All Questions
Tagged with ap.analysis-of-pdes asymptotics
21 questions
1
vote
0
answers
31
views
Explicit rate of decay of the positive standing wave of the subcritical nonlinear Schrödinger equation
Consider the following semilinear problem:
$$
\begin{cases}
- \Delta u + u = u |u|^{p - 2}
&\text{in} ~ \mathbb{R}^N;
\\
u (x) \to 0 &\text{as} ~ |x| \to \infty,
\end{cases}
$$
where $N \geq 2$...
- 144
2
votes
0
answers
81
views
Asymptotic expansion of the solution of a nonlinear wave equation
I am reading the article Recovery of a cubic non-linearity in the wave equation in the weakly non-linear regime (arXiv link) by Barreto, Plamen, where they consider the modified cubic NLWE
$$-\...
- 840
2
votes
1
answer
102
views
Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function
Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
- 23
0
votes
0
answers
115
views
Existence of Green functions and some properties
Let $\Omega$ be a smooth domain in $\mathbb{R}^N$， $N\geq 3$, $p\in \Omega$ is a fixed point, $\lambda$ is a parameter (can be 0,>0,<0), if there exisits a Green function $G_{\lambda}(x,p)$ ...
- 471
8
votes
2
answers
670
views
Asymptotic behavior of a certain oscillatory integral
Let $x>0$ and consider the integral
$$I(x):=\int_0^\infty \frac{e^{i r}}{r^{\frac{1}{2}}} \int_0^\infty \frac{e^{-s}}{s^{\frac{1}{2}}} \frac{r}{sx+\sqrt{sxr}+r} \, ds \, dr.$$
I am trying to ...
- 852
3
votes
1
answer
251
views
Asymptotic behavior of a double oscillatory integral
Let $0<\theta_1,\theta_2<\pi/2$. Suppose $\psi$ is a smooth real-valued function with compact support.
Consider the oscillatory integral
$$I(t):=\int_{0}^{1}\frac{1}{(y-e^{\dot{\imath}\theta_1})
...
- 852
2
votes
1
answer
94
views
Decay rate for a small perturbation of a simple linear ODE
MOTIVATION.
Let $f:[0,+\infty)\to \mathbb{R}$.
Solutions to $\partial_tf(t) = -\lambda f(t)$, $f(0)\neq 0$ approach zero exactly as $e^{-\lambda t}$.
This property is preserved if we apply an ...
- 2,533
5
votes
1
answer
161
views
Asymptotics for repulsive aggregation(-diffusion) equation
Consider the aggregation-diffusion equation
$$
\frac{\partial \rho}{\partial t} = \nabla (\rho \nabla(W\star \rho)) + \nu \Delta \rho,
$$
where $W:\mathbb{R}^d \to \mathbb{R}$ is a twice continuously ...
- 223
3
votes
1
answer
162
views
$\Delta_g f = 0$ on the Riemannian Manifold $(\mathbb{R}^3 \setminus B , g)$ with conditions on the boundary and at infinity
Consider the manifold $\mathbb{R}^3 \setminus B$ where $B$ is the ball with radius 1 with riemannian metric $g$ (not necessarily the euclidean metric).
I am looking for solutions to $\Delta_g f = 0$ ...
- 969
3
votes
1
answer
373
views
Asymptotic expansion of heat operator $e^{-\Delta{t}}$ and $e^{-\mathcal{D}t}$ of Dirac operator
For a closed Riemannian manifold $M$ of $n$-dimension, we consider the Laplace-Beltrami operator $\Delta$.
It is known that we have an asymptotic expansion for the trace of heat operator $e^{-\Delta{...
3
votes
0
answers
103
views
Ratio of solutions to two heat equations
Let $u(x,t)$ and $v(x,t)$ respectively solve the two one-dimensional heat equations with different (real) diffusion coefficients on the same domain $D$ and the same initial & boundary conditions, ...
- 103
1
vote
0
answers
88
views
Parabolic (heat) PDE Green's function spatial asymptote at infinity
Consider a general parabolic partial differential equation with its spatial dimensions on $R^n$, such as a heat equation, with the diffusion coefficient dependent on the spacial variables. Does its ...
- 2,239
4
votes
0
answers
144
views
Asymptotic expansion of a Gaussian integral and heat kernel
When considering the heat kernel of a Schr\"odinger operator
$$- \Delta + V(x) $$
where $\Delta$ is the standard Laplacian on ${\mathbb R}^n$ and $V$ is a nonnegative potential function that has ...
- 1,207
5
votes
0
answers
172
views
Has anyone studied a transport equation of this form?
Let $L\colon \mathbb{R}^2 \times \mathbb{R}^+\to \mathbb R$ satisfy
$$
\frac{\partial L}{\partial t} (x,t) = \max\left\{ \frac{\partial L}{\partial x_1}, \frac{\partial L}{\partial x_2} \right\}
$$
...
- 941
1
vote
0
answers
86
views
Asymptotics of a elliptic pde when exponent gets large
I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...
- 1,385
2
votes
0
answers
201
views
geometric irregularities in pde's
The following question is intended for a person more acquainted with the works of Yves Laurent.
see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French)
http://link.springer.com/...
- 113
2
votes
0
answers
104
views
Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform II
This is a modification of a previous question.
Question: Suppose $a(s)\in C^\infty([0,1])$ and $H(s,x)\in C^\infty([0,1]\times [0,1])$ with $H(s,x)>0$, $\forall s,x\in [0,1]$. Suppose,
$$\sup_{\...
- 468
4
votes
2
answers
534
views
Quantitative Weierstrass Approximation and Paley-Wiener for the Laplace Transform
Question: Suppose $a(x,y)\in C^\infty([0,1]\times [0,1])$ and suppose
$$\sup_{\lambda>1} \bigg|\lambda\int_0^1 e^{\lambda x} a(x,1/\lambda)dx\bigg|<\infty.$$
Is $a(x,0)=0$, $\forall x\in[0,1]$?
...
- 468
5
votes
1
answer
810
views
Asymptotics of the TBA equation
The Thermodynamic Bethe Ansatz equation is an integral equation that was derived by Yang and Yang to study some interacting systems. In the simplest case, it is
$$\epsilon(\beta)=R\cosh\beta-\int\frac{...
- 3,357
3
votes
1
answer
1k
views
Long time behavior of the heat equation on R
Let $\mu\in\mathcal{S}'(R)$ be a Schwartz distribution. The solution of a heat equation with $\mu$ as the initial data is
$$
u(t,x)= \int_R \frac{e^{-\frac{(x-y)^2}{2t}}}{\sqrt{2\pi t}} \mu(d y)
$$
...
- 1,649
2
votes
1
answer
376
views
Asymptotic behaviour near the boundary in the Dirichlet problem for the Laplacian.
Perturbative behaviour of solutions of the solutions of the Dirichlet problem for the Laplacian:
Lets consider $ B = B(0, 1) \in \mathbb{R}^2$ be the unit circle with center at $0\in\mathbb{R}^2$. ...
- 85