# Tagged Questions

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

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### 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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### 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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### 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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### 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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### 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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### 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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### 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 -\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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### 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 ...
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 =...