Determination of the stabilization rate of the solution to one initial problem for the heat equation

Differential equations are intensively used as models for a wide range of natural science problems. For most differential equations, it is not possible to obtain solutions in quadratures expressed in terms of elementary or special functions, and if it is possible, then the representations of these solutions are often very cumbersome, which makes their practical application difficult. Therefore, the question of finding simple formulas that describe with a sufficient degree of accuracy the qualitative behavior of solutions to differential equations on a certain interval of variation of the independent variable is very acute. Asymptotic methods are employed to determine the qualitative behavior of solutions to differential equations on a certain interval of change of the independent variable. Asymptotic methods are more preferable than numerical methods when one needs to know the behavior of the solution to a differential equation considered on an unbounded interval. This is explained by the fact that the discrepancy of the solution to a differential equation (the modulus of the difference between the true solution and the numerical solution) is usually estimated from above through a value proportional to the length of the interval on which the numerical method is applied. The paper considers the one-dimensional Cauchy problem for an inhomogeneous heat equation with a homogeneous initial condition. Using an explicit representation of the solution to the Cauchy problem, an exact uniform estimate and an exact pointwise estimate of the stabilization rate of the solution to the Cauchy problem to zero for a long time were constructed.

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