Title Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on Mathbb{R}

Published Providence : American Mathematical Society, 2020

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Description 1 online resource (100 p.) Series Memoirs of the American Mathematical Society Ser. ; v.264 Memoirs of the American Mathematical Society ; no. 1278. Contents Cover -- Title page -- Chapter 1. Introduction -- Chapter 2. Main results -- 2.1. Minimal systems of waves and propagating terraces -- 2.2. The case where 0 and are both stable -- 2.3. The case where one of the steady states 0, is unstable -- 2.4. The \om-limit set and quasiconvergence -- 2.5. Locally uniform convergence to a specific front and exponential convergence -- Chapter 3. Phase plane analysis -- 3.1. Basic properties of the trajectories -- 3.2. A more detailed description of the minimal system of waves -- 3.3. Some trajectories out of the minimal system of waves 6.7. Completion of the proofs of Theorems 2.7, 2.9, 2.17 -- 6.8. Completion of the proofs of Theorems 2.11 and 2.19 -- 6.9. Proof of Theorem 2.22 -- Bibliography -- Back Cover Summary The author considers semilinear parabolic equations of the form u_t=u_xx+f(u),\quad x\in \mathbb R,t>0, where f a Ĉ1 function. Assuming that 0 and \gamma >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 Bibliography Includes bibliographical references Notes Print version record Subject Reaction-diffusion equations. Differential equations, Parabolic. Differential equations, Partial. Ecuaciones diferenciales Ecuaciones de reacción-difusión Differential equations, Parabolic Differential equations, Partial Reaction-diffusion equations Partial differential equations -- Parabolic equations and systems [See also 35Bxx, 35Dxx, 35R30, 35R35, 58J35] -- Initial value problems for second-order parabolic equations. Partial differential equations -- Qualitative properties of solutions -- Asymptotic behavior of solutions. Partial differential equations -- Qualitative properties of solutions -- Stability. Partial differential equations -- Qualitative properties of solutions -- Oscillation, zeros of solutions, mean value theorems, etc.. Form Electronic book ISBN 9781470458065 1470458063

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