Propagating Terraces and the Dynamics of Front-Like Solutions of Reaction-Diffusion Equations on

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 -- Chapter 4. Proofs of Propositions 2.8, 2.12 -- Chapter 5. Preliminaries on the limit sets and zero number -- 5.1. Properties of Ω( ) -- 5.2. Zero number -- Chapter 6. Proofs of the main theorems -- 6.1. Some estimates: behavior at =±∞ and propagation -- 6.2. A key lemma: no intersection of spatial trajectories -- 6.3. The spatial trajectories of the functions in \Om( ) -- 6.4. \Om( ) contains the minimal propagating terrace -- 6.5. Ruling out other points from _{\Om}( ) -- 6.6. Completion of the proofs of Theorems 2.5, 2.13, and 2.15 -- 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.

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 C^1 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 proves this result without requiring monotonicity of u(\cdot ,0) or the nondegeneracy of zeros of f. The 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>0, of the solutions in question.

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