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In this paper it is established that in an infinite angular domain for Dirichlet problem of the heat conduction equation the unique (up to a constant factor) non-trivial solution exists, which does not belong to the class of summable functions with the found weight. It is shown that for the adjoint boundary value problem the unique (up to a constant factor) non-trivial solution exists, which belongs to the class of essentially bounded functions with the weight found in the work. It is proved that the operator of a boundary value problem of heat conductivity in an infinite angular domain in a class of growing functions is Noetherian with an index which is equal to minus one. MSC: Primary 35D05; 35K20; secondary 45D05

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In this paper we investigate the first boundary value problem for essentially loaded equation of heat conduction, i.e. when laden terms are derivatives for any finite order. It is shown that if the point of load is fixed, this problem is uniquely solvable. The stated boundary problem is reduced to the Volterra integral equation of the second kind. Estimates of the kernel of the integral equation are made, which indicate a weak singularity of the kernel. It is shown that if the point of load is fixed, then the stated boundary problem is uniquely solvable.

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