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It seems you had a course on linear 2nd order scalar PDEs. All these words are meaningfull meaningful but restrictive. Nowadays, the interesting PDEs are non-linear (for instance the 1M dollars prize for the Navier-Stokes equations). The type of a nonlinear equation is usually obtained by linearizing it around a constant solution. But it can display strange behaviour that linear ones don't. Example: harmonic maps with values in manifolds whose topology is non-trivial. Also, the linear theory is essentially a part of functional analysis (linear operators over topological vector spaces), but the nonlinear theory is much more involved. Sometimes, there is no theory at all! Example: hyperbolic systems of conservation laws in several space variables (my speciality).

Going back to your questions:

Question 1. You are correct as long as your unkown is scalar. When it is vector valued, it is slightly more involved. A system of PDE writes $A(\nabla)\vec u=0$, where $A$ is a matrix of differential operators. The principal symbol $A_0(\xi)$ has some homogeneity: $a_{ij}(\xi)$ is a homogeneous polynomial of degree $\mu_i-\nu_j$. The system is elliptic if $A_0(\xi)$ is non-singular for every $\xi\ne0$.

Question 2. A parabolic operator is more delicate to define because its principal part is not of homogeneous order. I am not sure that there is a general enough definition of it. Hyperbolicity is very interesting. You have to distinguish a direction $\vec e$ which you call time-like. Let me focus on scalar equations and consider the principal part, say of order $n$. Its symbol $P(X)$ is a homogeneous polynomial of degree $n$. It is hyperbolic if $P(\vec e)\ne0$ and for every $\xi$, the univariate polynomial $t\mapsto P(t\vec e+\xi)$ has its $n$ roots real. Therefore hyperbolicity has a lot to do with real algebraic geometry. The Petrowsky school (in particular O. A. Oleinik) is famous in this field.

Question 3. This is correct.

Question 4. The $x$-dependence of the coefficients makes the analysis more difficult, but it does not change the classification above. The situation is different when you consider operators that do not fall in these three categories. The case of sub-elliptic operators is especially interesting. Hormander remarked that an operator of the form $X_0+X_1^2+\cdots+X_r^2$, where the $X_j$'s are smooth vector fields, has ''elliptic properties'' if the Lie algebra spanned by the vector fields has full rank at every point $x$. This applies for instance to the Fokker-Planck operator $\partial_t-\Delta_v-v\cdot\nabla_x$.

Question 5. It is like medecine medicine in a hospital. One theory is not enough. You need to know a lot of different branches of mathematics. As much as you can. Recently, I worked on a problem from gas dynamics and I reproved the existence of complete minimal surfaces in a Riemann space of negative curvature... without knowing it. Until a colleague pointed it out to me.

1

It seems you had a course on linear 2nd order scalar PDEs. All these words are meaningfull but restrictive. Nowadays, the interesting PDEs are non-linear (for instance the 1M dollars prize for the Navier-Stokes equations). The type of a nonlinear equation is usually obtained by linearizing it around a constant solution. But it can display strange behaviour that linear ones don't. Example: harmonic maps with values in manifolds whose topology is non-trivial. Also, the linear theory is essentially a part of functional analysis (linear operators over topological vector spaces), but the nonlinear theory is much more involved. Sometimes, there is no theory at all! Example: hyperbolic systems of conservation laws in several space variables (my speciality).

Going back to your questions:

Question 1. You are correct as long as your unkown is scalar. When it is vector valued, it is slightly more involved. A system of PDE writes $A(\nabla)\vec u=0$, where $A$ is a matrix of differential operators. The principal symbol $A_0(\xi)$ has some homogeneity: $a_{ij}(\xi)$ is a homogeneous polynomial of degree $\mu_i-\nu_j$. The system is elliptic if $A_0(\xi)$ is non-singular for every $\xi\ne0$.

Question 2. A parabolic operator is more delicate to define because its principal part is not of homogeneous order. I am not sure that there is a general enough definition of it. Hyperbolicity is very interesting. You have to distinguish a direction $\vec e$ which you call time-like. Let me focus on scalar equations and consider the principal part, say of order $n$. Its symbol $P(X)$ is a homogeneous polynomial of degree $n$. It is hyperbolic if $P(\vec e)\ne0$ and for every $\xi$, the univariate polynomial $t\mapsto P(t\vec e+\xi)$ has its $n$ roots real. Therefore hyperbolicity has a lot to do with real algebraic geometry. The Petrowsky school (in particular O. A. Oleinik) is famous in this field.

Question 3. This is correct.

Question 4. The $x$-dependence of the coefficients makes the analysis more difficult, but it does not change the classification above. The situation is different when you consider operators that do not fall in these three categories. The case of sub-elliptic operators is especially interesting. Hormander remarked that an operator of the form $X_0+X_1^2+\cdots+X_r^2$, where the $X_j$'s are smooth vector fields, has ''elliptic properties'' if the Lie algebra spanned by the vector fields has full rank at every point $x$. This applies for instance to the Fokker-Planck operator $\partial_t-\Delta_v-v\cdot\nabla_x$.

Question 5. It is like medecine in a hospital. One theory is not enough. You need to know a lot of different branches of mathematics. As much as you can. Recently, I worked on a problem from gas dynamics and I reproved the existence of complete minimal surfaces in a Riemann space of negative curvature... without knowing it. Until a colleague pointed it out to me.