Could you though take a look at http://math.stackexchange.com/questions/1205836/how-could-we-find-the-energy/1206110?noredirect=1#comment2454114_1206110 . I have an answer about how to apply the energy metho but I don't know how to conclude that $u=0$. Do you maybe have an idea??


Here is an other example of the energy method: http://math.stackexchange.com/questions/1199161/energy-method-to-show-uniqueness-of-solution-of-pde

@CarlMummert

The notions “upper bound”, ”lower bound”, “least”, and “greatest” make sense
whenever we have an order, not just for numbers. Consider a fixed set X and its
subsets. We define an order by saying that A is less than B if A ⊂ B. a) every Y subset P(x) is bounded above and bounded below.

Hello @quid @robjohn @ThomasKlimpel !!

I am looking at this http://math.stackexchange.com/questions/1205836/how-could-we-find-the-energy . I have an answer about how to apply the method. Now I got stuck how to show that $\partial_{t} E(t) \leq 0$.
($\partial_{t} E(t)$ is defined in the answer of the post)

Do you maybe have an idea??

Here is an other example of the energy method: http://math.stackexchange.com/questions/1199161/energy-method-to-show-uniqueness-of-solution-of-pde

At the anwer there is the energy given by the formula $$E(t) = \frac{1}{2} \int_{a}^{b} u_{t}^{2} \, dx$$

Then the derivative is:
$$\partial_{t} E(t) = \frac{1}{2} \left[ u_{t}^{2}(b,t) - u_{t}^{2}(a,t) \right]$$

we want to show that $\partial_{t} E(t) \leq 0$ so that we can conclude that $u=0$. But how could we do that?? @robjohn

@robjohn http://i.stack.imgur.com/MaKfP.jpg

on 2c I need to rewrite the model in terms of the rescaled version
so it already gives me the subsitutions
X_t = B_t/k, b = ac, and Y_t = E_k/ka
so I need (x_t)(k) = b_t, and Y_t(ak)= E_t and put it inside these equations B_{t+1}=e^{-cE_t/B_t} and $E_{t+1} = aB_t

right?

So my new equations are

Y_{t+1} = X_t and X_{t+1} = 1/K e^{-bY_t/X_t}