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Zbl 1024.35051
Dal Passo, Roberta; Giacomelli, Lorenzo; Grün, Günther
A waiting time phenomenon for thin film equations.
(English)
[J] Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 30, No.2, 437-463 (2001). ISSN 0391-173X

The authors consider the fourth-order degenerate parabolic equation $$u_t + \nabla \cdot (|u|^n \nabla \Delta u)=0, \quad t>0, \ x \in \Omega, \tag 1$$ where $\Omega \subseteq \bbfR^N$, $N \in \{ 1,2,3\}$. Equation (1) is supplemented with suitable initial condition $u_0$ with compact support strictly contained in $\Omega$ and Neumann boundary conditions on $u$ and $\Delta u$ on $\partial \Omega$. Specifically, for $n \in (0,3)$ they ask the following question: Can one formulate conditions on the initial data $u_0$ so that the solution of (1), $u(t)$ exhibits the waiting time phenomenon, which in the case of $N=1$ means the following: Let $x_0 \in \partial \operatorname {supp} u_0$. Then $u(t)$ exhibits the waiting time phenomenon at $x_0$ if there is a time $T^*$ and a neighbourhood $B(x_0)$ of $x_0$ such that $\operatorname {supp} u(t) \cap B(x_0) = \operatorname {supp} u_0 \cap B(x_0)$; for the multidimensional version (employing an external cone condition) see section 5 of the paper.\par The main results of the paper are: For $0<n<2$ and $N=1$ or if $1/8<n<2$, $N=2,3$, the waiting time phenomenon occurs at a point $x_0 \in \partial\operatorname {supp} u_0$ if $u_0(x)$ grows at most as $|x-x_0|^{4/n}$ in a neighbourhood of $x_0$ (Theorems 4.1 and 5.1). For $2\leq n < 3$, $N=1$ the waiting time phenomenon occurs at $x_0$ if $u_0x$ grows ar most like $|x-x_0|^{4/n-1}$ (Theorem 6.1). Finally, in section 7 optimality of these exponents is considered and results of numerics suggesting such optimality are presented.\par The main tools used in the proofs are a version of the Gagliardo-Nirenberg inequalities (Theorem 2.3), entropy (for $n<2$) and weighted energy estimates (for $n \geq 2$), and a crucial extension of an iteration lemma due to Stampacchia (Lemma 3.1).
[Michael Grinfeld (Glasgow)]
MSC 2000:
*35K65 Parabolic equations of degenerate type
35K35 Higher order parabolic equations, boundary value problems
35R35 Free boundary problems for PDE
76A20 Thin fluid films
74K35 Thin films

Keywords: fourth-order degenerate parabolic equation; weighted energy estimates; Neumann boundary conditions; Gagliardo-Nirenberg inequalities; entropy

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