# Questions tagged [pde]

Questions on partial (as opposed to ordinary) differential equations which involving partial derivatives of one or more dependent variables with respect to more than one independent variables.

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### Some questions on Alfréd Rényi's paper ON THE THEORY OF ORDER STATISTICS

when I read ON THE THEORY OF ORDER STATISTICS written by Alfréd Rényi, I found this statement, and I can't figure out why this satisfied, could anyone give me some idea? A necessary condition of ...
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### How does one show that $\biggl(\dfrac{\partial}{\partial{t}}-\dfrac{\partial^{2}}{\partial x^{2}}\biggr)$ is a linear operator?

How does one show that $\dfrac{\partial{u}}{\partial{t}}-\dfrac{\partial^{2}{u}}{\partial{x^{2}}}$ is a linear operator? At first, I was thinking that there might be some distributive property that I'...
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### Boundedness of travelling front norm

For a reaction diffusion equation $u_t=u_{xx}+f(u)$, assume that $\mu_1<\mu_2$ are equilibria and $V(\xi), \xi=x-ct$ with $c>0$ is a traveling front, where $\lim_{\xi\to -\infty}V(\xi)=\mu_2$ ...
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### How do we get the function $f(y')=f(bx-ay)$?

The following is from Partial Differential Equations by Strauss: Let us solve $au_{x} + bu_{y} = 0$, where $a$ and $b$ are constants not both zero. Coordinate Method Change variables (or “...
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### Steady states of $u_t= u_{xx}+\pi^2u$

I just put the following one-dimensional reaction-diffusion equation in Mathematica: $$u_t= u_{xx}+au$$ with $\Omega=(0,1)$ with Dirichlet boundary conditions. When $a<9$, no matter the initial ...
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### Long time stability in nested Bochner spaces

Let $V\subset H$ be densely, compactly and continuously embedded, which, among others, means that $$\|x\|_H \leq \|x\|_V\quad (*)$$ for all $x\in V$. I have the following stability estimates for a ...