Questions tagged [cauchy-problem]
Use this tag for questions about partial differential equations that satisfy certain conditions given on a hypersurface in the domain.
314 questions
3
votes
2
answers
79
views
Problem on choosing the sign of the solution of a PDE using Fourier series
I'm trying to solve the following PDE using the Fourier series method:
\begin{align}
&\partial_t u(t, x) - t\partial^2 u(t, x) = 0 && x \in [0, \pi], \; t\in\mathbb{R}^+ \\
&u(t, 0) = ...
- 492
0
votes
1
answer
80
views
Existence and uniquess of a global solution
Consider che Cauchy problem:
$\begin{cases} u'=f(u) \\
u(0)=0
\end{cases}$
where $f: \mathbb{R} \to \mathbb{R}$ is continuous. Prove that if $0$ is an isolated zero of $f$ and $f$ is differentiable at ...
- 409
0
votes
0
answers
39
views
Global solution for ODE problem
Is is true that if $f:\mathbb{R}\to\mathbb{R}$ is a $C^1$ function then the problem:
$$\begin{cases}y'(t)=f(y(t)), t\geq 0\\ y(0)=y_0\in\mathbb{R} \end{cases}$$
has a unique global solution $y:[0,\...
- 2,172
0
votes
0
answers
82
views
Cauchy Problem (ODE) and Picard Theorem
In $y^2 + x^2 \leq 2^2 $ we have the following Cauchy problem $$y' = \cos(x) - 1 - y^2 ,y(0)=1$$
a. Show that there is a $x^* \in [-1,1] $ such that $\displaystyle \lim_{x \to x^*} y(x) = \infty $.
b. ...
- 325
0
votes
0
answers
34
views
Uniform bound for the spatial derivative of the solution of a semilinear parabolic PDE
Consider the semilinear, second-order, parabolic PDE
\begin{align*}
-u_t + a(t, x)u_{xx} + b(t,x,u)u_x &= 0, \\
u(0, x) &= f(x),
\end{align*}
for $(t,x)\in[0, T]\times\mathbb{R}$.
I am ...
- 165
3
votes
1
answer
85
views
Extension of the solution of a Cauchy problem
I am trying to study the solution of the following Cauchy problem:
\begin{align*}
y'&=\frac{x^2y^2}{1-y}e^{-xy}\\
y(0)&=\frac{1}{2}
\end{align*}
The existence and uniqueness of the local ...
- 871
0
votes
2
answers
123
views
A proof on a Cauchy problem
I am trying to prove the following statement:
Let $y(x):I\rightarrow \mathbb{R}$ be the solution, defined on the maximal interval $I\subset \mathbb{R}$ of the Cauchy problem
\begin{equation*}
y'=y^2+f(...
- 871
0
votes
0
answers
58
views
Exercise on Cauchy problem
I am trying to study the following Cauchy problem:
\begin{equation*}
\begin{cases}
&y'=(1+\arctan(ty))\cos(y)(1+\sin(y)) \\
&y(0)=0
\end{cases}
\end{equation*}
I found that the solution $y^*$ ...
- 871
1
vote
0
answers
41
views
On the IVP $\frac{dy}{dx} = x^2 + e^{-y^2}, \quad y(0) = 0. $
Using Picard's Existence and Uniqueness Theorem, find the largest interval of existence and uniqueness of the solution for the initial value problem (IVP): $$
\frac{dy}{dx} = x^2 + e^{-y^2}, \quad y(...
- 6,580
0
votes
0
answers
74
views
Solving PDE with conditions $x^2 + y^2 = a^2$, $u = h$
We consider the Cauchy problem associated to the linear PDE
$$x\left(x^2 + y^2\right)u_x + 2y^2\left(xu_x + yu_y - u\right) = 0,$$
$$x^2 + y^2 = a^2, \text{ } u = h.$$
We must study the existence of ...
0
votes
1
answer
56
views
Existence of solution of Cauchy problem in a neighbourhood of $\left(x_0, 0\right)$
We consider the Cauchy problem
$$x(u - x)u_x + u(u - y)u_y = 0,$$
$$u(x, 0) = 2x.$$
We have to study the existence of solution in a neighbourhood of each $\left(x_0, 0\right)$, $x_0 \in \mathbb{R}$.
...
0
votes
0
answers
26
views
Cramer family of distribution
Show that one parameter Cauchy distribution with pdf,
f(x,theta)=(1/pi)*(1/(1+(x-theta)^2)), x and theta are real valued
Belongs to Cramer family, Prove all 5 ...
2
votes
0
answers
55
views
Dense subset of $C^{1}_{c}([0, T); V)$, where $V$ is a Hilbert Space and $0< T < \infty$. [closed]
I was trying to understand the definition of a weak solution of an evolution equation of first order. And I got confused with the following questions-
Question 1. Can we say $C^{1}_{c}([0, T)\times \...
