Diffusion Equation and Maximum Principle

[Tsunoyukami]

"Consider a solution of the diffusion equation $u_{t} = u_{xx}$ in {0 ≤ x ≤ L, 0 ≤ t ≤ ∞}.
a) Let M(T) = the maximum of u(t,x) in the closed rectangle {0 ≤ x ≤ L, 0 ≤ t ≤ T}. Does M(T) increase or decrease as a function of T?
b) Let m(T) = the minimum of u(t,x) in the closed rectangle {0 ≤ x ≤ L, 0 ≤ t ≤ T}. Does m(T) increase or decrease as a function of T?" (Partial Differential Equations An Introduction, 2nd Edition by Walter Strauss)


I've been thinking about this problem over the past week but I'm not convinced by the solutions I've been able to find online - I just don't understand them properly. First I will outline my thinking and then I will provide a link to one solution I found online (there are many others, though they are similar).


One of my first thoughts was that the minimum of u(x,t) should be the maximum of -u(x,t); therefore, if we have solved part a we have, in essence, solved part b as well.

By the maximum principle we know that the maximum occurs on one of the sides x = 0, x = L or t = 0.

Consider the rectangular regions $R_{1}$ = {0 ≤ x ≤ L, 0 ≤ t ≤ $T_{1}$} and $R_{2}$ = {0 ≤ x ≤ L, 0 ≤ t ≤ $T_{2}$} with $T_{1} < T_{2}$ and maxima $M(T_{1}) and M(T_{2})$, respectively. Clearly $R_{1}$ is contained in $R_{2}$. Next I need to show some relationship between $M(T_{1}) and M(T_{2})$ that will allow me to say that M(T) is either increasing or decreasing (or neither).

Initially my intuition led me to believe that if I hit play on my mental simulation of this that the heat flow would smooth out as a function of time and therefore the maximum would decreasing as T grew larger but all the solutions online suggest that M(T) is an increasing function.

I'm not too sure how to proceed from here. Any hints would be appreciated, as for an explanation as to how my intuition is misguided. Thanks a lot in advance!

EDIT: I forgot to add the link to a solution but am unsure whether or not that is permitted by the rules - in essence they all assert that $M(T_{2})$ is greater than $M(T_{1})$ without justification...I don't see why...

 

[vanhees71]

What are the assumed initial and boundary conditions? If none are given, the question doesn't make much sense to me.

 

[pasmith]

"Consider a solution of the diffusion equation $u_{t} = u_{xx}$ in {0 ≤ x ≤ L, 0 ≤ t ≤ ∞}.
a) Let M(T) = the maximum of u(t,x) in the closed rectangle {0 ≤ x ≤ L, 0 ≤ t ≤ T}. Does M(T) increase or decrease as a function of T?
b) Let m(T) = the minimum of u(t,x) in the closed rectangle {0 ≤ x ≤ L, 0 ≤ t ≤ T}. Does m(T) increase or decrease as a function of T?" (Partial Differential Equations An Introduction, 2nd Edition by Walter Strauss)


I've been thinking about this problem over the past week but I'm not convinced by the solutions I've been able to find online - I just don't understand them properly. First I will outline my thinking and then I will provide a link to one solution I found online (there are many others, though they are similar).


One of my first thoughts was that the minimum of u(x,t) should be the maximum of -u(x,t); therefore, if we have solved part a we have, in essence, solved part b as well.

By the maximum principle we know that the maximum occurs on one of the sides x = 0, x = L or t = 0.

Consider the rectangular regions $R_{1}$ = {0 ≤ x ≤ L, 0 ≤ t ≤ $T_{1}$} and $R_{2}$ = {0 ≤ x ≤ L, 0 ≤ t ≤ $T_{2}$} with $T_{1} < T_{2}$ and maxima $M(T_{1}) and M(T_{2})$, respectively. Clearly $R_{1}$ is contained in $R_{2}$. Next I need to show some relationship between $M(T_{1}) and M(T_{2})$ that will allow me to say that M(T) is either increasing or decreasing (or neither).

Initially my intuition led me to believe that if I hit play on my mental simulation of this that the heat flow would smooth out as a function of time and therefore the maximum would decreasing as T grew larger but all the solutions online suggest that M(T) is an increasing function.

I'm not too sure how to proceed from here. Any hints would be appreciated, as for an explanation as to how my intuition is misguided. Thanks a lot in advance!

EDIT: I forgot to add the link to a solution but am unsure whether or not that is permitted by the rules - in essence they all assert that $M(T_{2})$ is greater than $M(T_{1})$ without justification...I don't see why...

The maximum of a set is greater than or equal to the maximum of any subset of that set.

Since $R_1 \subset R_2$, it follows that $M(T_1) \leq M(T_2)$ and $M(T)$ is increasing.

 

[Tsunoyukami]

@vanhees71: no initial or boundary conditions are specified
@pasmith: I think I've got it.

Suppose we have a subset of a set (ie. $R_{1}$ is a subset of $R_{2}$). Then because $R_{2}$ contains $R_{1}$ it's maximum is at least the maximum of $R_{1}$, $M(T_{1})$ but it is possible for it to be larger than this.

What I mean is something like this. Suppose we have a set $U_{1} = [1,2,3]$ where $max (U) = max [1,2,3] = 3$. Then suppose we have another set $V$ such that $U$ is a subset of $V$. $V$ might contain no elements larger than 3 (in which case $max V = max U$) but it might contain others that are larger (in which case $max V > max U$.

Thanks! I think I was too busy thinking about turning time on and watching the heat flow outward in time as opposed to just looking at a larger rectangle to begin with.