A functional analytic approach towards nonlinear dissipative well-posed systems

The aim of this paper is to develop a functional analytic approach towards nonlinear systems. For linear systems this is well known and the resulting class of well-posed and regular linear systems is well studied. Our approach is based on the theory of nonlinear semigroup and we explain it by means of an example, namely equations of quasi-hyperbolic type. 1 Equations of quasi-hyperbolic type Let ψ : R → R be a strictly monotone decreasing surjective function with ψ(0) = 0. Associated to this function, we consider the following partial differential equation on the spatial interval [0, 1], ∂f ∂t (t, η) = ∂ψ(f) ∂η (t, η), 0 < η < 1, t > 0, (1) f(0, η) = f0(η), 0 < η < 1, f(t, 0) = 0, t > 0. As output we define y(t) = ψ(f(t, 1)). (2) We want to show that (1)–(2) defines a dissipative system. In particular, this will imply that the output y is a function in L1(0,∞). Department of Mathematics, University of Dortmund, D-44221 Dortmund, Germany, [email protected] Department of Applied Mathematics, University of Twente, P.O. 217, 7500 AE Enschede, The Netherlands, [email protected]