Blow-up of the Solution for a Class of Porous Medium Equation with Positive Initial Energy

This paper deals with a class of porous medium equation

ut = Δum + f(u)$ut\hspace{0.17em}=\hspace{0.17em}\text{Δ}{u}^m\hspace{0.17em}+\hspace{0.17em}f\left(u\right)$

with homogeneous Dirichlet boundary conditions. The blow-up criteria is established by using the method of energy under the suitable condition on the function f(u).     

Sectorial oscillation theory of linear differential equations

In this paper, the sectorial oscillation of the solutions of higher order homogeneous linear differential equations

f(k) + A_n-2(z)f(k-2) + ? + A₁(z)f^/ + A₀(z)f = 0$f^{\left(k\right)}\hspace{0.17em}+\hspace{0.17em}{A}_{n-2}\left(z\right)f^{\left(k-2\right)}\hspace{0.17em}+\hspace{0.17em}?\hspace{0.17em}+\hspace{0.17em}{A}_1\left(z\right)f^{/}\hspace{0.17em}+\hspace{0.17em}{A}_0\left(z\right)f\hspace{0.17em}=\hspace{0.17em}0$

with infinite order entire function coefficients is studied. Results are obtained to extend some results in [19] and [18].     

On zeros of solutions of higher order homogeneous linear differential equations

In this article, we investigate the exponent of convergence of zeros of solutions for some higher-order homogeneous linear differential equation, and prove that if A_k−1 is the dominant coefficient, then every transcendental solution f(z) of equation

f(k)+A_k-1 f(k-1)+?+A₀ f=0$f^{(k)}+A_{k-1}\hspace{0.17em}{f}^{(k-1)}+?+A_0\hspace{0.17em}f=0$

satisfies λ(f) = ∞, where λ(f) denotes the exponent of convergence of zeros of the meromorphic function f(z).