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

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Properties of Certain Example of Nonuniqueness to Heat equation

We say that a function $g\in C^{\infty}(\mathbb{R}^{n})$ is of Gevrey class $G^{\sigma}(\mathbb{R}^{n})$ if for each compact $K\subset\mathbb{R}^{n}$, there exist $C,R>0$ such that $$\sup_{x\in ...
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1answer
24 views

Equivalence of Dirichlet problems. Gilbarg & Trudinger

I do not understand the proof of theorem 11.4 in the book "Elliptic Partial Differential Equations of Second Order" by Gilbarg & Trudinger. The reason is that I do not understand the text right ...
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1answer
29 views

If $u \in H^1(0,\infty)$, does $u_x(\infty) = 0$?

Let $u \in H^1(0,\infty)$. Then is it true that $u'(x) \to 0$ as $x \to \infty$? I am wondering by Green's formula/IBP in this setting; do I get something like $$\int_0^\infty u_{xx}v = ...
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1answer
34 views

How to address multiple cases in this BVP? (Laplace equation in quarter-annulus)

The original problem: $$\nabla^2 u =0 \ \ \ \ for \ \ \ 0<a<r<b\ \ \ ,\ \ \ 0<\theta <\frac \pi 2$$ $$u(r,0)=0,\ \ u(r,\frac \pi 2)=f(r),\ \ u(a,\theta)=u(b,\theta)=0$$ My ...
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1answer
18 views

Exponential decay estimate involving $C_{0}$-semigroup and the principle eigenvalue

Consider the heat equation defined as $$\begin{cases} \partial_{t}u(x,t)=\Delta_{x}u(x,t) &\mbox{ } t>0\mbox{, } u\in U \\ u(x,t)=0 &\mbox{ } t\ge 0\mbox{, } x\in\partial U \\ u(x,0)=g(x) ...
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1answer
17 views

Solving PDE using normal mode

Given a linearised PDE $u_t=u_{xx}+\mu u$ where $x\in[0,1]$. A hint given is $u=V(x)\exp(st+ikx)$, where $s$ can be complex and $k$ is real. When I substituted into the PDE, I get ...
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0answers
88 views

How to find an ODE with prescribed terminal values? [on hold]

Let us consider an ODE $$\frac{dx_t^y}{dt}=g(x_t^y),$$ where y is the initial condition i.e. $x_0^y=y$. Now, given a function $f$ (increasing and smooth) is it possible to find $g$ (i.e. an ODE) ...
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1answer
31 views

How to solve a particular PDE (which reminds of heat equation)

I suddenly ran into this equation: Let $u:[a,b]\times \mathbb{R} \rightarrow \mathbb{R}$ be a function satisfying: $$\partial_t u = -u' + \frac{1}{2}u''$$ with bountary conditions $u(0,x)=g(x)$ where ...
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1answer
44 views

Good sources to learn about Geometric Analysis

So far the topics that have captured my interest the most are analysis and geometry. I want to learn a bit more about geometric and global analysis. There seems to be no shortage of geometry books but ...
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0answers
21 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 ...
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3answers
21 views

inhomogeneous pdes by separation of variables

This is the problem: $u_t=c^2 u_{xx}+g(x,t),0<x<l,\text{ and } t>0$ $u(0,t)=0=u(l,t)$, $t\ge 0$ $u(x,0)=f(x)$ I have trouble passing this problem to homogeneous form
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1answer
43 views

Maximum principle-estimation

Let $S=\{x \in \mathbb{R}^2 \mid |x| <1\}$. Using the maximum principle I have to show that the solution of the problem $$-\Delta u(x)=f(x), x \in S \\ u(x)=0, x \in \partial{S}$$ satisfies the ...
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1answer
21 views

A problem with a simple PDE

My task is to find a general solution to such a PDE: $xu_x+yu_y=0$. My approach: Such an equation is constant on its characteristics. So at first I want to find out what they look like. ...
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0answers
15 views

Poisson equation with nonlinear Neumann conditions

Let $\beta:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function such that $0<a\leq \beta^\prime\leq b$, for some constants $a,b$. Give the weak formulation of the problem \begin{equation} ...
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1answer
57 views

heat conduction problem

Find the solution of the heat conduction problem $U_{xx} =4U_t , 0 < x < 2, t>0;$ $U(0,t)=0, U(2,t)=0, t>0$; $U(x,0)=2\sin(\frac\pi2x)-\sin(\pi x) + 4\sin(3\pi x), 0 \le x \le 2 $ ok ...
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1answer
24 views

Laplacian operator on $L^2(\Omega)$?

