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Results tagged with ap.analysis-of-pdes
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user 176379
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
1
vote
0
answers
68
views
How to check that $f(t,x)\ge 0$ for any $x\ge 0$?
Suppose $f(t,x)$ satisfices $\partial_t f=\Lambda^{-\alpha}f\partial_xf$, for $0<\alpha<1$ and where $\Lambda=(-\partial_{xx})^{1/2}$, $f_0(x)=f(0,x)$ is odd, and $f_0(x)\ge 0$, $\forall x\ge 0$. We c …
- 171
3
votes
1
answer
204
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How to use comparison principle to prove the following inequality about Laplace equation?
Assume that $\Omega$ is a bounded connected domain and $\partial \Omega \in C^{\infty}$. Denote $\Gamma_1,\Gamma_2,\cdots,\Gamma_n$ are $n$ connected components of $\partial\Omega$. This notation lead …
- 171
2
votes
0
answers
59
views
How to prove this this integral equality which contain nonlocal operator, $(-\partial_{xx})^...
Suppose that $\theta(t,x)$ is even about $x$ and is smooth. $0\le \gamma<1/2$, $0<\delta<1-2\gamma$. $\Lambda=(-\partial_{xx})^{1/2}$
My Question: How to prove that
$$
\int_0^{\infty} \frac{(\Lambda^{ …
- 171