Let $u:\Omega\times(0,\infty)\to(0,\infty)$ satisfies the heat equation $$ u_t=\Delta u, $$ then does the function $v(x)=u(\cdot,t)$ defined for every fixed $t$ satisfy the equation $$ \Delta u=0 \text{ in }\Omega? $$
2026-03-25 20:33:37.1774470817
Parabolic to Elliptic
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No, only the steady states of the equation have to satisfy this condition (Don't forget the boundary conditions!). Informally, you can view the parabolic equation as describing the evolution towards the solution of the elliptic equation.