Can the Marshak Wave equation be solved using separation of variables?

[MathematicalPhysicist]

I have this PD equation:
$$T_t=\nabla ^2 T^4$$
where T=T(r,t) the above laplacian is spherical symmetrical (i.e only the spherical radial coordinate of the operator should be taken into account).
and $$Q_0=\int_{0}^{\infty}T(r,t=0)dr$$.

So I tried solving it by separation of variables but I get a tough ODE of the radial part.

Here's what I got
$$T(r,t)=F(r)G(t) \newline \frac{\frac{dG}{dt}}{G^4}=\frac{\nabla ^2 F^4}{F}=\lambda$$
Now after some manipulations I get for the radial equation the next equation:
$$F''(r)+3(F'(r))^2/F(r)+2F'(r)/r-\lambda/(2F(r))^2=0$$

And this is where I am stuck, any suggestion as to how to untie this equation, is even possible?

 

[jvicens]

Thank you for sharing your attempt at solving this PD equation. It is definitely a challenging problem and I can see why you are stuck. The ODE that you have obtained for the radial part is indeed difficult to solve. However, I believe there is still a way to approach this problem.

One possible approach is to use numerical methods to solve the ODE. Since the equation is nonlinear, it may not have an analytical solution. Therefore, numerical methods such as Euler's method, Runge-Kutta method, or finite difference methods can be used to approximate the solution. These methods involve breaking down the domain of the problem into smaller intervals and approximating the solution at each interval. With enough intervals, a good approximation of the solution can be obtained.

Another approach is to use perturbation methods. This involves expanding the solution in terms of a small parameter and solving the resulting equations using a series solution. This method can be used when the parameter in the equation is small, which may be the case in this problem.

I hope these suggestions can help you make progress in solving this PD equation. Best of luck to you!