Integro-differential Burgers equation. Solvability and homogenization (Q965004)

The authors' investigate the integro-differential equation of Burgers' type \[ {\partial u\over\partial x}- \beta{\partial^2 u\over\partial y^2}+ \alpha{\partial\over\partial y} f(u)= v{\partial\over\partial y} \int^y_{-\infty} (x,y') e^{(y'-y)/\tau} dy', \] the integral in the right-hand side describing the memory's relaxation. The initial condition is \(u(0,y)=\varphi(y)\) on \(\mathbb{R}\), and the periodicity condition is \(u(x,y+1)= u(x,y)\) for \((x,y)\in\mathbb{R}\times (0,X)\). Under various assumptions, one proves the existence, uniqueness in certain Sobolev spaces, and estimates are found for the solution in terms of the data. In order to obtain the results, technical procedures are involved, including the study of the integral operator in the right-hand side and its dual.