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A family of nonlinear fourth order equations of gradient flow type. (English) Zbl 1187.35131

The authors study the initial value problem for the following family of degenerate fourth order parabolic problems:
\[ u_{t}+\operatorname {div}\left( u\nabla \left( u^{\alpha-1}\Delta u^{\alpha}\right)\right) -\lambda \operatorname {div}(xu) =0\;\text{ in } \mathbb R^n\times \left[ 0,\infty \right),\quad u\left( x,0\right) =u_{0}(x)\;\text{ in }\mathbb R^n, \tag \(*\) \]
where \(n\geq 1\), \(u_{0}\geq 0\), \(u_{0}\in L^1 (\mathbb{R}^{n})\), \(\int_{\mathbb R^n}(1+|x|^2 )u_0(x)\, dx<\infty \) and \(\int_{\mathbb R^n}u_0(x)\log u_0(x)\, dx<\infty \). The parameters are subject to the conditions \(1/2\leq \alpha\leq1\) and \(\lambda \geq 0\). If \(\lambda=0\), the limit cases \(\alpha=1/2\) and \(\alpha=1\) are the Derrida-Lebowitz-Speer-Spohn and the thin film equation, respectively, where for pioneering existence results one may refer to P. M. Bleher, J. L. Lebowitz and E. R. Speer [Commun. Pure Appl. Math. 47, 923–942 (1994; Zbl 0806.35059)] and F. Bernis and A. Friedman [J. Differ. Equations 83, 179–206 (1990; Zbl 0702.35143)], respectively.
The present paper investigates systematically connections with related second order nonlinear Fokker-Planck-equations, which F. Otto [Commun. Partial Differ. Equations 26, 101–174 (2001; Zbl 0984.35089)] observed to be gradient flows of suitable entropy functionals with respect to an \(L^2\)-Wasserstein distance. Exploiting this approach, the authors construct global nonnegative weak solutions to (\(*\)) for which they precisely describe the asymptotic behaviour for \(t\to\infty\). If \(\lambda>0\), independently of \(u_0\), one has convergence to Barenblatt profiles \(x\mapsto (a-b|x|^2)^{1/(\alpha- 1/2)}_+\) with suitable constants \(a,b\) depending only on \(\alpha,\lambda\) and \(\int_{\mathbb R^n}u_0(x)\, dx\). The same limit profiles arise also in the Fokker-Planck-equations mentioned before. If \(\lambda=0\), one has convergence to \(0\) and the authors specify the asymptotic shape of a suitably rescaled solution \(u(t,\, .\, )\).

MSC:

35K65 Degenerate parabolic equations
35K30 Initial value problems for higher-order parabolic equations
35B36 Pattern formations in context of PDEs
49J45 Methods involving semicontinuity and convergence; relaxation
35B40 Asymptotic behavior of solutions to PDEs
35K59 Quasilinear parabolic equations
76A20 Thin fluid films
35Q84 Fokker-Planck equations
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