All Questions
Tagged with semigroups-of-operators parabolic-pde
25
questions
1
vote
0
answers
66
views
Parabolic regularity for weak solution with $L^2$ data
I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions:
$$\begin{cases}\...
- 1,434
3
votes
1
answer
228
views
$L^{\infty}$ estimate for heat equation with $L^2$ initial data
Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem:
$$\begin{...
- 1,434
0
votes
0
answers
140
views
Convergence of Solutions of Integral Equations with Weakly Converging Forcing Terms
Let $\Omega$ be a bounded interval of $\mathbb{R}$ and let $y\in L^\infty(\Omega \times (0,T))$ be a mild solution of the integral equation
$$
y(\cdot,t)=S(t) y_0+\int_0^t S(t-s) \left[u(\cdot,s)y(...
- 55
1
vote
0
answers
59
views
Solution to $u_t = A(t)u + f(t)$ on bounded domain
I am dealing with the problem
\begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\
\partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...
- 11
2
votes
0
answers
54
views
Are there results that relate the restriction of the heat semigroup in $\mathbb{R}^n$ and the semigroup in a domain?
I'm thinking about the following situation:0 suppose that
$$
S_{\Omega}(t)f = \int_{\Omega} K_{\Omega}(x,y,t)f(y)dy
$$ where $K_\Omega(x,y ,t)$ is the Dirichlet heat kernel in the domain $\Omega$. It ...
- 645
2
votes
1
answer
248
views
Periodic solution for linear parabolic equation - existence, regularity
I am interested in proving the existence and regularity of solution to the following problem:
$$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...
- 1,434
4
votes
1
answer
341
views
Strong positivity of Neumann Laplacian
There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...
- 1,434
0
votes
0
answers
49
views
How was this heat semigroup estimate made in a paper on reaction–diffusion systems?
In Yamauchi - Blow-up results for a reaction–diffusion system, in the proof of Lemma 3.3, there is the passage
$$S(t-s)|x|^{\sigma/1-k} \geq C_1(t-s)^{\sigma/2(1-k)}.$$
Here $S(t)$ denotes the heat ...
- 645
1
vote
0
answers
90
views
Is there any class of initial data for which the heat semigroup is increasing in time?
Lee and Ni proved in the paper Global existence, large time behavior and life span of solutions of semi linear parabolic Cauchy problem that the heat semigroup decays as $t$ tends to infinity, that is
...
- 645
2
votes
0
answers
120
views
Gradient $L^\infty$-estimate for heat equation with homogeneous Dirichlet boundary condition
$\Omega\subset \mathbb{R}^N$ is a bounded smooth domain. Consider the homogeneous heat equation with zero boundary condition in $\Omega$
\begin{cases}
\partial_t u-\Delta u=0 \quad(x,t)\in \Omega\...
- 121
4
votes
1
answer
390
views
Contractivity of Neumann Laplacian
I have an intriguing and probably simple question: reading the articles and books of Wolfgang Arendt on semigroups of linear operators, I found on many places properties of the Neumann Laplacian.
In ...
- 1,434
0
votes
0
answers
90
views
Results about Schrödinger equations
Does anyone know any paper or book that deals with Schrodinger equations, specifically on asymptotic properties like blowup or limitation of solution when time goes to infinity using Schrödinger ...
- 645
1
vote
0
answers
36
views
Existence and uniqueness for fractional parabolic equation with transport term
Let us consider the problem
\begin{equation}
\begin{cases}
u_t+(-\Delta)^{\sigma}u+\mathrm{div}(a(t,x)u) = 0 & \text{in } \mathbb{R}^n \times [0, T) \\
u(x,0)=u_0(x) & \text{in } \...
- 99
5
votes
1
answer
157
views
Spectrum of an elliptic operator in divergence form with a reflecting boundary condition
Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$
\begin{align}
L ...
- 53
3
votes
1
answer
367
views
Neumann/Robin Laplacian semigroup well-known estimate
Let $\Delta_R:D(\Delta_R)\to L^2(\Omega)$ the Robin Laplacian defined on:
$$D(\Delta_R)=\left\{u\in H^1(\Omega)\ \big |\ \Delta u\in L^2(\Omega),\ \dfrac{\partial u}{\partial\nu}+bu=0 \ \text{on}\ \...
- 1,434