Solving partial differential equations according to the molding method (semigroup theory). Which form (a, j) does the operator A have?

We have the equations

$$ rho (x) frac { partial ^ 2u} { partial ^ 2} (t, x) = div (A (x) degrees u + B (x) grad frac { partial u} { partial t}) (t, x) -div ( gamma (x) v (t, x)) $$

$$ beta (x) frac { partial v} { partial t} (t, x) = div (C (x) degree v) (t, x) – gamma (x) .grad frac { partial u} { partial t} (t, x) $$

This leads to an operator

$$ A = begin {pmatrix} 0 &, I &, 0 \ frac {1} { rho (x)} div A (x) grad &, frac {1} { rho (x)} div B (x) grad &, frac {-1} { rho (x)} div gamma (x) \ 0 &, frac {-1} { beta (x)} gamma (x) grad &, frac {1} { beta (x)} div C (x) grad end {pmatrix} $$

Now A continues to work $ (u_1, u_2, u_3) ^ T $How do I find the associated form of A?
Help Thanks

Analysis of pdes – A question of Schrödinginger Semigroup – By B. Simon

The question comes from the newspaper: B. Simon, Schrödinger Semigroups, Bull A.M., (1982) Vol. 7 (3).

Theorem C.1.2 (underestimation estimate) of the paper states: If $ Hu = 0 $, from where $ H = – Delta + V $ for a limited continuous function $ V $, Then
$$ | u (x) | leq C int_ {B_r (x)} | u (y) | dy, $$
from where $ C $ depends on $ r $ and norm of $ V $ and $ B_r (x) $ takes place within the domain of $ u $,

On page 499 it writes: If $ e ^ {a | x |} u in L ^ 2 $ for all $ | a | <M $then from the Subsolution estimate $ e ^ {a | x |} u in L ^ infty $,

Q How can the result be deduced? $ e ^ {a | x |} u in L ^ 2 $ for all $ | a | <M $, then $ u in L ^ infty $,

PS: What I do not understand is that $ e ^ {a | x |} u $ Does not the equation satisfy, is there a result to solve this problem?

Monoids – Can the nilpotent relationships in the semigroup $ ( wp (X ^ 2), circ) $ be generated by elements of $ 3 $?

In every finite sentence $ X $ the set of relationships $ wp (X times X) = {R: R subseteq X times X } $ forms a semigroup in relation to the relationship composition $ circ $, With this statement, every nilpotent in $ wp (X times X) $ be generated by $ 3 $ Elements of $ wp (X times X) $? I can prove it is possible with only four, though I think that's possible 3% $ Elements. If someone could correct me or quote relevant articles, I would be happy.


Edit: To repeat a relationship $ R subseteq X times X $ is nilpotent iff $ exists n in mathbb {N}: underscore {R circ R circ cdots circ R} _ { emptyset} = emptyset $

fa.functional analysis – convolution with an analytical semigroup

To let $ e ^ {At} $ denote an analytic semigroup in Hilbert space $ X $ generate by $ A: D (A) to X $, Leave too $ f in L ^ 1 (0, tau; X) $, I want to show that folding
$$ g (t) = int_0 ^ t e ^ {A (t-s)} f (s) ds $$
belongs $ W ^ {1,1} (0, tau; X) cap L ^ 1 (0, tau; D (A)) $, When I sit down $ f (s) = e ^ {As} x $ from where $ x in X $ then $ g (t) = te ^ {At} x $, Because of the analyticity of $ e ^ {At} $ we have
$$ sup_ {t> 0} | tAe ^ {At} | _ {L (X)} < infty $$
thus $ g (t) in L ^ { infty} (0, tau; D (A)) $ and $ dot {g} (t) = e ^ {At} x + tAe ^ {At} x in L ^ { infty} (0, tau; X) $, Does that prove that?