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Softcover ISBN:  9781470441128 
Product Code:  MEMO/264/1278 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
eBook ISBN:  9781470458065 
Product Code:  MEMO/264/1278.E 
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MAA Member Price:  $76.50 
AMS Member Price:  $51.00 
Softcover ISBN:  9781470441128 
eBook ISBN:  9781470458065 
Product Code:  MEMO/264/1278.B 
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MAA Member Price:  $153.00 $114.75 
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 264; 2020; 87 ppMSC: Primary 35
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 largetime behavior of the frontlike 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 frontlike 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.

Table of Contents

Chapters

1. Introduction

2. Main results

3. Phase plane analysis

4. Proofs of Propositions 2.8, 2.12

5. Preliminaries on the limit sets and zero number

6. Proofs of the main theorems


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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 largetime behavior of the frontlike 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 frontlike 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.

Chapters

1. Introduction

2. Main results

3. Phase plane analysis

4. Proofs of Propositions 2.8, 2.12

5. Preliminaries on the limit sets and zero number

6. Proofs of the main theorems