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Results tagged with ap.analysis-of-pdes
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user 1004
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
3
votes
Accepted
Nonharmonic solutions of Laplace's equation
Here's a counterexample in $n\ge3$ dimensions. With $\lVert\cdot\rVert$ denoting the Euclidean norm, set
$$
f(x)=\begin{cases}
x_1x_2x_3\lVert x\rVert^{-n-4},&{\rm if\ }x\not=0,\cr
0,&{\rm if\ }x=0.
\ …
- 17.1k
56
votes
Accepted
Unconditional nonexistence for the heat equation with rapidly growing data?
It is true that for any initial datum $u_0\in C^\infty(\mathbb{R})$ there exists a solution $u\in C^\infty(\mathbb{R}^+\times\mathbb{R})$ to the heat equation with initial condition $u(0,x)=u_0(x)$. A …
- 17.1k