$L^\infty$ stability property of an ODE

Suppose we have the initial-value problem on $$(0,L)$$:

$$\frac{d u(x)}{d x} = f(x) u(x),\, \qquad x\in\Omega,\,~~ u(0) = u_0,$$

I am reading a claim that says if we multiply the ODE by $$u$$ and integrate over $$(0,L)$$, we have

$$\frac{1}{2}u^2(L) - \frac{1}{2} u^2_0 = \int_{0}^{L} f(x)u^2(x) \,dx$$

"from which the $$L^\infty$$-stability of the solution follows." I agree that the equation is correct, but:

1. Why does this guarantee stability?
2. What exactly is meant by $$L^\infty$$-stability? I interpret stability in the context that the numerical solution will remain bounded as the the step size is reduced, but here, we do not have a discretization yet...

This discussion is given in the context of discontinuous Galerkin methods.

• Can you provide the reference? Jun 8 '19 at 16:33
• The $L^\infty$ norm of a function is its maximum absolute pointwise value, as in $\|f\|_\infty = \max_{x\in\Omega} |f(x)|$. Stability here means that if $\|f\|_\infty < \infty$, then $\|u\|_\infty < \infty$. The reasoning is a bit circular to me, because you can't just multiply pointwise by $u$ if $\|u\|_\infty = \infty$, but I can't immediately think of a counterexample. Jun 8 '19 at 16:49
• A reference is: ima.umn.edu/sites/default/files/1921.pdf and the specific statement is on page 2. I also don't understand the argument, but it is important to how the author formulates stability of the DG methods within the paper.
– cm2
Apr 7 at 21:11

The $$L^\infty$$-stabilité would be the stability in the sense that the $$L^\infty$$ norm of the state $$u$$ of the system is smaller than the $$L^\infty$$ norm of the initial data $$u_0$$ : $$\lVert u \rVert_{\infty} < \lVert u_0 \rVert_{\infty}.$$ With $$\lVert u \rVert_{\infty} = \sup_{t>0}|u(t)|$$. To obtain this property, we need to make the assumption $$f(T)<0$$ for all $$T>0$$. In fact, if $$f<0$$, $$\frac{1}{2}u^2(T)-\frac{1}{2}u^2_0 \leq 0 \quad \forall T>0$$ This can be interpreted as the fact that the energy of the system decreases with time. Then we deduce that $$|u(T)|\leq|u_0|, \, \forall T>0$$, which allows us to conclude.
Furthermore, the author defines here 1997 C.I.M.E. Lecture Notes Proposition 3.1, the $$L^2$$ (discrete) stability, which seems to confirm that this is what it is all about.