Partial Differential Equation (PDE)

Camillo
Let $u(x,t) $ be the solution of the problem :

$ u_t(x,t) -u_[x x] (x,t) =0 $ for $00 $
$u(x,0) = sin pix $ for $0<=x<=1 $
$u(0,t)=2te^(1-t) ; u(1,t) = 1-cos pit $ for $ t> 0 $

$u(x,t)$ is continuous in the half strip $S =[0,1]xx[0,oo) $.

a) Prove that $ u(x,t) $ is not negative.
b) Find an upper limit for the values $u(1/2,1/8)$ and $ u(1/2,3) $ .

Risposte
Camillo
Your try is correct, of course :D

gugo82
A solution was missing, so here's my try.

"Camillo":
Let $u(x,t) $ be the solution of the problem :

$ \{ (u_t(x,t) -u_[x x] (x,t) =0, " for " 00 ),( u(x,0) = sin pix , " for " 0<=x<=1 ),( u(0,t)=2te^(1-t) " and " u(1,t) = 1-cos pit , " for " t> 0 ):}$

$u(x,t)$ is continuous in the half strip $S =[0,1]xx[0,oo) $.

The PDE is of parabolic type (in fact it's the classical heat equation); the initial and boundary data, i.e. the functions $f(x):=sinpix $ and $g_1(t):=2t"e"^(1-t),\ g_2(t):=1-cos pi t$, are non negative respectively in $[0,1]$ and $[0,+oo[$.

According to standard nomenclature, I will call $I_T:=]0,1[\times ]0,T]$ parabolic cylinder; $\bar(I_T):=[0,1]\times[0,T]$ closed parabolic cylider, $Gamma_T:=\bar(I_T) \setminus I_T$ parabolic boundary at time $T>0$.

"Camillo":
a) Prove that $ u(x,t) $ is not negative.

a) For any fixed $T>0$, we can say that a solution $u$ of the Cauchy problem is $>=0$ in the closed parabolic cylinder $bar(I_T)$: in fact, the minimum principle for parabolic PDE states that $min_(\bar(I_T)) u =min_(Gamma_T) u$ (the minimum of $u$ in the closed parabolic cylinder coincides with the minimum in the parabolic boundary $Gamma_T$); we actually have $min_(Gamma_T) u =min \{min f , min g_1, min g_2\} =0$, so $u>=0$ in $\bar(I_T)=I_T\cup Gamma_T$.
Since $T>0$ was arbitrarily chosen, we have $u>=0$ in the whole of $S$.

"Camillo":
b) Find an upper bound for the values $u(1/2,1/8)$ and $ u(1/2,3) $.

b) By maximum principle, we have:

$u(1/2,1/8)<=max_(Gamma_(1/8)) u$
$\quad \quad =max \{ max_([0,1]) f, max_([0,1/8]) g_1, max_([0,1/8]) g_2\}$
$\quad \quad =max \{ 1, 1/4"e"^(7/8), 1-cos(pi/8)\}$
$\quad \quad =1$

and also:

$u(1/2, 3)<=max_(Gamma_3) u$
$\quad \quad =max \{ max_([0,1]) f, max_([0,3]) g_1, max_([0,3]) g_2\}$
$\quad \quad =max \{ 1, 2, 2\}$
$\quad \quad =2$.

Note that $u<=2$ in the whole of $S$.

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