Viscous scalar conservation law with nonlinear flux feedback and global attractors
The authors study the forced viscous scalar conservation law on \((0,1)\) \[ u_t-\nu u_{xx}+f(u)_x=F, \;u=u(x,t), \;x\in (0,1), \;t>0 \] with initial data \(u(x,0)=u_0(x)\) and nonlinear boundary feedback control \[ u_x(0,t)-g_0(u(0,t))=u_x(1,t)+g_1(u(1,t))=0 \] where \(g_0\), \(g_1\) are continuous nondecreasing functions with \(g_0(0)=g_1(0)=0\). On the base of nonlinear semigroup theory global existence and uniqueness of the solution are established for any bounded measurable initial and forced functions. Under appropriate growth conditions on \((f,g_0,g_1)\) existence of an absorbing set and a nonempty compact global attractor in \(L^\infty\)-topology are also proved.
scientific article; zbMATH DE number 1236175
scientific article