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Existence and global stability of traveling curved fronts in the Allen-Cahn equations. (English) Zbl 1159.35378

Summary: This paper is concerned with existence and stability of traveling curved fronts for the Allen-Cahn equation in the two-dimensional space \[ D_tu(t,x,y)=\Delta u(t,x,y)+f(u(t,x,y)),\quad t>0,\;(x,y)\in\mathbb R^2.\tag{1} \] where the nonlinear term \(f\) satisfies the following conditions: (i) \(f(1)=f(-1)=0\), \(f\) is positive in \((-\infty,-1)\) and negative in \((1,+\infty)\), (ii) \(\int_{-1}^1 f\,ds>0\), (iii) \(f'\) is negative in \((-\infty,-1]\cup [1,+\infty)\).
By using the supersolution and the subsolution, we construct a traveling curved front, and show that it is the unique traveling wave solution between them. Our supersolution can be taken arbitrarily large, which implies some global asymptotic stability for the traveling curved front.

MSC:

35K55 Nonlinear parabolic equations
35B35 Stability in context of PDEs
35K15 Initial value problems for second-order parabolic equations
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