Introduction to Nonlinear PDE - Key Properties

Non-uniqueness: The nonlinear ODE $u' = u^{1/2}$ with $u(0) = 0$ has infinitely many solutions. Similarly, nonlinear PDEs may have multiple weak solutions, requiring additional criteria (entropy, viscosity, variational) to select physical solutions.

Blowup: Solutions to $u_t = \Delta u + u^p$ with $p > 1 + 2/n$ can blow up in finite time: $\max u(t) \to \infty$ as $t \to T^*$. Critical exponent $p_c = 1 + 2/n$ separates global existence from finite-time blowup.

Scaling and Self-Similarity: Many nonlinear PDEs admit self-similar solutions $u(x,t) = t^{-\alpha}f(x/t^\beta)$. For Burgers: $u(x,t) = \frac{x}{t}$ is the rarefaction wave. For porous medium: Barenblatt profiles are exact self-similar solutions.

Conservation Laws and Entropy: Physically derived nonlinear PDEs often conserve quantities (mass, energy, momentum). For Burgers $u_t + (u^2/2)_x = 0$, entropy $\eta(u) = u^2/2$ with flux $q(u) = u^3/3$ satisfies $\eta_t + q_x \leq 0$ (entropy inequality, selecting physical shocks).