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Results tagged with ap.analysis-of-pdes
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user 475539
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
1
answer
160
views
A priori estimates to $u_t - \Delta u = u^2$ [closed]
My research is now considering the a priori estimates on the equation
$$
\begin{cases}u_t - \Delta u = u \min(u,c) \\
u(0,y) = u(1,y)\\
u_x(0,y) = u_x(1,y)\\
\partial_n u(x,0) = \partial_n u(x,1) = 0
…
- 159
3
votes
A priori estimates to $u_t - \Delta u = u^2$
I figured out a way to show it, basically by separating the space and consider where $A^+ := \{u>0\}$ and $A^- := \{u<0\}$. Then the RHS reads
\begin{equation}
\int u^2 \min(u,c) = \int_{\{u\geq 0\}} …
- 159