Laplace Transform of $t^{t^2}$

Question

I have been trying to model a dynamic system and I came up with a differential equation which involved me finding the laplace transform of $t^{t^2}$. I have tried all the theorems I am familiar with, and I have not been able to emerge at a solution. Can someone please put me on the right path ?

Answer 1

$\mathcal L\{f(t)\}=$$\int_0^\infty f(t) e^{-st} dt\\ \int_0^\infty t^{t^2} e^{-st} dt\\ \int_0^\infty (e^{t^2\ln t})(e^{-st}) dt\\ \int_0^\infty e^{t^2\ln t-st} dt\\ $

That is not going to converge.

$t^{t^2}$ is growing too fast.