to PDEs in closed form
35B25 Singular perturbations in context of PDEs