Let $\Omega\subseteq \mathbb R^n$ be an open subset and $\displaystyle \Delta:=-\sum_{j=1}^n D_j^2$ be the Laplacian operator. I have some questions concearning this operator: $(i)$ Does it map ...
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0answers
10 views

Nonlinear Schrodinger equation - modified

I have a nonlinear Schrodinger equation: $ia_1\dfrac{\partial A}{\partial x}-a_2\dfrac{\partial^2 A}{\partial t^2}+|A|^2A=0$, $A$ is the amplitude and the above equation governs the slow modulation of ...
3
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0answers
32 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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1answer
15 views

Mean Value Property/Bulk Mean Value Property: conceptual issues (PDEs)

So my lecturer proves the mean value property which poses that under a few conditions on $u$ and $\overline{B_r(x_0)} \subset \Omega \subset \mathbb{R}^n$: $$u(x_0) = \dfrac{1}{\text{Vol}(\partial ...
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0answers
15 views

PDE with Robin boundary condition [on hold]

I need any advice (sugesseted book or a method etc..) to solve it
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1answer
75 views

Can I combine the wave and heat equations?

I have this equation $$\frac{\partial^2u}{\partial x^2} = 2\frac{\partial u}{\partial t} + \frac{ \partial^2u}{\partial t^2}$$ Is it possible for me to use both the wave and heat equations to solve ...
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1answer
12 views

Transforming the diffusion equation.

We need to transform the diffusion equation : $u_t = k(u_{xx} + u_{yy})$ into axisymmetric form : $u_t = k(u_{rr} + \frac{u_{r}}{r})$ , I first converted the laplace equation $u_{xx} + u_{yy}$ = 0 ...
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1answer
23 views

Proof of the Harnack inequality

Let $\Omega\subseteq\mathbb{R}^n$ be a domain, $\Omega'\subset\subset\Omega$ be a domain and $u\in C^0(\overline{\Omega})$. Suppose we know $$\sup_{\Omega'}u\le 3^n\inf_{\Omega'}u\tag{1}$$ if ...
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1answer
17 views

Deciding when to use $\mu^2$ or $-\mu^2$ in separation of variable (PDE'S)

I have the following question Consider the two-dimensional PDE on $u = u(x,y)$ $$u_{xx}-u_{yy}=0$$ $$u(x,0)=\phi(x)$$ $$u_y(x,0)=0$$ where $\phi:\mathbb{R}\rightarrow \mathbb{R}$ is a function ...
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2answers
19 views

Show that lamda is greater than or equal to zero for a sturm liouville problem

To show that this problem can be put into S-L form for an eigenvalue problem, Observe that The S-L form is of $$\text{p'(x)}\phi _x\text{+p(x)}\phi _{\text{xx}}\text{+q(x)$\phi $+$\lambda \phi ...
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1answer
32 views

Need help with this S-L problem

Consider the problem: $$\frac{\text{$\delta $u}}{\text{$\delta $t}}=\frac{\delta ^2u}{\text{$\delta $x}^2}\text{-u+x(1-x)}$$ The IC are given as: $$u(0,t)=1$$ $$u_x\text{(1,t)=0}$$ ...
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1answer
25 views

Partial Differential equation with laplacian and gradient.

Suppose that $\Omega\subset \mathbb{R}^n$ is open and bounded. Let $u\in C^2(\Omega)\cap C^0(\bar\Omega)$ is a solution for the equation $\triangle u+\sum_{k=1}^na_ku_{x_k}+c(x)u=0$ where ...
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1answer
9 views

Vector Identity that appears in the Poisson Kernel of the $B(0,1)$

Given that $y\in \partial B(0,1)$ how do i get that $|x|^2|y-\frac{x}{|x|^2}|=|y-x|$? It may be a silly question but I don't see why? Thanks!
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0answers
28 views

First order partial differential equation.

We need to solve the given first order partial differential equation : $(y-xu)u_x$ + $(x+yu)u_y$ = $x^{2} + y^{2}$ . I tried this : $\frac{dx}{y-xu}$ = $\frac{dy}{x+yu}$ = $\frac{du}{x^{2} + ...
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1answer
40 views

Radial Solution to the wave equation in terms of odd functions

Show that if $f \in C^3_c (\mathbb{R})$ is an odd function then for $|x|=r$ define \begin{equation} u(x,t) = \frac{f(r+t)+f(r-t)}{2r} \end{equation} then $u$ extends as a $C^2$ function that solves ...
2
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1answer
26 views

Inversion map is a Conformal map

I'm studying PDE by Evans book and I need to show that the inversion map $f:\mathbb{R}^n-\{0\}\to \mathbb{R}^n$, defined by $$f(x)=\frac{x}{\|x\|^2}$$ is conformal. So I have a hint, show that ...
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14 views

