## 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)

### MSC:

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