# Questions tagged [partial-differential-equations]

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

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### Solving an IVBP for a metal rod dipped from warm to cold water

I have to solve the diffusion equation for a solid rod, which is in a bath of 100 degrees. At $t=0$ it is moved to a second bath, where the water temperature is 0 degrees. I prepare the diffusion ...
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I have just finished 2nd year of my maths degree and I was wondering if anyone had any good recommendations for summer reading, I want to make sure I keep practising so that I don't feel underprepared ...
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### The method of separation of variables in solving a PDE

Suppose we have a PDE and we can solve it with the separation of variables and let's say the variables are $x,y,z$. Is the factorized function $f(x,y,z)=g(x)h(y)l(z)$ the only possible solution? Are ...
1 vote
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### Weak convergence in $H^1(\mathbb{R}^N)$ implies weak convergente in $H^1(B)$

Let $\{u_n\}$ be a bounded sequence in the sobolev space $H^1(\mathbb{R}^N)$ converging weakly to some $u \in H^{1}(\mathbb{R}^N)$. How can I prove that some subsequence converges to the same limite ...
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### Does Homogenity implies first order PDE

I have a pde say $$f(x,y,z, a,b)=0$$ with $x,y$ being independent variable and $z$ being the dependent variable. If i can write $z=f(x,y,a,b)$ and if $f$ is a homogeneous function then can i say that ...
1 vote
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### Proof of the embedding of time dependent Sobolev spaces

Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. $H^{-1}(\Omega)$ is the dual of $H_0^1(\Omega)$. For shorthand I write $\mathcal{H} = H^1(0,T,H_0^1(\Omega),H^{-1}(\Omega))$. I want to ...
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### does $u(1/2,1/2)>v(1/2,1/2)$

given $u_t=u_{xx}+e^t\sin(x)$ $u(x,0)=x^2$ $u(0,t)=u(1,t)=t^2$ and $v_t=v_{xx}+e^t\sin(x)$ $v(x,0)=x$ $v(0,t)=v(1,t)=t$ Does $u(1/2,1/2)>v(1/2,1/2)$? My attempt: Define $w=u-v$ and the system of ...
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### solve the pde without any initial condition $xu_x-xyu_y=u$

Solve of the following PDE using method of characteristics: $$(1)\qquad xu_x-xyu_y=u \qquad u(x,x)=x^2 e^x$$ $$(2)\qquad xu_x-xyu_y=u, \forall x,y$$ I assume that $(x,y)$ are functions of a parameter, ...
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### A $C^2$ boundary implies Interior sphere condition

I have problem when I try to read evan’s PDE in chapter 6. That is a $C^2$ boundary implies the interior sphere condition. Could anyone give some details of this proof? I think $C^2$ is used when we ...
1 vote
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### Similar to energy estimate in PDE

Let $B_1(0)$ be the unit ball in $\mathbb R^3$ centered at the origin. Assume that the function $v$ is a smooth function defined on $\mathbb R^3$ with $v_r = \frac{x\cdot\nabla v}{|x|}\in L^2(B_1(0))$....
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### Liouville's equation $\Delta u=K e^{ u}$ when $K<0$.

I'm quite interested in the elliptic PDE like $\Delta u = K e^{u}$ when $K<0$, when $K>0$, it's very easy talk about the existence of the solution but when $K<0$ it seems that we lose some ...
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### Symbol and principal symbol of differential operator

The symbol of a differential operator is defined as $L(x,p):=\sum_{|\alpha| \leq m}a_\alpha(x)p^\alpha$ and the principal symbol $L^p(x,p)$ is defined similar but with $|a|=m$. What would be the ...
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### Prove that a linear partial differential equation of first order can't be elliptic

A linear PDE of first order has the general form $$a(x,y)u_x+b(x,y)u_y-c(x,y)u-d(x,y)=0$$ A PDE is elliptic if the symmetric matrix of the coefficients of the highest derivatives has a determinant ...
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### Inequality $\theta\|{-\Delta u}\|_{L^2(U)}^2 \leq (Lu,-\Delta u),$ for elliptic operator

Let $U$ be the bounded smooth open subset of $\Bbb{R}^n$, with $u \in H^2 \cap H^1_0$. Let $L = \sum_{ij} (a_{ij}(x) u_{x^i})_{x^j} + \sum_k b_k(x) u_{x^k} + c(x) u$ be a general linear differential ...
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### Compact supports of initial data implies compact supports for all $t$ in semi-linear wave equation $-\frac{\partial^2}{\partial t^2}u+\Delta u=u^3$.

Problem: Let $u(t,x,y)$ be a smooth real function defined on $\mathbb R \times \mathbb R^2$ where $t \in\mathbb R$ and $(x,y) \in\mathbb R^2$. We assume that it solves the following semi-linear wave ...
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### Solve partial equation

$x z_x + y z_y =x+y+z$ Solve for z if $z(x,x+1)=2x+1$. I have used characteristic method to get $y=cx$. How to find the second solution to get the general solution?
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