Coupled reaction-diffusion equations with degenerate diffusivity: wavefront analysis

Abstract We investigate traveling wave solutions for a nonlinear system of two coupled reaction-diffusion equations characterized by double degenerate diffusivity: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:mrow> <mml:mtable columnalign="left" displaystyle="true"> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi>n</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>−</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>,</mml:mo> <mml:mstyle scriptlevel="0"/> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:msub> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>g</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>h</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msub> <mml:mi>b</mml:mi> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>x</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> These systems mainly appear in modeling spatio-temporal patterns during bacterial growth. Central to our study is the diffusion term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>g</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mi>h</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , which degenerates at n = 0 and b = 0; and the reaction term <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> , which is positive, except for n = 0 or b = 0. Specifically, the existence of traveling wave solutions composed by a couple of strictly monotone functions for every wave speed in a closed half-line is proved, and some threshold speed estimates are given. Moreover, the regularity of the traveling wave solutions is discussed in connection with the wave speed.