## 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

## Abstract algebra – In a semigroup, the product of two subgroups is always a subgroup

In a semigroup, the product of two subgroups is always a subgroup.
To let $$A$$.$$B$$ be two subset of semigroup G.
$$AB = {ab: a in A, b inB }$$

I try to prove that result, but I have no idea. So please give me a hint. Please do not give me any answers.

## 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 $$| a | then from the Subsolution estimate $$e ^ {a | x |} u in L ^ infty$$,

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