I'm solving:
$$\frac{\partial^2f}{\partial x_1^2}+\sum_{i=1}^{n}a_{i}\frac{\partial f}{\partial x_i}=0
\label{1}\tag{1}$$
wit $a_i$ some complicated functions of $x_i's$. If we didn't have the second order derivative, we could apply the method of characteristics and we would have the desired results. Instead, if $n=2$, we could apply the theory of ellipctic, hyperbolic and parabolic PDE's to proceed. However, in the general case of arbitrary $n$ it looks like the equation will be difficult to solve.
Questions
- Can we classify this second order, linear, homogeneous PDE further?
EDIT: Yes, we could say it is a parabolic PDE. - What techniques can be used for solving such equations?
In the case of $a_i=const.$ the equation can be solved using the Laplace-Fourier transforms assuming we have some adequate boundary/initial conditions. This technique can be even used for certain simple $a_i$, but to my understanding it fails in the general case. The same applies to the Green's function method.
Best Answer
Recently I have answered, in a different context, to a question involving the class to which this equation belongs thus, by using formerly produced materials, I think I can give a comprehensive answer to this one.
Precisely, this is a degenerate parabolic equation, since the matrix of the coefficients of the second order derivatives is evidently positive semi-definite, as it is $$ \mathbf{A}\triangleq(a_{ij})_{1,j=1,\ldots,n}= \begin{pmatrix} 1 & 0 & \ldots & 0 \\ 0 & 0 & \ldots & 0 \\ \vdots & \vdots &\ddots &\vdots\\ 0 & 0 & \ldots & 0 \end{pmatrix} $$ thus we have $$ \sum_{i,j=1}^n a_{ij} \xi_i\xi_j \ge 0\label{2}\tag{2} $$ for all real vectors $\boldsymbol{\xi}=(\xi_1, \ldots,\xi_n)\in\Bbb R^n.$ The degeneracy comes from the fact in \eqref{2} equality holds for at least one $\boldsymbol{\xi}\neq \mathbf 0$ (as a matter of fact, in our case this happens for all the vectors belonging to the hyperplane $\{\boldsymbol{\xi}\in\Bbb R^n: \xi_1=0\}$), thus it is degenerate as the Tricomi equation is.
Equations of this kind are called equations with nonnegative characteristics, and their theory is comprehensively addressed in the monograph [1] by Oleĭnik and Radkevič.
The main technique, which is de facto a bundle of techniques, is called elliptic regularization: a parameter-dependent, say $\varepsilon$-dependent, part is added to the the equation under analysis in order to transform the problem to an elliptic one, which can be studied with more or less standard techniques. The solution of the original problem is then recovered by letting $\varepsilon \to 0$ in the family of solution to the (elliptic-)regularized problem, ad done in [1], §1.5, pp. 30-41. Specifically, in our case we may construct the solutions to \eqref{1} by using the solutions to the following elliptic differential equation: $$ \frac{\partial^2f_\varepsilon}{\partial x_1^2}+ \varepsilon \sum_{i=2}^{n}\frac{\partial^2f_\varepsilon}{\partial x_i^2}+\sum_{i=1}^{n}a_{i}\frac{\partial f_\varepsilon}{\partial x_i}=0\quad \varepsilon >0 \label{3}\tag{3} $$ However, despite the apparent simplicity of the methodology described above, the study of equations with non-positive characteristics offers some hidden difficulties: for example, when solving the boundary problem for \eqref{1} in a sufficiently regular domain $\Omega$, there are (possibly finite measure) subsets of the boundary where it is not necessary to specify the boundary condition at all. These subsets are defined by the so called Fichera's function ([1], §1.1, p. 17) (named after Gaetano Fichera) and for example they influence also the statement of the maximum principle ([1], §1.1, theorem 1.1.2, pp. 21-22) for degenerate equations of nonpositive characteristics as \eqref{1} is. Definitely, in order to analyze such kind of equations, reference [1] is worth a look.
References
[1] Ol’ga Arsen’evna Oleĭnik, Evgeniĭ Vladimirovich Radkevič, "Second order equations with nonnegative characteristic form", (Russian) Itogi Nauki i Tekhniki. Seriya "Matematicheskii Analiz" 1969, 7-252 (1971), Zbl 0217.41502. Translated as the book:
Ol’ga Arsen’evna Oleĭnik, Evgeniĭ Vladimirovich Radkevič, Second order equations with nonnegative characteristic form, Translated from the Russian by Paul C. Fife. New York-London: Plenum Press, 1973, pp VII+259, ISBN: 0-306-30751-0, MR0457908.