- 189
0
votes
1
answer
76
views
Global existence for quadratic ODE
It is known systems of ODEs with a locally Lipschitz vector field can only have local existence results, as the solutions may blow up in finite time. I wonder if anything can be said on selected ...
- 127
1
vote
0
answers
34
views
What is the Propagator associated to a homogeneous Cauchy problem?
I came across this problem:
Consider the following Cauchy problem on $\mathbb{R}_t\times\mathbb{R}_x^n$
$$\begin{cases}\partial_tu+\omega\cdot\nabla_xu=f(t,x),\\ u|_{t=0}=u_0(x)\end{cases}$$
where $\...
- 1,173
3
votes
0
answers
160
views
Non-autonomous system of two nonlinear ordinary differential equations with conditions
Consider the ODE system:
$$
\frac{df}{dx}= -\sqrt{g},\tag{1}
$$
$$
\frac{dg}{dx}= -\sqrt{x}f,\tag{2}
$$
where $f=f\left(x\right)$ and $g=g\left(x\right)$ are the functions on the interval $x\in\left[0,...
0
votes
1
answer
81
views
Issue about the Cauchy problem of separable variables differential equation
Consider the Cauchy problem
$$\begin{cases}x'(t)=f(x) \\ x(0)=x_0\end{cases}$$
with $t\in \mathbb{R}$ and $f:I\subseteq\mathbb{R} \longrightarrow\mathbb{R}$.
If $f(x_0)=0$, then $x(t)=x_0$ is the ...
- 1,256
4
votes
0
answers
125
views
Attenuation estimation of the solution of the two-dimensional wave equation Cauchy problem
This is the equation given,
$$\begin{array}{l}
u_{tt}=a^{2}\left(u_{x x}+u_{y y}\right), \\
\left\{\begin{array}{l}
\left.u\right|_{t=0}=\varphi(x, y), \\
\left.u_{t}\right|_{t=0}=\psi(x, y) .
\end{...
- 51
1
vote
1
answer
60
views
Extreme points of set of bounded measures satisfying continuity equation
$\textbf{Ordinary Differential Equation:}$
Let $x(\cdot) \in C([0,1];\mathbb{R}^n)$ (C($\cdot$) denotes the set of continuous functions) be the trajectories satisfying the following differential ...
- 11
1
vote
0
answers
68
views
Solving Cauchy problem PDE and drawing the solution graph
I want to solve and draw the solution of PDE such that
\begin{align}
\rho_t +2\rho\rho_x=0
\end{align}
and with initial condition
\begin{align}
\rho(x,0)=
\begin{cases}
1 & \text{if $x<0 $}\\ ...
- 717
1
vote
1
answer
62
views
Logistic equation and Cauchy - Lipschitz theorem
I consider the following differential equation
$$
x’(t)= rx(t)(1-\frac{x(t)}{K})
$$
Where $r$ and $K$ are constant.
I consider an initial condition $x(0)=x_0\in (0,K)$.
In my lecture notes, it is ...
- 1,657
0
votes
1
answer
121
views
Elementary proof of Cauchy-Lipschitz's theorem in a simple case
Let $a,b,c \in \mathbb{R}$, with $a \neq 0$. we consider the linear homogeneous differential equation : $(E) : ay''+by'+cy=0$.
Is there an "elementary" way (i.e. without invoking "big&...
1
vote
0
answers
117
views
Local Lipschitz continuity: existence and uniqueness of a solution to a ODE Cauchy problem
The version of the Picard–Lindelöf that I know, which is possibly one of the most common statements of the theorem states that, given a closed rectangle $D\subset \mathbb{R}\times\mathbb{R}^n$ and let ...
- 528
1
vote
0
answers
26
views
Qualitative study of the solution of the Cauchy problem: $x' = \log x$, $x(0) = a$ where $a > 0$
I have the following exercise:
Consider the Cauchy problem: $$\begin{cases}x' = \log x \\ x(0) = a \end{cases} \quad \text{where } a > 0 .$$ Perform a qualitative study of the solution of the ...
- 626
0
votes
0
answers
50
views
Extending solutions of a differential equation
I have a following exercise:
Prove that every solution of an equation $x' = \sin(x^2+t^2) + 3|x|$ can be extended to all $t$ in real numbers.
How do I go about proving this? I know theorems stating ...
0
votes
0
answers
44
views
Show supremum norm of solution to ode is less than $k$ i.e. $||e^{At}|| \leq k$.
So I'm trying to make an exercise in ode's.
