Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms
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| Format: | Preprint |
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2020
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| _version_ | 1866910451288244224 |
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| author | Folino, Raffaele Plaza, Ramón G. Strani, Marta |
| author_facet | Folino, Raffaele Plaza, Ramón G. Strani, Marta |
| contents | This paper establishes the emergence of slowly moving transition layer solutions for the $p$-Laplacian (nonlinear) evolution equation, \[ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, \] where $\varepsilon>0$ and $p>1$ are constants, driven by the action of a family of double-well potentials of the form \[ F(u)=\frac{1}{2n} |1-u^2|^{n}, \] indexed by $n>1$, $n\in\mathbb{R}$ with minima at two pure phases $u = \pm 1$. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to $\pm 1$ except at a finite number of thin transitions of width $\varepsilon$, persist for an exponentially long time in the critical case with $n=p$, and for an algebraically long time in the supercritical (or degenerate) case with $n > p$. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg-Landau type are established. In contrast, in the subcritical case with $n<p$, the transition layer solutions are stationary. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2005_04784 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms Folino, Raffaele Plaza, Ramón G. Strani, Marta Analysis of PDEs 35K91, 35K57, 35B36, 35B40, 35K59 This paper establishes the emergence of slowly moving transition layer solutions for the $p$-Laplacian (nonlinear) evolution equation, \[ u_t = \varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), \qquad x \in (a,b), \; t > 0, \] where $\varepsilon>0$ and $p>1$ are constants, driven by the action of a family of double-well potentials of the form \[ F(u)=\frac{1}{2n} |1-u^2|^{n}, \] indexed by $n>1$, $n\in\mathbb{R}$ with minima at two pure phases $u = \pm 1$. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to $\pm 1$ except at a finite number of thin transitions of width $\varepsilon$, persist for an exponentially long time in the critical case with $n=p$, and for an algebraically long time in the supercritical (or degenerate) case with $n > p$. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg-Landau type are established. In contrast, in the subcritical case with $n<p$, the transition layer solutions are stationary. |
| title | Long time dynamics of solutions to $p$-Laplacian diffusion problems with bistable reaction terms |
| topic | Analysis of PDEs 35K91, 35K57, 35B36, 35B40, 35K59 |
| url | https://arxiv.org/abs/2005.04784 |