Boundary and extremum problems for the nonlinear reaction–diffusion–convection equation under the Dirichlet condition

The global solvability of a boundary value problem for the reaction–diffusion–convection equation in which the reaction coefficient nonlinearly depends on the solution is proved. An inhomogeneous Dirichlet boundary condition for concentration is considered. In this case, the nonlinearity caused by the reaction coefficient is not monotonic in the entire domain. The solvability of a control problem with boundary, distributed, and multiplicative controls is proved. In the case when the reaction coefficient and quality functionals are Fréchet differentiable, optimality systems for extremum problems are derived. Based on their analysis, a stationary analogue of the bang–bang principle for specific control problems is established.