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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.
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 …
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.
\ …