Nonlinear wave equations and applications from control theory

In 1961 Joergens [14]  rst proved that the Cauchy problem to the wave equation with
power type nonlinearity
(1.1) utt − u + uSuSp−1 = 0 inRn × R
in the case of R3 and when p < 5 admits of a unique regular solution u > C2 for any Cauchy
data u0 > C3, u1 > C2.
Our research in this paper will be centred on  nding the connection between the above
results and the closely related equations from the control theory in a more abstract setting.
In particular we examine the quasipotential formula in [7] and investigate whether
the result is applicable to (1.1) or its generalisations. To this end we  rst extend the
existence theorems in [10] to controlled equations, and then establish the  nding that the
quasipotential admits of analogous expression for our well-posed problem. Along the way
there is also a remarkable discovery concerning the large time behaviour of the solution.