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- W3113032120 abstract "<p style='text-indent:20px;'>This paper establishes the emergence of slowly moving transition layer solutions for the <inline-formula><tex-math id=M2>begin{document}$ p $end{document}</tex-math></inline-formula>-Laplacian (nonlinear) evolution equation, <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=FE1> begin{document}$ u_t = varepsilon^p(|u_x|^{p-2}u_x)_x - F'(u), qquad x in (a,b), ; t > 0, $end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id=M3>begin{document}$ varepsilon>0 $end{document}</tex-math></inline-formula> and <inline-formula><tex-math id=M4>begin{document}$ p>1 $end{document}</tex-math></inline-formula> are constants, driven by the action of a family of double-well potentials of the form <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id=FE2> begin{document}$ F(u) = frac{1}{2theta}|1-u^2|^theta, $end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>indexed by <inline-formula><tex-math id=M5>begin{document}$ theta>1 $end{document}</tex-math></inline-formula>, <inline-formula><tex-math id=M6>begin{document}$ thetain mathbb{R} $end{document}</tex-math></inline-formula> with minima at two pure phases <inline-formula><tex-math id=M7>begin{document}$ u = pm1 $end{document}</tex-math></inline-formula>. The equation is endowed with initial conditions and boundary conditions of Neumann type. It is shown that interface layers, or solutions which initially are equal to <inline-formula><tex-math id=M8>begin{document}$ pm 1 $end{document}</tex-math></inline-formula> except at a finite number of thin transitions of width <inline-formula><tex-math id=M9>begin{document}$ varepsilon $end{document}</tex-math></inline-formula>, persist for an exponentially long time in the critical case with <inline-formula><tex-math id=M10>begin{document}$ theta = p $end{document}</tex-math></inline-formula>, and for an algebraically long time in the supercritical (or degenerate) case with <inline-formula><tex-math id=M11>begin{document}$ theta>p $end{document}</tex-math></inline-formula>. For that purpose, energy bounds for a renormalized effective energy potential of Ginzburg–Landau type are established. In contrast, in the subcritical case with <inline-formula><tex-math id=M12>begin{document}$ theta<p $end{document}</tex-math></inline-formula>, the transition layer solutions are stationary." @default.
- W3113032120 created "2020-12-21" @default.
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- W3113032120 date "2021-01-01" @default.
- W3113032120 modified "2023-10-18" @default.
- W3113032120 title "Long time dynamics of solutions to $ p $-Laplacian diffusion problems with bistable reaction terms" @default.
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- W3113032120 doi "https://doi.org/10.3934/dcds.2020403" @default.
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