Let $A \in M_5(\mathbb{R})$ be a matrix with 3 eigen-values $\lambda_1 = -1$ with multiplicity $3$ and $\dim(\text{Ker}(A+I))=1$, $\lambda_{2,3} = \pm 2i$. ...
- 59
1
vote
0
answers
38
views
Help with the solution to the Cauchy problem of the firdst order partial differential equation (quasi linear case)
I am looking for the help with the solution to the following PDE (along with IC):
$$ uu_x + u_t = u, u(x, 0) = 2x $$
Solution (without parameterization):
Applying the method of characteristics one can ...
-2
votes
1
answer
61
views
How do you solve Cauchy problem to the first order PDE [closed]
How do you solve Cauchy problem to the first order PDE of the form:
$y^{-1}u_x+u_y=u^2, u(x,1)=x^2$
0
votes
0
answers
34
views
Cauchy conversion Criteria Integral
I have the following question: I want to show, that the improper integral exists:
$$\int_{0}^{b}\tfrac{\sqrt{1+y'(x)^2}}{\sqrt{y(x)}}$$
I stated: For $$[t,b] \subset(0,b]$$, all t must be in the ...
1
vote
0
answers
64
views
A particular Cauchy problem for first-order PDE
Is the following result true?
Suppose $M$ is a smooth manifold, $W\subseteq J^1M\cong M\times T^\ast M$ is an open subset, $F\colon W\to \mathbb R$ is a smooth function, and $(x_0,u_0,p_0)\in W$. Let ...
- 428
1
vote
1
answer
148
views
Linear kinetic PDE: Characteristics of the transport operator are given by the flow a Hamiltonian
I am trying to read and understand the article "Hypocoercivity for linear kinetic equations conserving mass." by Dolbeault, Mouhot, Schmeiser. doi: 10.1090/s0002-9947-2015-06012-7 (https://...
- 281
3
votes
1
answer
64
views
About differentiable dependence in a Cauchy problem
Let $\epsilon >0$, consider the Cauchy problem: $$\epsilon x' = x^2 + (1-\epsilon)t \quad , x(0)=1$$
If $x(t;\epsilon)$ denotes the solution (defined on the maximum interval) of the problem, I'm ...
- 195
2
votes
1
answer
260
views
On number of solutions of ODE $y’=-y^a, y(0)=0$.
How many solutions the ODE of the type $y’=-y^a, y(0)=0, 0<a<1$ have? I am able to prove that the IVP $y’=y^a, y(0)=0$, where $0<a<1$, has infinite
number of solutions by finding it’s ...
- 6,580
1
vote
0
answers
34
views
When can we prescribe the speed of a normal flow from a hypersurface?
Suppose that $X_0:M^n \rightarrow \mathbb{R}^{n+1}$ is a smooth embedding of a compact hypersurface. I'm given a scalar function $\eta \in C^{\infty}\left( M \times [0, T) \right)$ and I'd like to ...
- 11
1
vote
0
answers
38
views
Does this Cauchy problem $x'=tx^{2/3}$, $x(0)=0$ have unique solution? [duplicate]
Consider the following Cauchy problem:
$$
\left\{ \begin{array}{lcc}
x'=tx^{2/3} \\
\\ x(0)=0 \\
\end{array}
\right.
$$
I proceed as usual. First we notice that $x(t)=0 \ \forall t \in \mathbb{R}$ ...
- 1,374
0
votes
1
answer
118
views
Exercise1.1 of Tao's book <Nonlinear Dispersive equation> which is about Cauchy Kowalevski Theorem
In the book of Tao's nonlinear dispersive equations Exercise1.1, the author want us to prove $$\|\partial^{m}_{t}u(0)\|_{\mathcal{D}}\leq K^{m+1}m!$$ use the equation $\partial_{t}u=F(u(t))$ where $F:\...
11
votes
1
answer
552
views
How to prove Picard's existence and uniqueness theorem by Tonelli sequence instead of Picard sequence? For O.D.E./ODE.
$\qquad$First of all, this question for O.D.E. comes from an end-of-book exercise with no answer.
$\qquad$Secondly, allow me to give the definitions of the relevant contents in the question to avoid ...
- 317
3
votes
0
answers
157
views
Problem with the differential equation $2u_{xx}-3u_{xy}+u_{yy}+u_x-u_y=1$
Specify the largest domain in which the given Cauchy problem has a single solution, and find this solution
$$2u_{xx}-3u_{xy}+u_{yy}+u_x-u_y=1, \; u\Bigg|_{x=0,y>0}=-2y, \; u_x\Bigg|_{x=0,y>0}=-1$...
user639597
4
votes
1
answer
166
views
A relation between Dirichlet problem and Brownian motion
I'm reading about Dirichlet problem and Brownian motion in these notes, i.e.,
Fact. Let $D$ be an open and bounded domain in $\mathbb{R}^n$ and $\partial D$ be its (smooth) boundary. Let $h \in \...
