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I am dealing with the Fisher's Equation:

Consider the PDE: $u_t=u_{xx}+u(1-u)$, where $-\infty<x<\infty$ and $t>0$.
Prove that there exists $c^*>0$ such that for each $c>c^*$ there exists a traveling wave solution $u(x,t)=U(x+ct)$ which satisfies $\lim_{z\to-\infty}U(z)=0$ and $\lim_{z\to\infty}U(z)=1$.

I totally have no idea how to start this problem. In fact we did not so understand what he was talking about in class about this kind of non-linear PDE.

In addition to helping with this problem, does anyone have any materials about the nonlinear PDE in the form of $u_t=u_{xx}+u(1-u)$ and $u_t=u_{xx}+u(a-u)(1-u)$ with $a\in(0,1/2)$? Would you please share with me? Thanks in advance.

Amzoti
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Scorpio19891119
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  • You should start assuming that $u(x,t)=U(x+ct)$, plugging this into your equation and analyzing the ODE equation after this. – Artem Apr 22 '13 at 02:01

1 Answers1

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This result is classical. You can go to the following link to get what you want: www.math.colostate.edu/~pauld/M546/TWS3.pdf

Cameron Williams
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xpaul
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