Abstract
We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: satisfying a specific decay rate: with and . Here and . Such a solution arises naturally when we study a very singular self-similar solution for a degenerate parabolic equation with nonlinear convection term defined on the half line .
Highlights
We consider a quasilinear degenerate diffusion equation with nonlinear convection term defined on the half line as ut |ux|p−2ux x uq x, x, t ∈ R × R, 1.1 with homogeneous Neumann boundary condition ux 0, t 0, 1.2 where p > 2, q > p − 1
We are mostly interested in nonnegative solutions of 1.1 having the form u x, t t−αf xt−β : t−αf r, 1.3 where α, β are positive numbers
It can be verified that u x, t is a self-similar solution to 1.1 if and only if u has the form 1.3
Summary
We here investigate the existence and uniqueness of the nontrivial, nonnegative solutions of a nonlinear ordinary differential equation: |f |p−2f βrf αf f q 0 satisfying a specific decay rate: limr → ∞rα/βf r 0 with α : p−1 / pq−2p 2 and β : q−p 1 / pq−2p 2.
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