Large-time structure of solutions of scalar conservation laws without convexity in the presence of a linear source field
The problem studied here is the scalar law \(u_ t+[f(u)]_ x+g(u)=0\), \(x\in R^ 1\), \(u(x,0)=u_ 0(x)\), \(u_ 0\) \(L\)-periodic, with mean \(0:\int_ 0^ L u_ 0(x)dx=0\). The model in mind has \(f(u)=| u|^{p-1}u\), \(p>2\). The author proves that, as \(t\to\infty\), the solution remains in a bounded set whose size is independent of the size of the initial data. Further, for sufficiently large times, the number of shocks per period will attain a finite value which then remains a constant number. The proofs involve a detailed use of the theory of generalized characteristics via the use of Fillipov's theory of differential inclusions.
nonlinear system of functional and delay-differential equations
Fillipov's theory of differential inclusions
scientific article; zbMATH DE number 93320
scientific article