## ABSTRACT

In this chapter, we consider principal spectrum and principal Lyapunov exponents of nonautonomous and random parabolic equations. First in Section 4.1, we introduce basic assumptions for a given random (nonautonomous) parabolic equation and associate a proper family of parabolic equations with the given equation. Then based on the notions introduced in Chapters 2 and 3 in the general setting, we introduce the concepts of principal spectrum and principal Lyapunov exponents for a given random (nonautonomous) parabolic equation in terms of the associated family of parabolic equations, which naturally extends the classical concept of principal eigenvalue for the elliptic and periodic parabolic problems. Also applying the general theories developed in Chapters 2 and 3, we present basic properties of principal spectrum and principal Lyapunov exponents of random (nonautonomous) parabolic equations. In addition, we provide some examples which satisfy the basic assumptions in this section. In Section 4.2, we investigate the monotonicity of principal spectrum and principal Lyapunov exponents of random (nonautonomous) parabolic equations with respect to zero order terms. We also study the relation among the principal spectrum and principal Lyapunov exponents for random (nonautonomous) parabolic equations with different types of boundary conditions. Sections 4.3 and 4.4 concern the continuous dependence of principal spectrum and principal Lyapunov exponents of random (nonautonomous) parabolic equations with respect to the coefficients. Because of the speciality of the zero order coefficients, the continuous dependence with respect to these coefficients are considered in Section 4.3 first. In Section 4.4, the general continuous dependence is then discussed. Throughout Section 4.1 to Section 4.4, many results and arguments for random and nonautonomous equations are similar. However, considering that different readers may be interested in different types of equations, for convenience, in each section from Section 4.1 to Section 4.4, we treat these two types of equations in different subsections and provide proofs for most similar results. This chapter ends up with some historical remarks in Section 4.5.