# Questions tagged [linear-pde]

Questions about linear partial differential equations. Often used in combination with the top-level tag ap.analysis-of-pdes.

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### Parabolic theory for singular coefficients on bounded domains (Reference Request)

In Evans, the theory for linear PDE of parabolic type with bounded coefficients is developed. There are nice results such as long-existence of weak solutions and the parabolic regularity theorems. Is ...
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### Using Darboux's to solve 2D system of first order linear PDEs with variable coefficients

I've spent some time over the last few days looking at the references suggested in this question and this question and I think the information therein is my best shot at solving this system that arose ...
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### Generalized functional for solution of PDEs

Asked this on Math Stack Exchange awhile ago but it got ignored then deleted. To solve a differential equation of one variable, you need constraints equal to the number of derivatives. For a partial ...
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### Maximum principle and linear transport

Let us consider the linear transport equation $$\partial_t u + \mathrm{div}(a(t,x)u)=0$$ with initial data $u(0,\cdot) = u_0$ in $\mathbb R^N$. Here we consider a smooth Lipschitz vector field $a$. ...
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### Solve a coupled PDE in a rectangle

We consider a coupled PDE in a rectangle $\Omega=(-1,1)\times(-1,1)$. For the simplicity, we assume that the functions are periodic in $x_{1}$ direction. \begin{cases} \nabla\cdot u=f_{1},\ & \...
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### Linear transport equation with unbounded coefficients

Consider the PDE $$\partial_t f(x,t) = \langle q(x), \nabla \rangle f(t,x) + p(x),$$ with Schwartz initial data $f(0,x) = f_0(x) \in \mathscr S(\mathbb R^n).$ I am wondering then if $q$ and all its ...
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### Is this equation of hyperbolic type?

I want to now whether this equation is of hyperbolic type: $$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$ with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$ I would say that the answer is yes. By ...
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### Fourier Transform; half space baby problem (new)

This question is related to a prior question i asked, see Fourier Transform ; half space elliptic baby problem. Essentially I am asking the same question now but taking a lot more care. So lets ...
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### Looking for a reference or the procedure on how to solve the parabolic equation with $L^2$-weight

Let $\zeta, u_0\in L^2(\Omega)$, with $\zeta \geq 0$ and $\Omega\subset \Bbb R^d$ open and bounded. \begin{equation}\label{Star-3.7} \begin{cases} \partial_t u -\Delta u + \zeta u=0 &\mbox{ in }\...
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### Solutions of constant coefficients differential operator on $\mathbb{R}^n$

Let $D$ be a constant coefficients linear differential operator on complex valued functions on $\mathbb{R}^n$. Let $\tilde D=\mathbb{F}^{-1}\circ D\circ \mathbb{F}$ be its conjugation with the Fourier ...
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### Calculating frequency of sound of ringing metal coin

I would like to reproduce the results of Manas - The music of gold: Can gold counterfeited coins be detected by ear?, but it skips a lot of steps, and the mathematics behind it is a bit advanced for ...
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### Perturbation in the equation $u_t=\epsilon Pu$, where $P$ is an elliptic partial differential operator
Let $Pu=\sum_{ij} \partial_j(a_{ij}(x) \partial_{i} u)$ be an elliptic operator. Consider the equation  (u=u_\epsilon)\\ \partial_t u=\epsilon Pu \text{ in } \mathbb R^+ \times \Omega,\\ u(0,x)=u_0(...