Questions on "Partial Differential Equations", as opposed to "ordinary differential equations".

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Minimum Knowledges to precisely calculate PDEs (integral equations)

Basically all I want to do is to calculate (or prove) precisely equations such as $$ \frac{1}{n\alpha(n)r^{n-1}}\int_{\partial B(x,r)} u(y) dS(y) = \frac{1}{n\alpha(n)} \int_{\partial B(x,r)} u(x+rz)...
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27 views

Proving parabolic PDE has maximum on boundary

Given $$u_t=u_{xx} - \frac{x}{2t}u_x - u^3$$ for $x,t>0$, with $u \rightarrow \frac{1}{2\sqrt t}$ as $x \rightarrow \infty$ for fixed any $t>0$, then I want to show that a maximum of $u(x,t)$ ...
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41 views

Resolvent estimate of hyperbolic Laplacian

Consider the Laplacian $-\Delta$ on the hyperbolic space $\mathbb{H}^n$. For $\lambda \in \mathbb{C} \setminus [0, \infty)$, do we have resolvent estimates of the form $$\Vert (-\Delta - \lambda I)^{-...
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40 views

$\Box u= | u |^2 u$ global solution in $C^\infty$

Let $u_0, u_1 \in C^\infty( \mathbb{R},\mathbb{R}^3)$. Consider the cubic defocusing NLW $$(\ast)\begin{cases} \Box u= |u|^2 u \\ (u,\partial_t u) \restriction_{t=0} = (u_0,u_1) ,\end{cases}$$ where $\...
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Is this derivative well defined?

Let $\Omega$ be an open bounded domain in $\mathbb{R}^n$. Let $z_1,z_2,\dots,z_k\in \bar \Omega $ be $k$ distinct points. Let $\bf Z$ denote the k-tuple $Z = (z_1,z_2, \dots, z_k)$ Define $R_i = \{x\...
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79 views

What type of equation is this?

Is this equation an ODE or PDE $$ \frac{d^3u}{dx^3}−αxu=0, x∈R $$ The only thing given is $\int_R u(x) =\pi $ and $α>0$ is some constant. I have to find the solution using fourier ...
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53 views

Estimating/approximating a very high dimensional unbounded poisson's equation

Consider the poisson equation on an unbounded domain. Suppose that the solution is known to exist. $$ \Delta u=f $$ I would like to estimate the solution of the this equation at a given point $x_0$. ...
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84 views

How do I apply this PDE as an image filter?

I'm trying to preprocess a height map image with a helmholtz-type equation as described in this paper. The equation is: $$ddx(h') + ddy(h') + y(h'-h) = 0$$ I solved for h and got: $$\dfrac{ddx(h')}{...
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196 views

Prove difference quotient converges to weak derivative in $L^p$

I am trying to solve the following exercise: Let $U$ be an open set in $\mathbb{R}^n$ and let $V$ be a compact subset with Lipschitz boundary. Assume that $f$ is in $L^p(U)$ with $1<p<\infty$ ...
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79 views

Regularity of weak solution for $u_t - \Delta u = f$ with $u(0) = u_0 \in L^1(\Omega)$

Let $\Omega$ be a bounded domain, and consider the equation $$u_t - \Delta u = f$$ $$u(0) = u_0 \in L^1(\Omega)$$ with Neumann BCs (or Dirichlet if convenient) where $f$ is smooth. Using energy/...
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72 views

Is $\Delta^{-1}$ a bounded operator?

Is the inverse Laplacian $\Delta^{-1}: H^{m+2}(M)\mapsto H^m(M)|1$ a bounded operator? Where $M$ is a compact manifold and $H^m(M)|1$ means its elements $f \in H^m(M)$ and $\langle1,f\rangle=\int_{M}...
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209 views

Wave equation for a string nonuniform (PDE)

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman , but I have been impossible these paragraphs. The displacement $u$ of a nonuniform string ...
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111 views

Small question about condition in the Bouchala paper's 2005

I have this condition from this paper: http://ejde.math.txstate.edu/Volumes/2005/08/bouchala.pdf (Strong resonance problems for the one-dimensional $p$-Laplacian. Bouchala, Jiri. Electronic Journal ...
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143 views

Finite difference solution of steady-state diffusion equation with variable material properties

I'm trying to use a finite difference method to solve the steady-state neutron diffusion equation in a nuclear reactor: $$ D(x) \nabla^2 \phi(x) + \left( \frac{\nu(x)}{k} \Sigma_f(x) - \Sigma_a(x) \...
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36 views

Solution to the cauchy problem: $u^{2}_{x}/2−u_{y} = −x^{2}/2 , u(x, 0) = x$

$\frac 12 u^{2}_{x}−u_{y} = −\frac 12x^{2} ;\\ u(x, 0) = x$ How do we show that that the solution blows up in finite time and explain this in terms of characteristic of the equation.
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How to derive the Sperically symmetric Wave Equation (in spherical coordinates) using first principles?

