calculation – Could this particularly symmetric nonlinear PDE provide a solution?

For a physics problem that interests me, I am looking for a (partial) solution to the following particularly symmetric EDP:

$$
fc {(1,0)} (t, x)} {f {{0,1}} (t, x)} = g_1 (t) h_1 (x) + g_2 (t) h_2 ( X)
$$

In this PDE, all functions and variables have a real value. $ x $ has the meaning of a spatial coordinate and $ t $ has sense of time. Exhibitors in parentheses indicate the order of differentiation. respective arguments. While the two functions of time $ g_1 (t) $ and $ g_2 (t) $ can at best be considered unspecified1, the two functions of space $ h_1 (x) $ and $ h_2 (x) $ are given by simple algebraic expressions:

  • $ h_1 (x) = + x (x ^ gamma-1) $
  • $ h_2 (x) = – x (x-1) $,

or $ gamma $ is a real positive number. Since the two time-dependent functions are left undetermined, some information about the form (in space) from $ f (t, x) $. Approximations, serial solutions or simply information on certain borderline cases would already be welcome.

Normally, when I solve particular nonlinear differential equations, I am able to discover a mathematical literature that deals with them. However, in this case, I am not sure of the term to look for. Therefore, I would also be grateful for any indication of sources of information that may help me to solve this problem on my own. In case this information is omnipresent and I could have found it, I apologize.

There are some constraints that $ f (t, x) $ should adhere to:

  • The domain of $ f (t, x) $ is $[0infty)$[0infty)$[0infty)$[0infty)$, $ (0,1)

    $ and his codomaine is $[0infty)$[0infty)$[0infty)$[0infty)$.

  • $ f (t, 1) = 0 $
  • $ lim_ {x downarrow 0} f (t, x) = + infty $.
  • $ f (t, x) $ is a monotonous decreasing function of $ x $.
  • $ lim_ {x downarrow 0} f ^ {(0,1)} (t, x) = – infty $
  • $ lim_ {x uparrow 1} f ^ {(0,1)} (t, x) = – infty $

In short, the inverse function of $ f (t, x) $ w.r.t. his second argument might look like a Gaussian or a Lorentzian. In fact, the EDP considered here is derived from a much more horrible PDE, formulated according to this inverse, as well as other considerations.


1 They each depend on their own set of non-linear evolution equations for which, I am quite sure, have no solution or approximation. I think drawing these equations in the context of this question probably would not improve it.