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Tsunoyukami

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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}##.

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

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...

*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?"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...

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