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

The nonlinear diffusion equation ut = (u −4/3 ux)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 Lamé equation, and that the general solution of this linear equation is expressed in terms of the Weierstrass ℘-function and its derivative. As a result, all exact solutions v(x, t) on a fivedimensional 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. MSC: 35C05, 35B06, 35B44.

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