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Apr
29
awarded  Popular Question
Apr
27
comment Writing an abstract
You don't mention if $B$ and $B'$ are somehow related. If they belong to two antipodal regions of mathematics, then probably $B$ need not be mentioned. But I agree with @ArturoMagidin, being slightly more informative (used techniques, new insights...) wouldn't harm.
Apr
25
comment Strong maximum principle for heat equation. Positivity of solution
Well, $e^{t\mathcal A}$ is analytic, therefore so is $u$ wrt time; but when it comes to its space regularity, the solution is a priori only in the domain of $\Delta^k$ ($\Delta$ being the distributional Laplacian without b.c.), so one cannot apply any boundary regularity result. I am pretty sure $u$ is $C^\infty$ wrt space on compact subsets of $\Omega$, but I cannot go further.
Apr
25
revised Strong maximum principle for heat equation. Positivity of solution
corrected the domain of $\mathcal A$, which was too small in the previous version
Apr
24
comment Strong maximum principle for heat equation. Positivity of solution
@ChristopherSail You are right! And, unfortunately, this actually means that I did not adressed your question (case of $a\ge 0$), but rather the question of what happens if $a$ is monotonically increasing in time.
Apr
24
answered Solution of Heat equation with Neumann BC in an arbitrary domain
Apr
24
answered Strong maximum principle for heat equation. Positivity of solution
Apr
23
comment Strong maximum principle for heat equation. Positivity of solution
Are you willing to assume that $a$ is differentiable wrt time?
Apr
22
comment Connection between solution for Schrödinger equation and solution for heat equation
@WillieWong I haven't read Hörmander's original paper: In the book I quote the relevant result is Theorem 3.9.4, which is then said to be found in: L. Hörmander. Estimates for translation invariant operators in L^p spaces. Acta Math. 104 (1960), 93–139.
Apr
21
comment Connection between solution for Schrödinger equation and solution for heat equation
Sure. The point in the quoted paper is that the result are stated not only for the Laplacian (in which case the theory is standard indeed), but for general generators of analytic semigroups; and a precise characterization of analytic semigroups that extend to "boundary groups" along $i\mathbb R$ in terms of their asymptotic behavior at the boundary of their analyticity domain is provided.
Apr
16
revised Connection between solution for Schrödinger equation and solution for heat equation
added 74 characters in body
Apr
16
reviewed No Action Needed Differential equations → predicate logic mapping
Apr
15
answered Connection between solution for Schrödinger equation and solution for heat equation
Apr
13
answered Hashimoto Matrix (Non-backtracking operator) and the Graph Laplacian
Apr
13
answered On extending a semigroup property
Apr
13
revised a variation on the theory of equitable partitions for graphs
improved formatting
Apr
11
reviewed No Action Needed Compact Vertical Cohomology and Euler Class of CP1
Apr
8
comment Are Ritt operators mean ergodic?
@user89334 $R(\lambda,T):=(\lambda-T)^{-1}$
Apr
7
comment Abstract ODE; PDE; uniqueness of solution
@AndrásBátkai please allow for some quibble :) you write "a parabolic PDE" but what you actually mean is a "linear parabolic PDE". It is well-known since the 1970s that certain nonlinear parabolic PDEs (for instance, for the $p$-Laplacian with $p<2$) may have finite time extinction.
Apr
7
revised Are Ritt operators mean ergodic?
deleted 4 characters in body