This problem is a type of Cauchy problem for a first-order linear partial differential equation with variable coefficients. The differential equation describes a balance between transport (advection) and a source term, while the boundary conditions give initial values of the function and its normal derivative along the x axis and t axis, respectively.

The problem does not have a standard name because it's not a well-known or classic problem like the Heat Equation or Wave Equation. If it is part of a research paper or a textbook, it would likely be referred to by context, such as "Equation (1)" or "the advection-source problem" or "the problem given in Section X".

You could potentially name it based on its characteristics. Given the solution method and the boundary conditions' nature, you might call it something like "The Method of Characteristics Problem with Mixed Boundary Conditions". However, please note that this is not a formal or standard name, but rather a descriptive name based on the nature of the problem.