The theories developed in the previous chapters can be used to study some parabolic equations as well. In this chapter, we first consider linear abstract Cauchy problems with non-densely defined and almost sectorial operators; that is, the part of this operator in the closure of its domain is sectorial. Such problems naturally arise for parabolic equations with nonhomogeneous boundary conditions. By using the integrated semigroup theory, we prove an existence and uniqueness result for integrated solutions. Moreover, we study the linear perturbation problem. Then in the second section we provide detailed stability and bifurcation analyses for a scalar reaction-diffusion equation, namely, a size-structured model.