## Advances in Differential Equations

- Adv. Differential Equations
- Volume 10, Number 6 (2005), 635-674.

### On evolution completeness of nonlinear eigenfunctions for the porous medium equation in the whole space

V. A. Galaktionov and P. J. Harwin

#### Abstract

We consider the porous medium equation (PME) $$ u_t = \Delta (|u|^{m-1}u) \quad \mbox{in
${{\mathbb R}^N} \times {{\mathbb R}^N}_+$}, \quad m>1, $$ with continuous,
compactly-supported initial data $\hat{u}$. The PME admits various similarity solutions of
the form $$ u_k(x,t)= t^{-{\alpha}_k} \psi_k(x/t^{{\beta}_k}), \quad k=0,1,..., $$ where
each $\psi_k \in C_0({{\mathbb R}^N})$ satisfies a quasilinear elliptic equation in
${{\mathbb R}^N}$ and the exponents $\{{\alpha}_k,{\beta}_k\}$ are determined from the
solubility of such a nonlinear eigenvalue problem. The nonlinear eigenfunction subset
$\Phi=\{\psi_k\}$ consisting of a countable number of continuous families is rather
complicated and is known in one dimension and in the radial setting in ${{\mathbb R}^N}$
(in ${{\mathbb R}^N},$ for $N \ge 2$, only the first two eigenfunctions from $\Phi$ are
known). We show that the eigenfunction subset $\Phi$ can be *evolutionary complete*,
i.e., describes the asymptotics of arbitrary global $C_0$-solutions of the PME. We prove
such a completeness in one dimension and in the radial ${{\mathbb R}^N}$ geometry. The
analysis uses Sturm's theorem on zero sets for parabolic equations, scaling techniques,
and the theory of gradient dynamical systems. For $m=1$, i.e., for the linear heat
equation, the evolution completeness is directly associated with the completeness and
closure of the eigenfunction subset for the linear self-adjoint operator $\Delta + \frac
12 y \cdot \nabla + \frac N2 I$ in a weighted $L^2$-space. These linear eigenfunctions are
used as branching points of nonlinear ones for $m \approx 1^+$.

#### Article information

**Source**

Adv. Differential Equations Volume 10, Number 6 (2005), 635-674.

**Dates**

First available in Project Euclid: 18 December 2012

**Permanent link to this document**

https://projecteuclid.org/euclid.ade/1355867838

**Mathematical Reviews number (MathSciNet)**

MR2133648

**Zentralblatt MATH identifier**

1284.35217

**Subjects**

Primary: 35K55: Nonlinear parabolic equations

Secondary: 35K15: Initial value problems for second-order parabolic equations 35P30: Nonlinear eigenvalue problems, nonlinear spectral theory 37L99: None of the above, but in this section 76S05: Flows in porous media; filtration; seepage

#### Citation

Galaktionov, V. A.; Harwin, P. J. On evolution completeness of nonlinear eigenfunctions for the porous medium equation in the whole space. Adv. Differential Equations 10 (2005), no. 6, 635--674. https://projecteuclid.org/euclid.ade/1355867838