Entire solutions for a reaction\u2013diffusion equation with doubly degenerate nonlinearity

Rui Yan,Xiaocui Li

doi: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 (x,t)in mathbb{R}^{2}. 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.

Highlights

In this paper, we consider the following scalar reaction–diffusion equation: ut = uxx + f (u), (1.1)where f satisfies (A) f ∈ C2([0, 2]), f (0) = f (1) = 0, f (0) = f (1) = 0, f (s) > 0, f (1 – s) < 0 for small s > 0, and f (u) > 0 for u ∈ (0, 1).From (A), it is easy to see that u = 0, u = 1 are two constant equilibria of (1.1)
From assumption (A), we mainly focus on the reaction–diffusion equation with doubly degenerate nonlinearity
An entire solution to (1.1) can be obtained by considering two traveling front solutions with critical speeds that come from both sides of the x-axis in Sect

Summary

Introduction

For Allen–Cahn equation ut = uxx + u(1 – u)(u – a), with a ∈ (0, 1), as a special example in [10], Fukao et al in [11] gave a proof for the existence of entire solutions by using the explicit expression of the traveling front and the comparison theorem. Wang in [15] investigated the entire solutions for the degenerate Fisher equation by considering two traveling front solutions with critical speeds.

Results

Conclusion