| Summary: | The main goal in this article is to justify that source-type and
other global-in-time similarity solutions of the Cauchy problem
for the fourth-order thin film equation
$$
u_t=-\nabla \cdot (|u|^n \nabla \Delta u) \quad
\text{in }\mathbb{R}^N \times \mathbb{R}_ + \text{where }n>0,\;
N \ge 1
$$
can be obtained by a continuous deformation (a homotopy path) as
$n \to 0^+$. This is done by reducing to similarity solutions
(given by eigenfunctions of a rescaled linear operator $\mathbf{B}$)
of the classic bi-harmonic equation
$$
u_t = - \Delta^2 u \quad\text{in }\mathbb{R}^N \times \mathbb{R}_ +,
\text{ where }
\mathbf{B}=-\Delta^2 +\frac 14 y \cdot \nabla+ \frac N4 I.
$$
This approach leads to a countable family of various global similarity patterns
of the thin film equation, and describes their oscillatory sign-changing
behav iour by using the known asymptotic properties of the fundamental
solution of bi-harmonic equation.
The branching from $n=0^+$ for thin film equation requires Hermitian spectral
theory for a pair $\{\mathbf{B}, \mathbf{B}^*\}$ of non-self adjoint operators
and leads to a number of difficult mathematical problems. These include, as
a key part, the problem of multiplicity of solutions, which is under
particular scrutiny.
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