Traveling waves in a convolution model for phase transitions. (English) Zbl 0889.45012

The authors consider the following evolution problem for functions \(u(x,t)\) defined on \(\mathbb{R}\times \mathbb{R}^+\): \[ u_t= J* u-u- f(u),\tag{1} \] where the kernel \(J\) of the convolution \[ J* u(x)= \int^{+\infty}_{- \infty} J(x- y)u(y)dy \] is nonnegative, even, with unit integral, and the function \(f\) is bistable. The paper uses the following assumption:
(H\(_1\)) \(J\in C^1(\mathbb{R})\), \(J(s)= J(-s)\geq 0\) for all \(s\), \(\int_{\mathbb{R}} J=1\), \(\int_{\mathbb{R}} J(y)|y|dy<+ \infty\), \(J'\in L^1(\mathbb{R})\),
(H\(_2\)) \(f\in C^2(\mathbb{R})\), \(f(\pm 1)= 0< f'(\pm 1)\), \(f\) has only one zero \(\alpha\in (-1,1)\) and nonzeros outside \([- 1,1]\).
The authors seek solutions to (1) of the form \(u(x, t)=\widehat u(x- ct)\) for some velocity \(c\), with \(u\) having limits \(\pm 1\) at \(\pm\infty\); thus, making the change of variables \(\xi= x-ct\), they seek a function \(\widehat u(\xi)\) and a constant \(c\) satisfying \[ J*\widehat u-\widehat u+ c\widehat u'- f(\widehat u)= 0\quad\text{on }\mathbb{R},\tag{2} \]
\[ \widehat u(\pm\infty)= \pm1.\tag{3} \] Then, they define \(g(u)= u+ f(u)\) and suppose:
(H\(_3\)) \(g'>0\) on \([-1,\beta)\cup (\gamma,1]\), \(g'<0\) on \((\beta,\gamma)\), \(\beta\leq\gamma\).
Under (H\(_1\)), (H\(_2\)) there exists a solution \((u, c)\) to \[ \int_{\mathbb{R}} [J* u-u- f(u)]\cdot \phi- c\int_{\mathbb{R}} u\cdot \phi'= 0,\quad \phi\in C^\infty_0(\mathbb{R}), \] satisfying (3); under (H\(_1\))–(H\(_3\)) and if \(u\) is monotone, then \(c\neq 0\) implies \(u\) is smooth, it satisfies (2) and \[ c= \Biggl(\int^1_{- 1}f\Biggr) \Biggl(\int^{+ \infty}_{- \infty}(u')^2\Biggr)^{-1}. \] The authors establish the uniqueness, stability and regularity properties of taveling-wave solutions of (1).
Reviewer: D.M.Bors (Iaşi)


45K05 Integro-partial differential equations
45G10 Other nonlinear integral equations
82C26 Dynamic and nonequilibrium phase transitions (general) in statistical mechanics
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