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Apply $\partial_2$ to the first, $\partial_1$ to the second and sum. You find $$(x_1+x_2-1)\partial_1\partial_2(f_1+f_2)=0.$$ Away from the line $L:\,x_1+x_2=1$, you have $\partial_1\partial_2(f_1+f_2 …

By analytical, I presume that you mean explicit, or in close form. The known case so far is when $A$ and $B$ can be diagonalized in the same basis. Notice that the case where some eigenvalues come as …

I think that you should read L. Gaarding's seminal paper Linear hyperbolic partial differential equations with constant coefficients, Acta Math 85:1-62 (1951). It explains why hyperbolicity is the app …

A solution exists if and only if the following compatibility conditions are satisfied: $$\partial_xf_1=\partial_tf_2,\quad g'(t)=g(t)f_1(0,t),\quad u_0'(x)=u_0(x)f_2(x,0).$$ For the existence, you can …

This is not a transport equation. It is a conservation law. The difference between these class is that a TE is of the form $\partial_tu+a(t,x)\cdot\nabla_xu=0$, for which the essential supremum/infimu …

The theory of linear Intial-boundary value problem proceeds the following way. In your case, because of the zero initial condition, you may replace ${\mathbb R}^+$ by $\mathbb R$ and define $u\equiv0$ …

You made a confusion between the symbol (here $-(1+\xi^2)\tau^2+\xi^4$), and the principal symbol, which gathers the monomials of highest degree. Since the latter is $\xi^4-\xi^2\tau^2$, which splits …

There are a few, not many, books on hyperbolic equations. You might have a look to that of S. Benzoni-Gavage and myself: Multi-dimensional hyperbolic partial differential equations. First order system …

It seems you had a course on linear 2nd order scalar PDEs. All these words are meaningful but restrictive. Nowadays, the interesting PDEs are non-linear (for instance the 1M dollars prize for the Navi …

Since $u$ is nothing but a $2\pi$-periodic solution of $u_t=u_{xx}$, looking at critical points amounts to looking at the zeroes of $v:=u_x$, which is another solution of the same equation. Then your …

This is kind of meta-theorem. It has several version, and if someone was willing to write one theorem containing all the situations, it would be unreadible. The fully non-linear case (example $\part …