I want to use first principles (of mass and momentum conservation) on a spherical shell and derive the wave equation given below, where $p'$ is the pressure perturbation : $\frac{\partial^2 p'}{\...
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68 views

if $\Delta u = |\nabla u|^{3/2}$ then $u \in C^{\infty}$

Suppose $u \in H^1_{\text{loc}}(\mathbb{R^2})$. I want to show that if $\Delta u = |\nabla u|^{3/2}$ in the distributional sense then $u \in C^{\infty}$. I know that since $\nabla u \in L^2$ (I'll ...
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Solving system of first-order PDEs with Frobenius theorem

I've been stuck trying to solve this system: $$\ \frac{\partial u}{\partial x} = \frac{-2xy^2}{u} + 3y $$ $$\ \frac{\partial u}{\partial y} = \frac{-2x^2y}{u} + 3x $$ Which must satisfy $ \ u(0,0) = ...
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116 views

Deriving a formula for an initial boundary-value problem

Given $g : [0,\infty) \to \mathbb{R}$, with $g(0)=0$, derive the formula $$u(x,t)=\frac{x}{\sqrt{4\pi}}\int_0^t \frac 1{(t-s)^{3/2}}e^{-\frac{x^2}{4(t-s)}}g(s)\,ds$$ for a solution of the initial/...
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mononotone and pseudomonotone operators in current research

I know that the following question is quite broad for this forum. But I am interested in references or any other ideas. Can anyone provide some examples of applications of monotone or pseudomonotone ...
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242 views

Numerical scheme to 1D advection equation

I am trying to numerically solve a system of equations which model the early universe in 1D. The equations I am stuck on are; $$ (1)\quad \partial\rho/\partial t + \partial(\rho v)/\partial x = 0 $...
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25 views

Compactness of a sequence

Let $\theta_n(x,t)$ a sequence such that $$\theta_n\rightarrow\theta\;\;\mbox{in}\;\;C((0,T],H^s)\;\;\mbox{where}\;\;s>1.$$ Consider $\phi \in C^{\infty}$ and $$g_n(t,\phi)=\langle\theta_n(t),\phi\...
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Compactness of solution space of semi-linear parabolic PDE

Under what conditions a closed and bounded subset of solution space of following parabolic PDE is compact? $$x_{t}=x_{zz}+f(x,z)$$ Thank you!
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59 views

The order of a differential operator on a manifold is well-defined

In these notes, the author defines a differential operator of order at most $n$ on a manifold $M$ as an element of $\operatorname{Diff}^n(M) := \operatorname{span}_{0\leq j \leq n} (C^\infty(M;TM))^j.$...
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176 views

Weak subsolution and composition with convex smooth function

I'm trying to solve a problem in Partial Differential Equations by Lawrence C. Evans. It goes like this Assume $u\in H^1(U)$ is a bounded weak solution of \begin{equation}-\sum_{i,j=1}^n (a^{ij}u_{...
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Banach spaces involving time

Let's suppoe $u\in L^2(0,T;H_0^1(\Omega))$ with $u'\in L^2(0,T;H^{-1}(\Omega))$. We know that $$u\in C([0,T];L^2(\Omega))$$. In this result can the set $\Omega$ be the whole $\mathbb{R}^n$ or we need ...
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An interesting proof using Green's representation formula?

Let $B_R(0)$ be a ball in $\mathbb R^3 $ and define $$u(x)=\int_{B_R(0)}\frac{1}{|y-x|}dy$$ Prove that $$ u(x) = \begin{cases} \frac{2}{3}\pi(3R^2-|x|^2) & \quad \text{for $0 \le|x|\le R$ ...
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24 views

Inverse continuous

According Dehman, Gérard and Lebeau in "Stabilization and control for the nonlinear Schrödinger equation on a compact surface" (Math. Z. (2006) 254:729–749, DOI 10.1007/s00209-006-0005-3) is claimed ...
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42 views

Existence of a harmonic function?

I really have no clue about this problem. Should I represent $u(x)$ using $\varphi(x')$? But how to represent that? Thanks for any help!
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73 views

A Free Boundary Problem

Is there any special way to solve such a problem. Any idea would be appreciated. At least does anybody know which method is useful to solve this problem numerically? Is it even solvable numerically? ...
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59 views

Is solving Poisson's equation in Polars different from Cartesian?

I'm having trouble figuring out how to separate variables in polar coordinates in 2D. In cartesian coordinates it is fairly simple to use eigenfunction ideas because I can group together the x, y ...
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63 views

How to derive the cigar soliton solution to the Ricci flow equation?

