We study the existence of optimal controls for problems governed by semilinear parabolic equations. The nonlinearities in the state equation need not be monotone and the data need not be regular. In particular, the control may be any bounded Radon measure. Our examples include problems with nonlinear boundary conditions and parabolic systems.

Using Leray-Schauder degree theory we obtain various existence results for nonlinear boundary-value problems \begin{eqnarray*} (\phi(u'))'=f(t, u, u'),\quad l(u, u')=0 \end{eqnarray*} where $l(u, u')=0$ denotes the periodic, Neumann or Dirichlet boundary conditions on $[0,T],$ $\phi:\mathbb{R}\rightarrow (-a,a)$ is a homeomorphism, $\phi(0)=0.$

This work deals with a class of evolution problems consisting of a pair of coupled equations for modelling propagation of elastic waves in fluid-saturated porous media. The type of the first equation depends on two physical parameters (density and secondary consolidation) which can vanish while the second one is always parabolic. In case the density never vanishes, the first equation is second-order hyperbolic type and a weak solution to the problem is constructed using a variational method in a Sobolev framework. Next, the proof of uniqueness involves Ladyzenskaja's test-functions used to compensate a lack of regularity that would be required in a standard energy method. This approach gives rise to a priori estimates which are useful to prove that the linearized thermoelasticity and the quasi-static systems are defined as asymptotic models of the Biot problem when the secondary consolidation coefficient or the density is small.