Existence Of Global Positive Solution Research Articles

Blow-up results and global existence of positive solutions for the inhomogeneous evolution P-Laplacian equations

This paper deals with the Cauchy problem of inhomogeneous evolution P-Laplacian equations ∂ t u − div ( | ∇ u | p − 2 ∇ u ) = u q + w ( x ) with nonnegative initial data, where p > 1 , q > max { 1 , p − 1 } , and w ( x ) ⁄ ≡ 0 is a nonnegative continuous functions in R n . We prove that q c = ( p − 1 ) n / ( n − p ) is its critical exponent provided that 2 n / ( n + 1 ) < p < n , i.e., if q ≤ q c , then every positive solution blows up in finite time; whereas for q > q c , the equation possesses a global positive solution for some w ( x ) and some initial data. Meanwhile, we also prove that its positive solutions blow up in finite time provided that n ≤ p .

Existence and non-existence of global solutions of diffusion systems with nonlinear boundary conditions

This paper deals with existence and non-existence of global solutions of certain fast–slow diffusion systems with nonlinear boundary conditions. Necessary and sufficient conditions for global existence of positive solutions are obtained in terms of various parameters which appear explicitly in the definition of the systems.

Global existence of positive solutions for semilinear parabolic equations in a half-space

We prove the global existence of continuous solutions of the semilinear parabolic problem $\Delta u- {\partial \over {\partial t}}u+ V u^p =0$ in $\mathbb{R}^n_+\times (0,\infty)$, where ${\mathbb{R}^n_+}$ is a half-space in ${\mathbb{R}^n},\, n\geq 3$ . The potential $V$ is in some functional class ${\mathcal K}^{\infty}$. Our approach uses the Shauder fixed-point theorem.

Some Notes on Reaction Diffusion Systems with Nonlinear Boundary Conditions

This paper deals with the existence and nonexistence of global positive solution to a semilinear reaction-diffusion system with nonlinear boundary conditions. For the heat diffusion case, the necessary and sufficient conditions on the global existence of all positive solutions are obtained. For the general fast diffusion case, we get some conditions on the global existence and nonexistence of positive solutions. The results of this paper fill the same gaps which were left in this field.

Local behavior and global existence of positive solutions of auλ⩽−Δu⩽uλ

We study the behavior near the origin of C2 positive solutions u(x) of (∗)auλ≤-Δu≤uλin a punctured neighborhood of the origin in Rn (n>2) where the constants λ and a satisfy nn-2<λ<n+2n-2 and 0<a<1. We also study the existence of C2 positive solutions of (*) in Rn. In both cases we show that changing a from one value in the open interval (0,1) to another value in (0,1) can have a dramatic effect.