I am trying to derive the cigar soliton solution to the Ricci flow equation. Such solution has the form $$ {\frac {{{\it dx}}^{2}+{{\it dy}}^{2}}{{{\rm e}^{4\,t}}+{x}^{2}+{y}^{2 }}} $$ I am ...
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214 views

Comparison and maximum principle for parabolic pde

I was told the comparison principle can also be understood as the fact that the difference between a subsolution and a supersolution satisfies the maximum principle. I also know comparison principle ...
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35 views

Integral form of Bessel's (n = 0)

While studying for a comprehensive exam, I have come across this old problem: Consider the Helmholtz equation in the $\mathbb{R}^2$ plane $$u_{xx} + u_{yy} + \omega^2u = 0$$ Derive an ...
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96 views

Form-invariant solution to PDEs

I'm trying to understand how to create form-invariant solutions to PDEs. Let $u(x,t): \mathbb{R}^2 \to \mathbb{C}$, which solves the differential equation $\hat{L}u(x,t)=0$. $u(x,t)$ is form-...
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118 views

quasilinear partial differential equation

Given a PDE $ f e^2 \frac{\partial f}{\partial x}- e f^2 \frac{\partial f}{\partial y} + M_1 f^4 + M_2 f^2 + M_3=0 $ Note that $M_1$ , $M_2$ and $M_3$ are functions of $\cos (x-y)$ and $\sin (x-y)$ ...
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Studying Hamiltonian PDEs - Where to start?

I first got in contact with Hamiltonian PDEs when I wrote my first thesis about the Nash-Moser-Theorem as some books mentioned them as a field of application of this theorem. As those examples were ...
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126 views

Solving 1D telegrapher's equation by reduction to two-dimensional wave equation

The solution $w : \mathbb R \times \mathbb R_{+} \to \mathbb R$ of the Cauchy problem for the telegrapher's equation $$ w_{tt} - c^2 w_{xx} + c^2 \lambda^2 w = 0 $$ with $$ w(x,0) = 0, \qquad w_t(x,...
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Solution to Schrödinger equation $ \partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t)V(x)f(x,t).$

I want to solve $$ i\partial_t f(x,t) = -\partial_x^2 f(x,t) + \delta(t-t_0)V(x)f(x,t),$$ for any $V \in C^{\infty}[-1,1]$ and $f: [-1,1] \times \mathbb{R_{\ge 0}} \rightarrow \mathbb{C}$. I would ...
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150 views

Regularity of a Weak Solution to Fokker-Planck Equation

Suppose that $\rho \in L^1(\mathbb{R}^n \times (0,T))$ for every $T < \infty$ is a weak solution of the PDE \begin{align} \partial_t\rho &= \Delta \rho + \text{div}(\rho\nabla\Psi(x))\\ \rho(t =...
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82 views

Which came first, energy minimization or pde?

I'm interested in a historical perspective on pde. I would like to know more about the original derivation of pde. It seems like d'Alembert was working on the one dimensional wave equation $$ \...
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50 views

Missing step in Evan's PDE?

In the attached proof, I am having trouble justifying the presence of "$x_n^{p-1}$" in eh line immediately below "Thus" and "Letting $m\to\infty$". I understand he uses Minkowski's inequality, and ...
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79 views

$L^2(S;W^{1,p})$-regularity for solution of parabolic pde in Hilbert space setting

I have a question regarding the regularity theory for parabolic pdes. I am considering the following "minimal working example": Let $\Omega\subset\mathbb{R}^n$, $n\in\mathbb{N}$, be and bounded ...
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Anyone help me with this PDE using Fourier Transform?

I have this: $$\frac{\partial c}{\partial t} + p\frac{\partial c}{\partial z}+\lambda p\frac{\partial^{2} c}{\partial z\partial t}-\frac{\partial^{2} c}{\partial z^2}=0\quad(1)$$ $$c(z,0)=\delta(z)$$ ...
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178 views

Hölder estimate of solution to linear parabolic PDE

Consider a standard Cauchy-Dirichlet problem for linear parabolic PDE: \begin{cases} u_t + \mathcal{L}u=-g, \ (t,x) \in [0,T)\times \Omega \\ u(T,x)=0 , \ x\in \Omega\\ u|_{\partial \Omega}=0 \end{...
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89 views

Tensor differential equations

I am reading Ringstrom's book The Cauchy problem in General Relativity, But I don't really understand Chapter 12 associating to tensor equations. I want to read some other material about this. Could ...
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73 views

Are there any results linking hyperbolic operators and Pseudo-Riemaniann geometry?

In the case of a $n$-dimensional pseudo-riemannian manifold one has a symmetric bilinear form, $h$, with signature $(1,n)$ and the analogous operator to the Laplace -Beltrami operator,$\Delta$, is the ...
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60 views

Variational Problem on a manifold - bounding my functional from below

Let $(M, g_{ij})$ be a compact Riemannian manifold. Let $f \in C^{0,\alpha}(M)$ be a given function. I have the linear operator $L = \Delta h - 12u^2h$, where $u \in C^{2,\alpha}$. I need to show ...
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186 views

Passing to the limit in a PDE; problem with subsequence (please check my answer)

For $w \in L^2(0,T;H^1)$, consider the PDE $$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$ where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and $f$...
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110 views

A regularity result for a parabolic PDE? Want $u' \in L^\infty((0,T)\times \Omega)$

Let $f \in L^\infty((0,T)\times \Omega)$ and let $g \in L^\infty((0,T)\times \Omega)$ satisfy $$0 < a \leq g(x,t) \leq b\quad\text{for all $(x,t)$}$$ $$\frac{dg}{dt} \in L^\infty((0,T)\times \...