- 18.8k
0
votes
1
answer
891
views
Solve the Cauchy problem by the method of characteristic $pz+q=1$ with initial data $y=x,z=x/2$ . Indicate the region where the solution is valid.
Solve the Cauchy problem by the method of characteristic $pz+q=1$ with initial data $y=x, z=x/2$. Indicate the region where the solution is valid.
How to solve this problem.
Lagrange's auxiliary ...
- 173
1
vote
0
answers
72
views
solution of EDO defined on a specific interval
How can I prove that$$
\left\{\begin{array}{l}
y^{\prime}(t)=1-(t+1) e^{y(t)} \\
y(0)=y_0
\end{array}\right.
$$
the solution is defined in $[0,+\infty)?$.
My teacher said first to prove that $e^{y(t)} ...
- 595
2
votes
0
answers
56
views
Can you solve $y'(x) = y^2(x)\;\land\;y(x_0) = y_0 $ on $]-1, 1[$ using the Banach contraction theorem?
I would like to solve the following Cauchy problem:
$$
\begin{cases}
y'(x) = y^2(x)\\
y(x_0) = y_0
\end{cases}\tag 1
$$
In my opinion, using the Banach contraction theorem it can only be solved in $I:=...
user1107523
-2
votes
1
answer
95
views
Application of Cauchy's integral formula [duplicate]
Let $f(z)$ be analytic in a neighbourhood of $z_0$, where $f'(z_0)$ does not equal $0$. Show that
$$\int_C\frac{\mathrm{d}z}{f(z)-f(z_0)} = \frac{2\pi i}{f'(z_0)}$$
where $C$ is a small (as small as ...
- 53
2
votes
1
answer
211
views
Unqueness of the weak solutions of transport equation
Let $a\in \mathbb{R}$ and consider the IVP
\begin{eqnarray}
u_t+au_x&=&0 \quad \quad (x,t) \in \mathbb{R} \times \mathbb{R}^+\\
u(x,0)&=&u_0(x) \quad \quad x \in \mathbb{R}.
\end{...
- 1,107
1
vote
1
answer
155
views
Find initial condition so that ODE has multiple solutions
Given Cauchy problem
$$y'=\dfrac{x^2}{y(1+x^3)}, \;y(x_0)=y_0.$$
Is it possible to find $(x_0,y_0)$ such that given Cauhcy problem has multiple solutions?
It is obvious that equation is with separable ...
- 356
2
votes
1
answer
178
views
Convergence of solution of non linear problem using heat equation
I am considering the following non linear problem:
$u_{t}(t,x)-u_{xx}(t,x)+(u_{x}(t,x))^2=f(t,x)$ for $t>0, x \in (0,1)$
$u(0,x)=u_{0}(x)$ for $x \in [0,1]$
$u(t,0)=u(t,1)=0$ for $t>0$
where f ...
- 373
1
vote
1
answer
174
views
How can I find the solution of the Hopf equation $u_t + u u_x = 0$ with the initial condition $u(0,x) = \rho(x)$?
Here, $\rho$ is equal to:
\begin{align}{\rho(x) = }
\begin{cases}
a & x<0 \\
\frac{a}{L} (L-x) & 0 \le x \le L \\
\frac{a}{L} (x-L) & L < x < 2L \\
a & x \ge 2L
\end{cases}
...
- 317
0
votes
2
answers
79
views
PDE question that hard to see second first integral
Solve the IVP. $x\dfrac{\partial z}{\partial x}+y\dfrac{\partial z}{\partial y} = z- x^2-y^2, \; z\vert_{y=-2} = x-x^2 $
$\underline{\text{My Attempt:}}$
I first wrote Characteristic Equation for the ...
4
votes
1
answer
231
views
Does the solution of $y' = (x^2 + y^2) e^{-(x^2+y^2)}$ have a limit for $x \to \infty$?
An old exam problem I am trying to solve is as follows:
Given the cauchy problem $y' = (x^2 + y^2) e^{-(x^2+y^2)}, y(x_0) = y_0$, do the following:
Show that there is a unique solution for all $x \in ...
- 1,908
3
votes
0
answers
71
views
Solve the Initial-Value Problem $\quad x\frac{\partial u}{\partial x}+ u\frac{\partial u}{\partial y}\,= u + 2x^2$
Solve the initial value problem:
$$\,$$
$$ x\;\frac{\delta u}{\delta x}\,+\, u\;\frac{\delta u}{\delta y}\,=\, u + 2x^2,\qquad \mathrm{ with\;initial\, conditions}:\,u\,(x,\,\frac{1}{4}-x^2)=x$$
$$\,$$...
