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

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5
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0answers
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Energy inequalities with negative sobolev number.

Let $\phi\in H^{s}$ such that the following energy inequality is true: $$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\cdot)\|_s \, dt $$ where $P$ is a strictly hyperbolic linear operator. For ...
6
votes
2answers
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Why do we take the odd extension?

When we have the initial and boundary value problem $$u_{tt}(x,t)-c^2u_{xx}(x,t)=0, x>0, t>0 \\ u(0,t)=0 \\ u(x,0)=f(x), x \geq 0 \\ u_t(x,0)=g(x), x \geq 0$$ can we apply Green's theorem or ...
7
votes
1answer
119 views
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Nonlinear heat equation $u_{t} = \Delta(u^{4})$

Consider the nonlinear heat equation $u_{t} = \Delta(u^{4})$ in $\{x \in \mathbb{R}^{3}: |x| < 1\}$ with $u = 0$ on $\{x \in \mathbb{R}^{3}: |x| = 1\}$. The problem I am working on is to show that ...