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The generalized Korteweg–De Vries equation is the nonlinear partial differential equationThe cases below the exponent $p=5$ are energy-subcritical in $H^1(\mathbb{R})$.

The porous medium equation $u_t=\Delta (u^m)$ with $m>1$ is a nonlinear partial differential equation whose diffusivity vanishes at $u=0$. Unlike the heat equation, it has compactly supported solutions with finite propagation speed.