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

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2
votes
1answer
31 views

Boundary conditions that yields no solutions to the coefficient?

I want to make sure I'm not overlooking certain steps as I've already spent an hour looking through. The heat equation is given as: $$ \frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2} ...
0
votes
1answer
25 views

Solving the heat equation

I have the heat equation $$\frac{\partial u}{\partial t}=\frac{\partial^2u}{\partial x^2}$$ where $0\leq t<\infty$ and $0\leq x\leq1$ The Boundary conditions are $u(0,t)=0$ and $u(1,t)=1$ with ...
1
vote
1answer
17 views

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$

Solve the following PDE: $(1+\sqrt{z-y-x})z'_x+z'_y=2$ given that $z(x,2x)=2x$. I want to explain to you how we were taught to solve these at class, and this method seemed to work with other ...
0
votes
1answer
10 views

Mollifiers and smooth path connetction

Generaly it is about smooth path connection. I have a two smooth paths f,g and a points x,y,z of open subset of $R^n$ such that f is smooth path from a to y and g is smooth path from y to z. I define ...
1
vote
0answers
10 views

Bounds for the solution of heat equation using convolutions

We know that the solution of the heat equation $$u_t=u_{xx}$$ with initial condition $ u(0,x)=u_0(x)$ is given by $$u(t,x)=(u_o * H_t)(x)$$ where the heat kernel is given by ...
3
votes
1answer
38 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 ...
1
vote
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 ...
0
votes
0answers
28 views

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 ...
3
votes
1answer
44 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 ...
0
votes
1answer
26 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 ...
0
votes
1answer
32 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 = ...
0
votes
1answer
20 views

Observe approximation order of numerical solution of a Partial Differential Equation

When solving a Partial Differential Equation numerically, I estimated the approximation orders theoretically as follows, $$ u(x,t)= u_{h,k} + C_1 h^{p} +C_2 k^{q}, $$ where $ u_{h,k} $ is a numerical ...
2
votes
2answers
334 views

solving a PDE in 2 variables without boundary conditions

how could i solve the PDE (without boundary or other initial conditions) $ 1= y\partial _{y}f(x,y) -x \partial _{x}f(x,y) $
1
vote
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) ...
0
votes
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 ...
0
votes
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) ...
2
votes
3answers
22 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
1
vote
1answer
94 views

Partial differential equation (heat equation with other terms)?

Can some one help me solve the following PDE with the given intial and boundary conditions? $\gamma t\frac{\partial^{2}f}{\partial x^{2}}=t\frac{\partial f}{\partial t}-\alpha f$ Initial condition: ...
1
vote
1answer
117 views

Solution to the heat equation with mixed boundary conditions and step function.

PDE with the given intial and boundary conditions $\gamma \frac{\partial^{2}p}{\partial x^{2}}=\frac{\partial p}{\partial t}$ Initial condition: $p(x,t=0)=0$ Outer Boundary Condition: ...
1
vote
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 ...
0
votes
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 ...
16
votes
2answers
177 views

Why are $L^p$-spaces so ubiquitous?

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient ...
4
votes
1answer
87 views

Technical question about Strichartz estimate's proof.

I was studying the proof of Strichartz estimates from the book "Semilinear Schrödinger equation" of T. Cazenave. The proof is divided in several steps. Here we can assume ...
1
vote
0answers
28 views

Strichartz estimates for wave equations

Let's consider the wave equation $\Box u=F $ with $u(0)=g_0$ and $\partial_t g(0)=g_1$. Strichartz estimates tell us that $$\Vert u\Vert_{L^q_tL^r_x}+\Vert u\Vert_{C^0_t\dot{H}_x^s}+\Vert\partial_t ...
2
votes
0answers
38 views

use of Strichartz estimates

I have been learning(understanding) the Strichartz estimates. My naive Questions: (1) Can you give some ideas, how Strichartz estimates plays role in solving nonlinear dispersive equations(e.g. ...
4
votes
1answer
45 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 ...
2
votes
0answers
22 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 ...
0
votes
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 ...
8
votes
2answers
309 views

Compute $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$.

Compute the Fourier series for $x^3$ and use it to compute the value of $\sum\limits_{n = 1}^{\infty} \frac{1}{n^4}$. I determined the coefficients of the Fourier series, which are $$a_0 = ...
0
votes
0answers
47 views

Equations in two variables for second-order hyperbolic PDE

From pages 418-419 of PDE Evans... We begin by considering a general linear second-order PDE in two variables $$\tag{82} \sum_{i,j=1}^2 a^{ij} u_{x_i x_j} + \sum_{i=1}^2 b^i u_{x_i} + cu = 0,$$ ...
3
votes
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 ...
0
votes
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. ...
-4
votes
0answers
28 views

sde problem, may require Ito? [closed]

Given $dU_t = -\gamma U_t \, dt + dX_t$; How do I solve this equation for $U_t$?
0
votes
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} ...
4
votes
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 ...
1
vote
1answer
379 views

Find the general solution of the PDE

Find the general solution of the PDE $ xu_x-xyu_y-y=0 $ for all $u(x,y)$ and find the parametric form of the solution of the PDE which follows the side condition $ **u(s^2,s)=s^3** $ I got part ...
0
votes
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 ...
1
vote
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 ...
-1
votes
0answers
15 views

PDE with Robin boundary condition [on hold]

I need any advice (sugesseted book or a method etc..) to solve it
9
votes
1answer
169 views

Growth of Tychonov's Counterexample for Heat Equation Uniqueness

Define a function $\varphi$ on $\mathbb{R}_{+}$ by $$\varphi(t):=\begin{cases}e^{-1/t^{2}}, & {t>0}\\ 0, & {t\leq 0}\end{cases}\tag{1}$$ It is well-known that $\varphi$ is ...
7
votes
0answers
152 views
+100

Overview of nonlinear analysis, differential equations (ODE and PDE), dynamical systems, and mathematical physics, and their relationships

The fields of (i) nonlinear analysis, (ii) ODE and PDE, (iii) dynamical systems, and (iv) mathematical physics are very huge, fertile, and, in a sense, unorganized (see Open problems in ...
0
votes
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 ...
2
votes
1answer
317 views

Find all of the equilibrium points and describe the behavior of the $x' = \sin(x), y' = \cos(y) $.

Find all of the equilibrium points and describe the behavior of the $$x' = \sin(x), \quad y' = \cos(y) .$$ It has been a while since I took DE...Do we first need to set $x' = y'$ to solve for their ...
1
vote
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 ...
2
votes
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} + ...
1
vote
1answer
447 views

A question about Fisher's Equation and the Traveling Wave Equation

I am dealing with the Fisher's Equation: Consider the PDE: $u_t=u_{xx}+u(1-u)$, where $-\infty<x<\infty$ and $t>0$.Prove that there exists $c^*>0$ such that for each $c>c^*$ there ...
0
votes
1answer
24 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 ...
0
votes
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: $$u(0,t)=1$$ $$u_x\text{(1,t)=0}$$ ...
7
votes
0answers
60 views
+100

Why are diffeomorphism-invariant PDE not elliptic?

In reading geometric analysis papers, I frequently encounter a statement of the form, "The PDE in question is diffeomorphism-invariant, and therefore cannot be elliptic." My vague understanding is ...
0
votes
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 ...