Nonlocal nonlinear systems of transport equations in weighted \(L^1\) spaces: An operator theoretic approach (Q1970893)

The paper is concerned with nonlocal nonlinear systems of the form \[ \partial_t {\mathbf u} + z'(t)\partial_x {\mathbf u}={\mathbf \phi}(t,x,{\mathbf u},z(t)),\quad (t,x)\in (0,T)\times{\mathbb R} \] \[ z(t) =L\int_{-\infty}^\infty {\mathbf w}(y) {\mathbf u}(t,y)dy, \quad t\in [0,T]. \] The initial-value problem for the above system is reformulated as an abstract evolution equation in certain weighted \(L^1\) spaces. The author studies the local and global existence, blowing-up phenomena and the global uniqueness of weak solutions to the Cauchy problem.