On Qualitative Behaviour of Solutions to a Thin Film Equation with a Source Term

Abstract

In this article, we study a one-dimensional degenerate fourth-order parabolic equation (a thin-film model) with a source term. We prove existence of generalized weak solutions for the case $n>0$ and study interface propagation properties like: finite speed propagation and waiting time phenomenon for the case $1<n<2$. Our analysis is based on applications of global and local energy-entropy a priori estimates. Also, we illustrate some of our main analytical results by numerical simulations.

Appendix

Lemma 2

(Nirenberg 1966) If $\varOmega \subset \mathbb {R}^N$ is a bounded domain with piecewise-smooth boundary, $a > 1$, $b \in (0, a),\ d > 1,$ and $0 \leqslant k < j,\ k,j \in \mathbb {N}$, then there exist positive constants $d_1$ and $d_2$ $(d_2 = 0 \text { if } \varOmega$ is unbounded) depending only on $\varOmega ,\ d,\ j,\ b,$ and N such that the following inequality is valid for every $v(x) \in W^{j,d} (\varOmega ) \cap L^b (\varOmega )$:

$$\begin{aligned} \left\| {D^k v} \right\| _{L^a (\varOmega )} \leqslant d_1 \left\| {D^j v} \right\| _{L^d (\varOmega )}^\theta \left\| v \right\| _{L^b (\varOmega )}^{1 - \theta } + d_2 \left\| v \right\| _{L^b (\varOmega )} ,\ \theta = \frac{{\tfrac{1}{b} + \tfrac{k}{N} - \tfrac{1}{a}}}{{\tfrac{1}{b} + \tfrac{j}{N} - \tfrac{1}{d}}} \in \left[ {\tfrac{k}{j},1} \right) . \end{aligned}$$

Note that if $\varOmega = B(0, R) {\setminus } B(0,r)$, where B(0, x) is ball with the radius x and the origin at 0, then $d_2 = c (R - r)^{ - \frac{(a - b)N}{a b} -k}$.

Lemma 3

(Stampacchia 1963, Lemme 4.1, p. 19) Let f(x) be nonnegative, non-increasing in $[x_0,+\infty )$ function. Assume that f satisfies

$$\begin{aligned} f(y) \leqslant \frac{C}{(y -x)^{\alpha }} f^{\beta }(x) \text { for } y > x \geqslant x_0, \end{aligned}$$

(68)

where $C,\,\alpha ,\, \beta$ are some positive constants. Then

• (i) if $\beta > 1$ we have

  $$\begin{aligned} f(y) = 0 \text { for all } y \geqslant x_0 + d, \end{aligned}$$

  where $d^{\alpha } = C f^{\beta -1}(x_0) 2^{\frac{\alpha \beta }{\beta -1}}$;

• (ii) if $\beta = 1$ we get

  $$\begin{aligned} f(y) \leqslant e^{1- \zeta (y -x_0)}f(x_0) \text { for all } y \geqslant x_0, \end{aligned}$$

  where $\zeta = (e\,C)^{- \frac{1}{\alpha }}$;

• (iii) if $\beta < 1$ we obtain

  $$\begin{aligned} f(y) \leqslant 2^{\frac{\mu }{1-\beta }} \bigl [C^{\frac{1}{1-\beta }} + (2\, x_0)^{\mu } f(x_0) \bigr ] y^{-\mu } \text { for all } y \geqslant x_0 > 0, \end{aligned}$$

  where $\mu = \frac{\alpha }{1- \beta }$.

Lemma 4

(Dal Passo et al. 2001, Lemma 3.1, p. 444; Shishkov and Shchelkov 1998, Lemma 4, p.624) Assume that a given nonnegative, non-decreasing function $f : ( d,D) \rightarrow \mathbb {R}^1$ satisfies

$$\begin{aligned} f(y) \leqslant \frac{C}{(x -y)^{\alpha }} (f(x) + (x -d)^{\sigma } ) ^{\beta } \text { for } 0 \leqslant d \leqslant y < x \leqslant D , \end{aligned}$$

(69)

where $C,\,\alpha ,\, \beta , \, \sigma$ are some positive constants such that

$$\begin{aligned} \beta > 1 \text { and } \sigma \geqslant \tfrac{\alpha }{\beta -1}. \end{aligned}$$

Assume further that

$$\begin{aligned} d^{\alpha } \geqslant C \, 2^{\frac{\alpha \beta }{\beta -1}} ( 1 + 2^{\frac{\alpha }{\beta -1} - \sigma } )^{\beta } ( f(D) + (D-d)^{\sigma })^{\beta -1} . \end{aligned}$$

(70)

Then,

$$\begin{aligned} f(d) = 0 . \end{aligned}$$