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First a comment: in the context of nonlinear wave and Klein-Gordon equations, the venerable "ABC method" of Cathleen Morawetz is literally "mucking around until you see something". (The A, B, and C re …

Some of them: Multiply by $u_x$ and integrate, you get $$ \int u_{tt} u_x ~dx = 0 $$ so $$ \partial_t \int u_{t} u_x ~dx - \int u_t u_{tx} ~dx = 0 $$ the second term integrates to zero. Multiply b …

I just quickly read the proof you mentioned, and I think what is meant is following: Note that in Section 4.3 it is noted that any weak solution $U$ may be renormalized to be a continuous (in weak* t …

Javier already gave some very good references. Let me just add one more if you are thinking about classical field theories: Demetrios Christodoulou, Action Principle and Partial Differential Equations …

Here's one possible solution, but this may or may not be what your professor had in mind. Since $\lambda$ is a constant, we can ignore it by absorbing it into $g$. Assume $u$ is scalar (takes va …

For simplicity I'll assume $u$ is scalar valued, but I am pretty sure the discussion also works for $u$ that is a section of some vector bundle over $M$ (if the wave operator is quasidiagonal). Additi …