Entire Solutions and Asymptotic Behavior to a Class of Parabolic $k$-Hessian Equations with More General Right-Hand Side Terms

Abstract

This paper investigates entire classical separable variable radial solutions to a class of parabolic k-Hessian equations -u_t (sigma_k(lambda(D^2u)))^alpha = f(|x|)g(t), where the right-hand side consists of positive continuous functions. Without assuming that -u_t is positively bounded, we extend previous work for the equations whose right-hand side is 1 to more general cases, including cases where the right-hand side terms are bounded and periodic. By employing the method of Euler's broken line, we obtain the existence and nonexistence results. Furthermore, we obtain the precise asymptotic power exponent.