{"product_id":"propagating-terraces-and-the-dynamics-of-front-like-solutions-of-reaction-diffusion-equations-on-mathbb-r","title":"Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on $\\mathbb {R}$","description":"The author considers semilinear parabolic equations of the form $u_t=u_xx f(u),\\quad x\\in \\mathbb R,t\u0026gt;0,$ where $f$ a $C^1$ function. Assuming that $0$ and $\\gamma \u0026gt;0$ are constant steady states, the author investigates the large-time behavior of the front-like solutions, that is, solutions $u$ whose initial values $u(x,0)$ are near $\\gamma $ for $x\\approx -\\infty $ and near $0$ for $x\\approx \\infty $. If the steady states $0$ and $\\gamma $ are both stable, the main theorem shows that at large times, the graph of $u(\\cdot ,t)$ is arbitrarily close to a propagating terrace (a system of stacked traveling fonts). The author proves this result without requiring monotonicity of $u(\\cdot ,0)$ or the nondegeneracy of zeros of $f$. \u003cbr\u003e\u003cbr\u003eThe case when one or both of the steady states $0$, $\\gamma $ is unstable is considered as well. As a corollary to the author's theorems, he shows that all front-like solutions are quasiconvergent: their $\\omega $-limit sets with respect to the locally uniform convergence consist of steady states. In the author's proofs he employs phase plane analysis, intersection comparison (or, zero number) arguments, and a geometric method involving the spatial trajectories $\\{(u(x,t),u_x(x,t)):x\\in \\mathbb R\\}$, $t\u0026gt;0$, of the solutions in question.","brand":"American Mathematical Society","offers":[{"title":"Default Title","offer_id":56744396030296,"sku":"9781470441128","price":85.99,"currency_code":"EUR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0278\/1295\/4195\/files\/9781470441128.jpg?v=1771999273","url":"https:\/\/agendabookshop.com\/products\/propagating-terraces-and-the-dynamics-of-front-like-solutions-of-reaction-diffusion-equations-on-mathbb-r","provider":"Agenda Bookshop","version":"1.0","type":"link"}