1
$\begingroup$

Prove the Lyapunov stability in $L^2([0, \ell])$ of the diffusion problem governed by $ \frac{\partial u}{\partial t} − a^2\frac{\partial^2 u}{\partial x^2}= f(x, t), 0 \leq x \leq \ell, t \geq 0 $ subject to Neumann boundary conditions $u_x(0, t) = 0, u_x(\ell, t) = 0$, with respect to variations in the initial condition, $u(x, 0) = \varphi(x)$.

I'm unsure about how to start this problem. I know i need to find a $C>0$ such that it satisfies $$\int_0^\ell [u_1(x, t) − u_2(x, t)]^2{\rm d}x ≤ C\int_0^\ell[\varphi_1(x) − \varphi_2(x)]^2{\rm d}x$$

but I am unsure how to do this.

  • 0
    If $u_1$ and $u_2$ are solutions corresponding to initial conditions $\varphi_1$ and $\varphi_2$ then $v = u_1 - u_2$ satisfy the diffusion (heat) equation $v_t - a^2v_{xx} = 0$ with $v_x(0,t) = v_x(\ell,t) = 0$ and $v(x,0) = \varphi_1(x) - \varphi_2(x)$. Apply the maximum principle. – 2017-02-24
  • 0
    Sorry but I'm not familiar with the maximum principle, I've only been studying PDE's for a short time – 2017-02-24
  • 0
    It's a very basic principle for PDEs (look for it in your book). The [maximum principle](https://www.google.co.uk/search?q=maximum+principle+heat+equation) says that if $u$ satisfy the heat equation in some domain $\Omega$ and $x_*$ is a maximum point of $u$ then $x_*$ has to lie on the boundary of $\Omega$. For your domain the relevant boundary here is where you set your initial condition and boundary conditions which gives us $\max_{x\in [0,1]} |u_1(x,t) - u_2(x,t)| = \max_{x\in[0,1]} |\varphi_1(x) - \varphi_2(x)|$. – 2017-02-24
  • 1
    A simpler way: $v_t - a^2 v_{xx} = 0 \implies v_t v = a^2v_{xx}v$. Now integrate this equation over $x\in [0,\ell]$ using integration by parts on the right hand side and show that $\int_0^\ell v^2(x,t){\rm d}x$ is maximized at $t=0$ (consider the sign of the RHS). If you can show this then you get $\int_0^\ell [u_1(x,t) - u_2(x,t)]^2{\rm d}x \leq \int_0^\ell[\varphi_1(x) - \varphi_2(x)]^2{\rm d}x$. – 2017-02-24
  • 0
    I'm confused how you get to $v_t−a^2v_{xx}=0$?. How have you set $v(x,t)$ so that the transformed PDE =0? Also to show the last equation I've learnt that you need a constant C as this affects your answer, although I am unsure how? – 2017-02-25

0 Answers 0