GODOUNOV scheme

Help me, i can't understand how to decribe godounov scheme with discontinuous initial conditions for nolinear PDE , is it related with Riemann problem ? \begin{equation} u_t+f(u)_x=0 \end{equation} ...
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classification of first order PDE hyperbolic and parabolic

how i can classify a first PDE \begin{equation} u_t+cu_x=0 , \end{equation} is it hyperbolic or parabolic? . with details .
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1answer
24 views

Eigenfunction and their orthogonality with respect to the weight function

The Eigenfunction and their orthogonality with respect to the weight function $$\sigma$$ is defined as $$\int _a{}^b\phi _n\text{(x)}\phi _m\text{(x)$\sigma $(x)dx=0}$$. Given that I have some ...
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0answers
28 views

sde problem, may require Ito? [on hold]

Given $dU_t = -\gamma U_t \, dt + dX_t$; How do I solve this equation for $U_t$?
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1answer
40 views

Computing Fréchet derivative

I am reading Methods in Nonlinear Analysis by Kung-Ching Chang and having trouble in obataining a Fréchet derivative in the text. For those who has the book, it is on page 37, which concern Euler ...
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3answers
29 views

Fourier series sketching

Whenever I am asked to draw fourier series, is it correct to first draw the function on the interval first (in this case 0<= x < pi), then extend the the graph to the desired interval ...
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0answers
35 views

PDE complex boundary condition

My attempt to this question was setting T''-lambda T=0 and try lambda=0, >0 and <0. However, I do not seem to have sufficient information to determent which case have non-trivial solution ( since ...
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0answers
15 views

How do I solve this system of PDEs numerically?

Suppose that I have a system of PDEs of the following form: \begin{eqnarray} (\frac{\partial}{\partial x} - i\frac{\partial}{\partial y}) f(x,y) = F(f,g,h) \\ (\frac{\partial}{\partial x} - ...
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2answers
32 views

Heat equation-unicity

We have the folllwing problem: $\begin{cases} & \dfrac{\partial u}{\partial t} = k \dfrac{\partial^2 u}{\partial x^2}, 0 < x < l, t > 0\\ & u(0,t)=0,\\ & \dfrac{\partial ...
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1answer
23 views

Neumann problem on $\Omega$. Does $U$ solving the problem imply $U+c$ does?

Let $U$ solve the Neumann${}$ problem${}$ for laplace's equation on a${}$ domain $\Omega$. Show that $U+c$ also solves this problem for any $c\in\Bbb R$. What is being asked of me? Does this mean ...
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1answer
43 views

How do you find the inverse Laplace transform of $\frac{1}{\sqrt{s}(s-a)} $

When I use the convolution method, I can't avoid getting a divergent integral.
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21 views

Classification of Differential Equations

I know that there is a theory of integrating (partial) differential equation by finding its symmetries (which form a Lie group) and making corresponding transformation of the domain. I also know ...
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1answer
24 views

Lagrangians independent of $x$

In PDE Evans, 2nd edition, the following formula is printed as equation $\text{(9)}$ in §8.6 (on page 514): $$\sum_{k=1}^n (L_{p_i}u_{x_k}-L\delta_{ik})_{x_i}=0 \quad (k=1,\ldots,n) \tag{9}$$ ...
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1answer
30 views

Inhomogeneous First Order PDE with boundary condition

I'm having some trouble solving the following PDE: \begin{equation} u_x+u_y=1, u(y,\frac{y}{2})=y \end{equation} I know I can use the total differential to find a general solution: \begin{equation} ...
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47 views

Confusion with differential partial equation [closed]

what is the solution of this equation $$4z = \left(\frac{dz}{dx}\right)^2 + \left(\frac{dz}{dy}\right)^2$$ with initial conditions $x=0,z=y^2$.
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1answer
30 views

Harmonic Function with linear growth

We want to find harmonic functions $w$ in (say) $\mathbb{R}^2$ that are zero on $\{y=0\}$ with the linear growth bound \begin{equation} \sup_{\mathbb{B}_R} |w| \leq C(1+R) \end{equation} where ...
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1answer
28 views

How could we continue to show the inequality?

Let $\Omega$ a bounded space. Let $u_1$ the solution of the problem $$-\Delta u_1(x)=f(x), x \in \Omega \\ u_1(x)=g_1(x), x \in \partial{\Omega}$$ and $u_2$ is the solution of the problem $$-\Delta ...
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1answer
21 views

Does the Laplace operator include the second derivative with respect to time variable?

Does the Laplacian of a function $f(x,z,t)$ equal $f_{xx} + f_{yy} + f_{tt}$? We aren't sure whether or not time is included in it or not.
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1answer
33 views

How could we continue to get a contradiction?

Let $\Omega$ a bounded space. Using the maximum principle I have to show that the following problem has an unique solution. $$-\Delta u(x)=f(x), x \in \Omega \\ u(x)=g(x), x \in \partial{\Omega}$$ ...