Questions: Nonlinear PDEs Introduction

Question 1

Which of the following is a semilinear PDE?

• A. -Δu + u³ = 0
• B. div(|∇u|^{p-2}∇u) = 0 (p-Laplacian)
• C. det(D²u) = f (Monge-Ampere)
• D. u_t + uu_x = 0 (Burgers)

Answer: (A)

Semilinear means the highest-order derivatives appear linearly (here, -Δu is linear in second derivatives) while lower-order terms are nonlinear (u³). The p-Laplacian is quasilinear, Monge-Ampere is fully nonlinear, and Burgers is quasilinear (first-order).

Question 2

The superposition principle holds for nonlinear PDEs.

• T. True
• F. False

Answer: False

If u₁ and u₂ are solutions of a nonlinear PDE, their sum u₁ + u₂ is generally NOT a solution. This is the fundamental difference from linear theory: we cannot build general solutions from particular ones. Each nonlinear problem must be attacked individually.

Question 3

What is finite-time blow-up in nonlinear PDEs?

Model answer: The solution becomes unbounded (||u(·,t)|| → ∞) at some finite time T* < ∞, so no global solution exists

Blow-up is a genuinely nonlinear phenomenon absent from linear PDEs. For example, the ODE u_t = u² has solutions that blow up at t = 1/u₀. Similarly, the semilinear heat equation u_t = Δu + u^p can have solutions that become infinite in finite time when p > 1, depending on the initial data.

Question 4

The p-Laplacian equation div(|∇u|^{p-2}∇u) = 0 for p ≠ 2 is:

• A. Quasilinear
• B. Semilinear
• C. Fully nonlinear
• D. Linear

Answer: (A)

The highest-order term involves second derivatives of u, but its coefficients depend on the first derivatives of u (through |∇u|^{p-2}). This makes it quasilinear. For p = 2 it reduces to Laplace's equation.