While doing some estimates for PDEs I came across the following equation: 



> 
> $$
> y'(t) = \alpha(t) + \left( \int\_0^t y(\tau) \, d\tau\right)^\gamma, \qquad t \in [0,1]
> $$
> 
> 
> 


where $\alpha \colon [0,1] \to \mathbb R$ is some given function and $\gamma \ge 1$ is a fixed real number. I have a non-negative, bounded function $y$ satisfying this relation and I would like to derive some pieces of information about it. 


Of course I do not expect any risolutive formulas to hold, but hope is the last to die... do you have some hints? Any ideas on which kind of estimates I could derive?



An (hopeless) integro-differential equation
If I define $f(t)=\int\_0^t y(\tau)d\tau$, I need to solve
$$f''(t)=\alpha(t)+f(t)^\gamma.$$
For $\alpha\equiv 0$ this has the implicit solution


$${({\gamma}+1) f(t)^2 \left(c\_1 {\gamma}+c\_1+2 f(t)^{{\gamma}+1}\right)^2 \, \_2F\_1\left(\frac{1}{2},\frac{1}{{\gamma}+1};1+\frac{1}{{\gamma}+1};-\frac{2 f(t)^{{\gamma}+1}}{{\gamma} c\_1+c\_1}\right)^2}{}=\left(c\_2+t\right)^2\,\left(c\_1 {\gamma}+c\_1\right) \left(c\_1 ({\gamma}+1)+2 f(t)^{{\gamma}+1}\right)^2$$


Closed-form expressions in terms of special functions (Weierstrass $\wp$ and Jacobi elliptic function sn, courtesy of Mathematica) are possible for small integer $\gamma$:


$$f(t)=c\_1 e^t+c\_2 e^{-t},\;\;\gamma=1$$
$$f(t)=\sqrt[3]{6}\; \wp \left(\frac{t+c\_1}{\sqrt[3]{6}};0,c\_2\right),\;\;\gamma=2$$
$$f(t)=\pm\sqrt[4]{2} \sqrt{ic\_1}\; \text{sn}\left(\left.\frac{(-1)^{3/4} \sqrt{\sqrt{2} \sqrt{c\_1} t^2+2 c\_2 \sqrt{2} \sqrt{c\_1} t+c\_2^2 \sqrt{2} \sqrt{c\_1}}}{\sqrt{2}}\right|-1\right),\;\;\gamma=3$$


More generally, for constant $\alpha$, the implicit solution is
$$\int\_1^{f(t)} \left({2\alpha x+\frac{2}{\gamma+1}x^{\gamma+1}+c\_1}\right)^{-1/2} \, dx=c\_2+t$$
with explicit solutions


$$f(t)=-\alpha+c\_1 e^t+c\_2 e^{-t},\;\;\gamma=1$$
$$f(t)=\sqrt[3]{6} \;\wp \left(\frac{t+c\_1}{\sqrt[3]{6}};-2 \sqrt[3]{6} \alpha,c\_2\right),\;\;\gamma=2$$


