Exact solutions of a nonlinear diffusion equation on polynomial invariant subspace of maximal dimension

TL;DR

This paper derives explicit exact solutions for a nonlinear diffusion equation by reducing it to a polynomial invariant subspace, revealing solutions expressed via Weierstrass functions and analyzing blow-up and decay behaviors.

Contribution

It constructs explicit solutions on a maximal polynomial invariant subspace for a nonlinear diffusion equation using reduction to a Lamé equation and Weierstrass functions.

Findings

Solutions expressed via Weierstrass $ ext{wp}$-function.

Identification of blow-up and finite-time fading solutions.

Explicit construction of solutions on invariant subspace.

Abstract

The nonlinear diffusion equation $u_t = (u^{- 4/3} u_x)_x$ is reduced by the substitution $u = v^{- 3/4}$ to an equation with quadratic nonlinearities possessing a polynomial invariant linear subspace of the maximal possible dimension equal to five. The dynamics of the solutions on this subspace is described by a fifth-order nonlinear dynamical system (V.A. Galaktionov). We found that, on differentiation, this system reduces to a single linear equation of the second order, which is a special case of the Lam\'e equation, and that the general solution of this linear equation is expressed in terms of the Weierstrass $\wp$-function and its derivative. As a result, all exact solutions $v(x,t)$ on a five-dimensional polynomial invariant subspace, as well as the corresponding solutions $u(x,t)$ of the original equation, are constructed explicitly. Using invariance condition, two families of non-invariant solutions are singled out. For one of these families, all types of solutions are considered in detail. Some of them describe peculiar blow-up regimes, while others fade out in finite time.