In this chapter we analyze the solution spaces of some differential equations. Mainly the local existence of solutions of nonlinear Timoshenko equation and the existence of solutions of sixth order Boussinesq equation are analyzed. The main results of this part are based on the papers [34, 35]. For further details we also cite [36, 37] and the reference therein.
Keywords: Cauchy problem, weak solution, sixth-order Boussinesq equation, Timoshenko Equation, Gronwalls inequality, blow up, contraction mapping, Young inequality, H¨older inequality, Sobolev-Poincar´e inequality.