Research Article: Entire solutions for a reaction–diffusion equation with doubly degenerate nonlinearity

Date Published: April 24, 2018

Publisher: Springer International Publishing

Author(s): Rui Yan, Xiaocui Li.

http://doi.org/10.1186/s13662-018-1606-y

Abstract

This paper is concerned with the existence of entire solutions for a reaction–diffusion equation with doubly degenerate nonlinearity. Here the entire solutions are the classical solutions that exist for all documentclass[12pt]{minimal}
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begin{document}$(x,t)in mathbb{R}^{2}$end{document}(x,t)∈R2. With the aid of the comparison theorem and the sup-sub solutions method, we construct some entire solutions that behave as two opposite traveling front solutions with critical speeds moving towards each other from both sides of x-axis and then annihilating. In addition, we apply the existence theorem to a specially doubly degenerate case.

Partial Text

In this paper, we consider the following scalar reaction–diffusion equation:
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begin{document}$$ u_{t}=u_{xx}+f(u), $$end{document}ut=uxx+f(u), where f satisfies documentclass[12pt]{minimal}
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begin{document}$(A)$end{document}(A)documentclass[12pt]{minimal}
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begin{document}$fin C^{2} ([0,2] ), f(0)=f(1)=0, f'(0)=f'(1)=0$end{document}f∈C2([0,2]),f(0)=f(1)=0,f′(0)=f′(1)=0, documentclass[12pt]{minimal}
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begin{document}$f'(s)>0, f'(1-s)<0$end{document}f′(s)>0,f′(1−s)<0 for small documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$s>0$end{document}s>0, and documentclass[12pt]{minimal}
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begin{document}$f(u)>0$end{document}f(u)>0 for documentclass[12pt]{minimal}
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begin{document}$uin(0,1)$end{document}u∈(0,1).

In this section, we will give some relevant preparations in order to obtain the main conclusions later. First of all, we state the definitions of supersolution and subsolution of (1.1) as follows.

In this section, we discuss the existence of entire solutions of (1.1). Firstly, we construct the supersolution.

For the double degenerated generalized Fisher-type equation,
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begin{document} $$begin{aligned} u_{t}=u_{xx}+u^{p}(1-u)^{q},quad p>1,q>1, end{aligned}$$ end{document}ut=uxx+up(1−u)q,p>1,q>1, where p and q are not necessarily integers. Now we show the conclusion of the existence of traveling wave front solutions of (4.1).

 

Source:

http://doi.org/10.1186/s13662-018-1606-y

 

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