Quasilinear parabolic equations with nonlinear flux boundary conditions and semigroups (Q1265566)

The existence and uniqueness of the integral solution of \[ {d\over dt} u(t)+ Au(t)= g(t),\quad t\in (0,T),\quad u(0)= u_0\tag{1} \] in the space \(L^1(0,L)\) is studied. The operator \(A\) in (1) is a nonlinear differential operator and is proved to be m-accretive. The theory of nonlinear contraction semigroup is used to achieve the continuous dependence on the data \(u_0\) and \(g\). It is also proved that the integral solution of (1) is a weak solution.